| # | ID | ODE | CAS classification |
Maple |
Mma |
Sympy |
time(sec) |
| \(1\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=6 x^{4} \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.316 |
|
| \(2\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} u^{\prime \prime }+u^{\prime }+\frac {u^{3}}{5}&=\cos \left (t \right )\\ u \left (0\right )&=2\\ u^{\prime }\left (0\right )&=0\\ \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.114 |
|
| \(3\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\left (6 x -8\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
22.958 |
|
| \(4\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (2 x +1\right ) y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (x +1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
34.150 |
|
| \(5\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\frac {{y^{\prime }}^{2}}{y^{2}}+\frac {y^{\prime \prime }}{y}+\frac {2 a \coth \left (2 a x \right ) y^{\prime }}{y}&=2 a^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
4.383 |
|
| \(6\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+a \right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.394 |
|
| \(7\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+a \right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.077 |
|
| \(8\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\left (x^{2}+a \right ) y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.077 |
|
| \(9\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (b^{2} x^{2}+a \right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.461 |
|
| \(10\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (c \,x^{2}+b x +a \right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.594 |
|
| \(11\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{4}+\operatorname {a1} \,x^{2}+\operatorname {a0} \right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.292 |
|
| \(12\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a +b \cos \left (2 x \right )\right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[_ellipsoidal] |
✓ |
✓ |
✗ |
1.617 |
|
| \(13\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a +b \cos \left (2 x \right )+k \cos \left (4 x \right )\right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[_ellipsoidal] |
✓ |
✗ |
✗ |
2.841 |
|
| \(14\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \csc \left (x \right )^{2} y+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.488 |
|
| \(15\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {a0} +\operatorname {a1} \cos \left (x \right )^{2}+\operatorname {a2} \csc \left (x \right )^{2}\right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
4.320 |
|
| \(16\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\left (a^{2}+\left (-1+p \right ) p \csc \left (x \right )^{2}+\left (-1+q \right ) q \sec \left (x \right )^{2}\right ) y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.294 |
|
| \(17\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a +b \sin \left (x \right )^{2}\right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[_ellipsoidal] |
✓ |
✓ |
✗ |
1.833 |
|
| \(18\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a +b \,{\mathrm e}^{x}+c \,{\mathrm e}^{2 x}\right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.226 |
|
| \(19\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a +b \cosh \left (x \right )^{2}\right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.808 |
|
| \(20\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a +b \sinh \left (x \right )^{2}\right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.902 |
|
| \(21\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a +b \sin \left (x \right )^{2}\right ) y+y^{\prime \prime }&=0 \end {array} \]
|
[_ellipsoidal] |
✓ |
✓ |
✗ |
1.617 |
|
| \(22\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (c \,x^{2}+b \right ) y+a y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.393 |
|
| \(23\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n y-y^{\prime } x +y^{\prime \prime }&=0 \end {array} \]
|
[_Hermite] |
✓ |
✓ |
✗ |
2.073 |
|
| \(24\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -a y-y^{\prime } x +y^{\prime \prime }&=0 \end {array} \]
|
[_Hermite] |
✓ |
✓ |
✗ |
2.063 |
|
| \(25\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 n y-2 y^{\prime } x +y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.139 |
|
| \(26\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b y+a x y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.236 |
|
| \(27\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y+\left (b x +a \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.426 |
|
| \(28\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b1} x +\operatorname {a1} \right ) y+\left (\operatorname {b0} x +\operatorname {a0} \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.780 |
|
| \(29\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {c1} \,x^{2}+\operatorname {b1} x +\operatorname {a1} \right ) y+\left (\operatorname {b0} x +\operatorname {a0} \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.635 |
|
| \(30\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b \,x^{-1+k} y+a \,x^{k} y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.944 |
|
| \(31\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} k \left (1+k \right ) y+\cot \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.987 |
|
| \(32\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (p \left (1+p \right )-k^{2} \csc \left (x \right )^{2}\right ) y+\cot \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.056 |
|
| \(33\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {a0} -\operatorname {a2} \csc \left (x \right )^{2}+4 \operatorname {a1} \sin \left (x \right )^{2}\right ) y+\cot \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.560 |
|
| \(34\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (b +k^{2} \cos \left (x \right )^{2}\right ) y+a \cot \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
4.758 |
|
| \(35\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \cot \left (x \right )^{2}+b \cot \left (x \right ) \csc \left (x \right )+c \csc \left (x \right )^{2}\right ) y+k \cot \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.379 |
|
| \(36\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y+a \cot \left (b x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
13.553 |
|
| \(37\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \csc \left (x \right )^{2} \left (2+\sin \left (x \right )^{2}\right ) y-\csc \left (2 x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
112.125 |
|
| \(38\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -a \left (1+a \right ) \csc \left (x \right )^{2} y-\tan \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.535 |
|
| \(39\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b y+a \tan \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.802 |
|
| \(40\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {a0} -\operatorname {a2} \operatorname {csch}\left (x \right )^{2}+4 \operatorname {a1} \sinh \left (x \right )^{2}\right ) y+\coth \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
35.490 |
|
| \(41\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {a0} +4 \operatorname {a1} \cosh \left (x \right )^{2}-\operatorname {a2} \operatorname {sech}\left (x \right )^{2}\right ) y+\tanh \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
29.761 |
|
| \(42\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b y+a \tanh \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.165 |
|
| \(43\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a k \,x^{-1+k} y+2 a \,x^{k} y^{\prime }+2 y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.662 |
|
| \(44\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 y^{\prime \prime }&=\left (x^{2}+a \right ) y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.478 |
|
| \(45\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+4 a +2\right ) y+4 y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.530 |
|
| \(46\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a +x \right ) y+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.386 |
|
| \(47\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left ({\mathrm e}^{x^{2}}-k^{2}\right ) x^{3} y-y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.792 |
|
| \(48\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y+\left (1+a \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✗ |
3.222 |
|
| \(49\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y+\left (1-a \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✗ |
3.130 |
|
| \(50\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -y+\left (1+a \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✗ |
2.881 |
|
| \(51\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b2} x +\operatorname {b1} \right ) y+a y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.796 |
|
| \(52\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {c1} \,x^{2}+\operatorname {b1} x +\operatorname {a1} \right ) y+a y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
3.955 |
|
| \(53\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n y+\left (1-x \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[_Laguerre] |
✓ |
✓ |
✗ |
3.340 |
|
| \(54\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n y+\left (1+k -x \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[_Laguerre] |
✓ |
✓ |
✗ |
27.156 |
|
| \(55\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b y+\left (a +x \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.781 |
|
| \(56\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -a y+\left (c -x \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[_Laguerre] |
✓ |
✓ |
✗ |
3.814 |
|
| \(57\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y+\left (b x +a \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.064 |
|
| \(58\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+\left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.299 |
|
| \(59\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (b x +2 a \right ) y-2 \left (b x +a \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.998 |
|
| \(60\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a b x +a n +b m \right ) y+\left (m +n +\left (a +b \right ) x \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.922 |
|
| \(61\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+\left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+\left (a +x \right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
15.510 |
|
| \(62\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (b x +a \right ) y+y^{\prime }+2 y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.303 |
|
| \(63\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y+4 \coth \left (x \right ) y^{\prime }+4 y^{\prime \prime } x&=0 \end {array} \]
|
[[_Emden, _Fowler]] |
✗ |
✗ |
✗ |
33.933 |
|
| \(64\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (b x +a \right ) y+8 y^{\prime }+16 y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.080 |
|
| \(65\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+\left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+\left (\operatorname {b0} x +\operatorname {a0} \right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
18.287 |
|
| \(66\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (c \,x^{2}+b x +a \right ) y+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.166 |
|
| \(67\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{k} \left (a +b \,x^{k}\right ) y+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.472 |
|
| \(68\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (c \,x^{2}+b x +a \right ) y+y^{\prime } x +x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
28.312 |
|
| \(69\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (-x^{4}+4 a \,x^{2}+n^{2}\right ) y+y^{\prime } x +x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
36.767 |
|
| \(70\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (c^{2} x^{4}+b^{2} x^{2}+a^{2}\right ) y+y^{\prime } x +x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
29.255 |
|
| \(71\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (m +1\right ) x^{m} a \left (m \right ) y+y^{\prime } x +x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
27.138 |
|
| \(72\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a y-2 \left (1-x \right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.706 |
|
| \(73\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {c2} \,x^{2}+\operatorname {b2} x +\operatorname {a2} \right ) y+\operatorname {a1} x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.776 |
|
| \(74\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (\operatorname {b1} \,x^{2}+\operatorname {a1} \right ) y+a x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.662 |
|
| \(75\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y+\left (b x +a \right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.594 |
|
| \(76\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (b \,x^{2}+a \right ) y+x^{2} y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.150 |
|
| \(77\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -y+x \left (x +3\right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
8.872 |
|
| \(78\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {c2} \,x^{2}+\operatorname {b2} x +\operatorname {a2} \right ) y+x \left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.178 |
|
| \(79\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {a1} +\operatorname {b1} \,x^{k}+\operatorname {c1} \,x^{2 k}\right ) y+x \left (\operatorname {a0} +\operatorname {b0} \,x^{k}\right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.372 |
|
| \(80\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a y+2 x^{2} \cot \left (x \right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.023 |
|
| \(81\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (a -x \cot \left (x \right )\right ) y+x \left (1+2 x \cot \left (x \right )\right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
33.696 |
|
| \(82\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a y-2 x^{2} \tan \left (x \right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.768 |
|
| \(83\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (a +x \tan \left (x \right )\right ) y+x \left (1-2 x \tan \left (x \right )\right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.734 |
|
| \(84\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y \left (\operatorname {a2} +\operatorname {b2} \,x^{k}+\operatorname {c2} \,x^{2 k}+\left (-1+\operatorname {a1} +\operatorname {b1} \,x^{k}\right ) f \left (x \right )+f \left (x \right )^{2}+f^{\prime }\left (x \right )\right )+x \left (\operatorname {a1} +\operatorname {b1} \,x^{k}+2 f \left (x \right )\right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
48.789 |
|
| \(85\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (b \,x^{2}+a \right ) y-y^{\prime } x +\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
27.204 |
|
| \(86\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n \left (n +1\right ) y-2 y^{\prime } x +\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
88.333 |
|
| \(87\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n \left (n +1\right ) y-2 y^{\prime } x +\left (-x^{2}+1\right ) y^{\prime \prime }&=\frac {2 \left (-1-n \right ) x \operatorname {LegendreP}\left (n , x\right )+2 \left (n +1\right ) \operatorname {LegendreP}\left (n +1, x\right )}{x^{2}-1} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
81.115 |
|
| \(88\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -p \left (1+p \right ) y+2 y^{\prime } x +\left (x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
19.013 |
|
| \(89\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} p \left (1+p \right ) y-2 y^{\prime } x +\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
78.080 |
|
| \(90\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n \left (1+a +b +n \right ) y+\left (-a +b -\left (2+a +b \right ) x \right ) y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
109.204 |
|
| \(91\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} p \left (2 k +p \right ) y-\left (1+2 k \right ) x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
41.002 |
|
| \(92\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} p \left (1+2 k +p \right ) y-2 \left (1+k \right ) x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
47.243 |
|
| \(93\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (k -p \right ) \left (1+k +p \right ) y-2 \left (1+k \right ) x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
43.842 |
|
| \(94\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b y+a x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
87.209 |
|
| \(95\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {c0} \,x^{2}+\operatorname {b0} x +\operatorname {a0} \right ) y+a x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
37.357 |
|
| \(96\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y+\left (b x +a \right ) y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
116.284 |
|
| \(97\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (c^{2} x^{2}+b^{2}\right ) y-y^{\prime } x +\left (a^{2}-x^{2}\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
48.630 |
|
| \(98\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y+2 y^{\prime }+\left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
18.532 |
|
| \(99\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} p \left (1+p \right ) y+\left (1-2 x \right ) y^{\prime }+\left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
32.111 |
|
| \(100\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y+\left (1-x \right ) y^{\prime }+\left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
44.406 |
|
| \(101\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-k +p \right ) \left (1+k +p \right ) y+\left (1+k \right ) \left (1-2 x \right ) y^{\prime }+\left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
86.051 |
|
| \(102\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n \left (a +n \right ) y+\left (c -\left (1+a \right ) x \right ) y^{\prime }+\left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
98.993 |
|
| \(103\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y+\left (b x +a \right ) y^{\prime }+x \left (x +1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
68.581 |
|
| \(104\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -a b y+\left (c -\left (a +b +1\right ) x \right ) y^{\prime }+\left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
120.240 |
|
| \(105\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y+\left (b x +a \right ) y^{\prime }+\left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
67.780 |
|
| \(106\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \operatorname {a2} y+\left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+x \left (\operatorname {a0} +x \right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
73.784 |
|
| \(107\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 a^{2} y-y^{\prime } x +2 \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
56.427 |
|
| \(108\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (b x +a \right ) y+\left (1-2 x \right ) y^{\prime }+2 \left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
33.905 |
|
| \(109\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 a \left (1+a \right ) y-\left (1+3 x \right ) y^{\prime }+2 \left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
32.207 |
|
| \(110\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (4 k x -4 p^{2}-x^{2}+1\right ) y+4 x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.175 |
|
| \(111\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -y-8 y^{\prime } x +4 \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
35.683 |
|
| \(112\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (4 p^{2}+1\right ) y-8 y^{\prime } x +4 \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
54.451 |
|
| \(113\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y+2 \left (1-x \right ) y^{\prime }+4 \left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
42.485 |
|
| \(114\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (b x +a \right ) y+2 \left (1-2 x \right ) y^{\prime }+4 \left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✗ |
✗ |
31.817 |
|
| \(115\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (c \,x^{2}+b x +a \right ) y+2 \left (1-2 x \right ) y^{\prime }+4 \left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
43.914 |
|
| \(116\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (k -p \right ) \left (1+k +p \right ) y+2 \left (1-\left (3-2 k \right ) x \right ) y^{\prime }+4 \left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
87.116 |
|
| \(117\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (k^{2} x +b \right ) y+2 \left (a x +1\right ) y^{\prime }+4 \left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✗ |
✗ |
66.185 |
|
| \(118\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y+b x y^{\prime }+\left (a \,x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
527.573 |
|
| \(119\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \operatorname {a2} y+\left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+\left (\operatorname {c0} \,x^{2}+\operatorname {b0} x +\operatorname {a0} \right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
121.245 |
|
| \(120\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -y+2 y^{\prime } x +x^{3} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.165 |
|
| \(121\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+\operatorname {a1} x y^{\prime }+x^{3} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
14.190 |
|
| \(122\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (c \,x^{2}+b x +a \right ) y+x^{2} y^{\prime }+x^{3} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
28.049 |
|
| \(123\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \operatorname {a2} x y+\left (\operatorname {b1} \,x^{2}+\operatorname {a1} \right ) y^{\prime }+x^{3} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
58.090 |
|
| \(124\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \operatorname {a2} y+x \left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+x^{3} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
57.398 |
|
| \(125\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+x \left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+x^{3} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
63.370 |
|
| \(126\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c x y+\left (b \,x^{2}+a \right ) y^{\prime }+x \left (x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
106.377 |
|
| \(127\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (1-b \right ) x y+\left (b \,x^{2}+a \right ) y^{\prime }+x \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
96.783 |
|
| \(128\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c x y+\left (a -\left (1+a \right ) x^{2}\right ) y^{\prime }+x \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
98.264 |
|
| \(129\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c x y+\left (b \,x^{2}+a \right ) y^{\prime }+x \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
101.770 |
|
| \(130\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \operatorname {a2} x y+\left (\operatorname {b1} \,x^{2}+\operatorname {a1} \right ) y^{\prime }+x \left (x^{2}+\operatorname {a0} \right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
113.211 |
|
| \(131\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+x \left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+\left (1-x \right ) x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
119.146 |
|
| \(132\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+x \left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+x^{2} \left (\operatorname {a0} +x \right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
121.591 |
|
| \(133\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y+x \left (x +1\right ) y^{\prime }+x \left (x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
68.553 |
|
| \(134\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \operatorname {a0} \operatorname {a1} \left (-k +x \right ) y+\left (1-\operatorname {a0} +\operatorname {a1} +\operatorname {a0} \operatorname {a2} -\operatorname {a3} +\left (\operatorname {a2} +\operatorname {a3} \right ) x +\left (1+\operatorname {a0} +\operatorname {a1} \right ) x^{2}\right ) y^{\prime }+\left (1-x \right ) \left (a -x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
316.703 |
|
| \(135\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {c1} x +\operatorname {c0} \right ) y+\left (\operatorname {b2} \,x^{2}+\operatorname {b1} x +\operatorname {b0} \right ) y^{\prime }+\left (\operatorname {a1} -x \right ) \left (\operatorname {a2} -x \right ) \left (\operatorname {a3} -x \right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
438.091 |
|
| \(136\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (b x +a \right ) y+2 \left (1-3 x \right ) \left (1-x \right ) y^{\prime }+4 \left (1-x \right ) x y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
48.581 |
|
| \(137\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b1} \,x^{2}+\operatorname {b0} \right ) y+\left (\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0} \right ) y^{\prime }+4 \left (1-x \right ) x \left (-a x +1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
290.758 |
|
| \(138\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-a^{2}+{\mathrm e}^{\frac {2}{x}}\right ) y+x^{4} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.352 |
|
| \(139\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (c \,x^{4}+b \,x^{2}+a \right ) y+x^{3} y^{\prime }+x^{4} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
16.438 |
|
| \(140\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y+x \left (x^{2}+1\right ) y^{\prime }+x^{4} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
12.675 |
|
| \(141\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \left (1+a \right ) y-2 x^{3} y^{\prime }+x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
72.220 |
|
| \(142\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -a^{2} y-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
36.016 |
|
| \(143\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (m^{2}-n \left (n +1\right ) \left (-x^{2}+1\right )\right ) y-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
77.567 |
|
| \(144\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (k^{2}-p \left (1+p \right ) \left (-x^{2}+1\right )\right ) y-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
73.348 |
|
| \(145\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (a^{2}-k \left (-x^{2}+1\right )\right ) y-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
61.502 |
|
| \(146\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {a4} \,x^{4}+\operatorname {a2} \,x^{2}+\operatorname {a0} \right ) y-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
77.238 |
|
| \(147\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b y+a x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
59.888 |
|
| \(148\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {c2} \,x^{2}+\operatorname {b2} x +\operatorname {a2} \right ) y+\operatorname {a1} x \left (-x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
91.485 |
|
| \(149\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b^{2} y+x \left (a^{2}+2 x^{2}\right ) y^{\prime }+x^{2} \left (a^{2}+x^{2}\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
120.530 |
|
| \(150\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (\operatorname {a4} \,x^{4}+\operatorname {a2} \,x^{2}+\operatorname {a0} \right ) y+2 x \left (a^{2}+2 x^{2}\right ) y^{\prime }+\left (a^{2}+x^{2}\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
100.226 |
|
| \(151\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {a4} \,x^{4}+\operatorname {a2} \,x^{2}+\operatorname {a0} \right ) y-2 x \left (a^{2}-x^{2}\right ) y^{\prime }+\left (a^{2}-x^{2}\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
92.225 |
|
| \(152\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {b1} \,x^{2}+\operatorname {a1} \right ) y+x \left (\operatorname {b0} \,x^{2}+\operatorname {a0} \right ) y^{\prime }+\left (a^{2}+x^{2}\right )^{2} \left (b^{2}+x^{2}\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
355.429 |
|
| \(153\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {c1} \,x^{4}+\operatorname {b1} \,x^{2}+\operatorname {a1} \right ) y+x \left (\operatorname {b0} \,x^{2}+\operatorname {a0} \right ) y^{\prime }+\left (a^{2}-x^{2}\right )^{2} \left (b^{2}-x^{2}\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
444.171 |
|
| \(154\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (c \,x^{2}+b x +a \right ) y+\left (1-x \right )^{2} x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.716 |
|
| \(155\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {c2} \,x^{2}+\operatorname {b2} x +\operatorname {a2} \right ) y+\left (1-x \right ) x \left (\operatorname {b2} x +\operatorname {a1} \right ) y^{\prime }+\left (1-x \right )^{2} x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
154.467 |
|
| \(156\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (4 k^{2}+\left (4 p^{2}+1\right ) \left (-x^{2}+1\right )\right ) y-8 x \left (-x^{2}+1\right ) y^{\prime }+4 \left (-x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
66.090 |
|
| \(157\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (4 k^{2}+\left (-4 p^{2}+1\right ) \left (-x^{2}+1\right )\right ) y-8 x \left (-x^{2}+1\right ) y^{\prime }+4 \left (-x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
65.088 |
|
| \(158\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (a \left (1+a \right ) \left (1-x \right )+b^{2} x \right ) y+2 \left (1-3 x \right ) \left (1-x \right ) x y^{\prime }+4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
227.941 |
|
| \(159\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {c2} \,x^{2}+\operatorname {b2} x +\operatorname {a2} \right ) y+\left (a -x \right ) \left (b -x \right ) \left (c -x \right ) \left (\operatorname {c1} \,x^{2}+\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+\left (a -x \right )^{2} \left (b -x \right )^{2} \left (c -x \right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1302.615 |
|
| \(160\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {a2} +\operatorname {b2} \,x^{k}\right ) y+x \left (\operatorname {a1} +\operatorname {b1} \,x^{k}\right ) y^{\prime }+x^{2} \left (\operatorname {a0} +\operatorname {b0} \,x^{k}\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
88.974 |
|
| \(161\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {a1} \cos \left (x \right )^{2}+\operatorname {a0} \right ) y+a^{2} \cos \left (x \right ) \sin \left (x \right ) y^{\prime }+\left (1-a^{2} \cos \left (x \right )^{2}\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
29.200 |
|
| \(162\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -\left (4 k^{2}-\left (-p^{2}+1\right ) \sinh \left (x \right )^{2}\right ) y+4 \cosh \left (x \right ) \sinh \left (x \right ) y^{\prime }+4 \sinh \left (x \right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
25.595 |
|
| \(163\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=x +6 y^{2} \end {array} \]
|
[[_Painleve, ‘1st‘]] |
✗ |
✗ |
✗ |
0.272 |
|
| \(164\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=a +b x +c y^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.309 |
|
| \(165\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=a +y x +2 y^{3} \end {array} \]
|
[[_Painleve, ‘2nd‘]] |
✗ |
✗ |
✗ |
0.312 |
|
| \(166\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=f \left (x \right )+g \left (x \right ) y+2 y^{3} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.438 |
|
| \(167\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=a -2 a b x y+2 b^{2} y^{3} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.327 |
|
| \(168\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\operatorname {a0} +\operatorname {a2} y+\operatorname {a1} x y+\operatorname {a3} y^{3} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.355 |
|
| \(169\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \,x^{r} y^{s}+y^{\prime \prime }&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.309 |
|
| \(170\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y \left (2 f \left (x \right )^{2}+f^{\prime }\left (x \right )\right )+3 f \left (x \right ) y^{\prime }+y^{\prime \prime }&=2 y^{3} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.633 |
|
| \(171\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime }+y^{\prime \prime }&=-12 f \left (x \right ) y+y^{3}+12 f^{\prime }\left (x \right ) \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.594 |
|
| \(172\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 a^{2} y+a y^{2}+\left (3 a +y\right ) y^{\prime }+y^{\prime \prime }&=y^{3} \end {array} \]
|
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✗ |
38.290 |
|
| \(173\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=f \left (x \right ) y^{2}+y^{3}+y \left (-2 f \left (x \right )^{2}+f^{\prime }\left (x \right )\right )+\left (3 f \left (x \right )-y\right ) y^{\prime } \end {array} \]
|
[NONE] |
✓ |
✗ |
✗ |
1.358 |
|
| \(174\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\operatorname {f2} \left (x \right )+\operatorname {f3} \left (x \right ) y+\operatorname {f1} \left (x \right ) y^{2}+y^{3}+\left (3 \operatorname {f1} \left (x \right )-y\right ) y^{\prime } \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.746 |
|
| \(175\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\operatorname {g3} \left (x \right )+\operatorname {g2} \left (x \right ) y+\operatorname {g1} \left (x \right ) y^{2}+\operatorname {g0} \left (x \right ) y^{3}+\left (\operatorname {f1} \left (x \right )+\operatorname {f0} \left (x \right ) y\right ) y^{\prime } \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.911 |
|
| \(176\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=y f^{\prime }\left (x \right )+\left (f \left (x \right )-2 y\right ) y^{\prime } \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
✗ |
3.140 |
|
| \(177\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=g \left (x \right )+f \left (x \right ) y^{2}+\left (f \left (x \right )-2 y\right ) y^{\prime } \end {array} \]
|
[[_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
✗ |
0.603 |
|
| \(178\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\operatorname {f3} \left (x \right )+\operatorname {f2} \left (x \right ) y^{2}+\left (\operatorname {f1} \left (x \right )-2 y\right ) y^{\prime } \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.654 |
|
| \(179\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\operatorname {f4} \left (x \right )+\operatorname {f3} \left (x \right ) y+\operatorname {f2} \left (x \right ) y^{2}+\left (\operatorname {f1} \left (x \right )-2 y\right ) y^{\prime } \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.788 |
|
| \(180\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=a +4 b^{2} y+3 b y^{2}+3 y y^{\prime } \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
34.803 |
|
| \(181\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 3 y y^{\prime }+y^{\prime \prime }&=f \left (x \right )+g \left (x \right ) y-y^{3} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.623 |
|
| \(182\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=f \left (x \right ) y^{2}-y^{3}+\left (f \left (x \right )-3 y\right ) y^{\prime } \end {array} \]
|
[[_2nd_order, _with_potential_symmetries]] |
✓ |
✗ |
✗ |
1.143 |
|
| \(183\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=a \left (1+2 y y^{\prime }\right ) \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
149.415 |
|
| \(184\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b y+a \left (-1+y^{2}\right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
13.169 |
|
| \(185\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} g \left (x , y\right )+f \left (x , y\right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.368 |
|
| \(186\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y+b y^{\prime }+a {y^{\prime }}^{2}+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
25.722 |
|
| \(187\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} h \left (y\right )+f \left (y\right ) y^{\prime }+g \left (y\right ) {y^{\prime }}^{2}+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
4.784 |
|
| \(188\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=a \left (-y+y^{\prime } x \right )^{k} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
0.736 |
|
| \(189\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=A \,x^{a} y^{b} {y^{\prime }}^{c} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.452 |
|
| \(190\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=a \left (b +c x +y\right ) \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
0.812 |
|
| \(191\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=f \left (a x +b y, y^{\prime }\right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.369 |
|
| \(192\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=f \left (x , \frac {y^{\prime }}{y}\right ) y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.328 |
|
| \(193\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=x^{-2+n} f \left (y x^{-n}, x^{1-n} y^{\prime }\right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.569 |
|
| \(194\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \,{\mathrm e}^{y} x +y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✗ |
0.544 |
|
| \(195\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y^{5}+2 y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[_Emden, [_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
0.481 |
|
| \(196\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y^{n}+2 y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[_Emden, [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.417 |
|
| \(197\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{m} y^{n}+2 y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.420 |
|
| \(198\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \,x^{m} y^{n}+2 y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.439 |
|
| \(199\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b \,{\mathrm e}^{y} x +a y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.420 |
|
| \(200\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x&=-y^{2}-2 y^{\prime }+x^{2} {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.573 |
|
| \(201\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-y+a x y^{\prime }\right )^{2}+y^{\prime \prime } x&=b \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.501 |
|
| \(202\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a y \left (1-y^{n}\right )+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.419 |
|
| \(203\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \,{\mathrm e}^{-1+y}+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.436 |
|
| \(204\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (1+a \right ) x y^{\prime }+x^{2} y^{\prime \prime }&=x^{k} f \left (x^{k} y, k y+y^{\prime } x \right ) \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.960 |
|
| \(205\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }&=6 y-4 y^{2} x^{2}+x^{4} {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.872 |
|
| \(206\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \left (-y+y^{\prime } x \right )^{2}+x^{2} y^{\prime \prime }&=b \,x^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
0.713 |
|
| \(207\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y x +a \,x^{4} {y^{\prime }}^{2}+x^{2} y^{\prime \prime }&=b \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.606 |
|
| \(208\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b x +a y {y^{\prime }}^{2}+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.410 |
|
| \(209\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }&=\sqrt {b y^{2}+a \,x^{2} {y^{\prime }}^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
1.020 |
|
| \(210\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }&=f \left (\frac {x y^{\prime }}{y}\right ) y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
22.712 |
|
| \(211\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y+a y^{3}+9 x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
0.459 |
|
| \(212\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 24+12 y x +x^{3} \left (-y^{3}+y y^{\prime }+y^{\prime \prime }\right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
0.872 |
|
| \(213\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }&=a \left (-y+y^{\prime } x \right )^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.777 |
|
| \(214\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -6+x y \left (12+3 y x -2 y^{2} x^{2}\right )+x^{2} \left (9+2 y x \right ) y^{\prime }+2 x^{3} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
1.064 |
|
| \(215\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }&=-4 y^{2}+x \left (x^{2}+2 y\right ) y^{\prime } \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.946 |
|
| \(216\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }&=-4 y^{2}+x^{2} y^{\prime } \left (x +y^{\prime }\right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.186 |
|
| \(217\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-y+y^{\prime } x \right )^{3}+x^{4} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.526 |
|
| \(218\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{b}+x^{a} y^{\prime \prime }&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.327 |
|
| \(219\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 24-48 y x +\left (-12 x^{2}+1\right ) \left (y^{2}+3 y^{\prime }\right )+2 x \left (-4 x^{2}+1\right ) \left (-y^{3}+y y^{\prime }+y^{\prime \prime }\right )&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.483 |
|
| \(220\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b +a x y-\left (-12 x^{2}+k \,x^{-1+k}\right ) \left (y^{2}+3 y^{\prime }\right )+2 \left (-4 x^{3}+x^{k}\right ) \left (-y^{3}+y y^{\prime }+y^{\prime \prime }\right )&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
2.621 |
|
| \(221\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sqrt {x}\, y^{\prime \prime }&=y^{{3}/{2}} \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.337 |
|
| \(222\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{{3}/{2}} y^{\prime \prime }&=f \left (\frac {y}{\sqrt {x}}\right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
16.459 |
|
| \(223\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} f \left (x \right ) f^{\prime }\left (x \right ) y^{\prime }+f \left (x \right )^{2} y^{\prime \prime }&=g \left (y, f \left (x \right ) y^{\prime }\right ) \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.536 |
|
| \(224\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} f \left (x \right )^{2} y^{\prime \prime }&=-24 f \left (x \right )^{4}+\left (3 f \left (x \right )^{3}-f \left (x \right )^{2} y+3 f \left (x \right ) f^{\prime }\left (x \right )\right ) y^{\prime } \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.614 |
|
| \(225\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 f \left (x \right )^{2} y^{\prime \prime }&=2 f \left (x \right )^{2} y^{3}+f \left (x \right ) y^{2} f^{\prime }\left (x \right )+f \left (x \right ) \left (-2 f \left (x \right ) y+3 f^{\prime }\left (x \right )\right ) y^{\prime }+y \left (-2 f \left (x \right )^{3}-2 {f^{\prime }\left (x \right )}^{2}+f \left (x \right ) f^{\prime \prime }\left (x \right )\right ) \end {array} \]
|
[NONE] |
✓ |
✗ |
✗ |
1.233 |
|
| \(226\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&={\mathrm e}^{x} y \left (\operatorname {a0} +\operatorname {a1} y^{2}\right )+{\mathrm e}^{2 x} \left (\operatorname {a2} +\operatorname {a3} y^{4}\right )+{y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.595 |
|
| \(227\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=-y^{2} x^{2}+\ln \left (y\right ) y^{2}+{y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.456 |
|
| \(228\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=y^{2} \left (f \left (x \right ) y+g^{\prime }\left (x \right )\right )+y^{\prime }+{y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.655 |
|
| \(229\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y-y^{\prime } x +{y^{\prime }}^{2}+y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.396 |
|
| \(230\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a x y^{\prime }+{y^{\prime }}^{2}+y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.368 |
|
| \(231\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=y^{3}-y f^{\prime }\left (x \right )+f \left (x \right ) y^{\prime }+{y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✓ |
✗ |
0.652 |
|
| \(232\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=-f \left (x \right ) y^{3}+y^{4}-f \left (x \right ) y^{\prime }+{y^{\prime }}^{2}+y f^{\prime \prime }\left (x \right ) \end {array} \]
|
[NONE] |
✗ |
✓ |
✗ |
0.783 |
|
| \(233\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=b y^{2}+y^{3}+a y y^{\prime }+{y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
17.912 |
|
| \(234\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=g \left (x \right ) y^{2}+f \left (x \right ) y y^{\prime }+{y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.183 |
|
| \(235\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=-y \left (f^{\prime }\left (x \right )-y^{2} g^{\prime }\left (x \right )\right )+\left (f \left (x \right )+g \left (x \right ) y^{2}\right ) y^{\prime }+{y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
✗ |
0.814 |
|
| \(236\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=\operatorname {a2} y^{2}+\operatorname {a3} y^{1+a}+\operatorname {a1} y y^{\prime }+a {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
231.881 |
|
| \(237\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} g \left (x \right ) y^{2}+f \left (x \right ) y y^{\prime }+a {y^{\prime }}^{2}+y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.672 |
|
| \(238\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y^{\prime } \left (1+y^{\prime }\right )+\left (x -y\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
0.559 |
|
| \(239\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x -y\right ) y^{\prime \prime }&=\left (1+y^{\prime }\right ) \left (1+{y^{\prime }}^{2}\right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
0.733 |
|
| \(240\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x -y\right ) y^{\prime \prime }&=f \left (y^{\prime }\right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
2.517 |
|
| \(241\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }&=4 y^{2} \left (x +2 y\right )+{y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.355 |
|
| \(242\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }&=-1-2 x y^{2}+a y^{3}+{y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.363 |
|
| \(243\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }&=y^{2} \left (a x +b y\right )+{y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.355 |
|
| \(244\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }&=-a^{2}-4 \left (-x^{2}+b \right ) y^{2}+8 x y^{3}+3 y^{4}+{y^{\prime }}^{2} \end {array} \]
|
[[_Painleve, ‘4th‘]] |
✗ |
✗ |
✗ |
0.486 |
|
| \(245\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }&=8 y^{3}-2 y^{2} \left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right )-3 f \left (x \right ) y y^{\prime }+{y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.639 |
|
| \(246\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }&=-1+2 x f \left (x \right ) y^{2}-y^{4}-4 y^{2} y^{\prime }+{y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.572 |
|
| \(247\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }&=f \left (x \right ) y^{2}+3 {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.356 |
|
| \(248\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \left (2+a \right )^{2} y y^{\prime \prime }&=-a^{2} f \left (x \right )^{2} y^{4}+a^{2} \left (2+a \right ) y^{3} f^{\prime }\left (x \right )+a \left (2+a \right )^{2} f \left (x \right ) y^{2} y^{\prime }+\left (-1+a \right ) \left (2+a \right )^{2} {y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✓ |
✗ |
0.983 |
|
| \(249\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }&=y \left (\operatorname {a2} +\operatorname {a3} y^{2}\right )+x \left (\operatorname {a0} +\operatorname {a1} y^{4}\right )-y y^{\prime }+x {y^{\prime }}^{2} \end {array} \]
|
[[_Painleve, ‘3rd‘]] |
✗ |
✗ |
✗ |
0.568 |
|
| \(250\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} f \left (x \right )+a y y^{\prime }+x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.267 |
|
| \(251\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }&=x y^{3}+a y y^{\prime }+x {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.455 |
|
| \(252\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }&=b^{2} x y^{3}+a y y^{\prime }+x {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.467 |
|
| \(253\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }&=-\left (1+y\right ) y^{\prime }+2 x {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.625 |
|
| \(254\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-y+y^{\prime } x \right )^{2}+x^{2} y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.626 |
|
| \(255\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y y^{\prime \prime }&=a y^{2}+a x y y^{\prime }+2 x^{2} {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.718 |
|
| \(256\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y^{2}+b x y y^{\prime }+a \,x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.708 |
|
| \(257\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (1-y\right )^{2} y-2 x \left (1-y\right ) y^{\prime }+2 x^{2} {y^{\prime }}^{2}+x^{2} \left (1-y\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.080 |
|
| \(258\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (x -y\right ) y^{\prime \prime }&=\left (-y+y^{\prime } x \right )^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.978 |
|
| \(259\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-y+y^{\prime } x \right )^{2}+x^{2} \left (x -y\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.875 |
|
| \(260\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (x -y\right ) y^{\prime \prime }&=a \left (-y+y^{\prime } x \right )^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.000 |
|
| \(261\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} y y^{\prime \prime }&=-y^{2}+x^{2} {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.530 |
|
| \(262\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} y y^{\prime \prime }&=-4 y^{2}+2 y y^{\prime } x +x^{2} {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.691 |
|
| \(263\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x +1\right )^{2} y y^{\prime \prime }&=a \left (2+x \right ) y^{2}-2 \left (x^{2}+1\right ) y y^{\prime }+x \left (x +1\right )^{2} {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.098 |
|
| \(264\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 3 x y^{2}-12 x^{2} y y^{\prime }+4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}+8 \left (-x^{3}+1\right ) y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.033 |
|
| \(265\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sqrt {a^{2}-x^{2}}\, \left (-y y^{\prime }-x {y^{\prime }}^{2}+x y y^{\prime \prime }\right )&=b x {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.399 |
|
| \(266\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \operatorname {f3} \left (x \right ) y^{2}+\operatorname {f2} \left (x \right ) y y^{\prime }+\operatorname {f1} \left (x \right ) {y^{\prime }}^{2}+\operatorname {f0} \left (x \right ) y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.846 |
|
| \(267\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 f \left (x \right ) y y^{\prime \prime }&=4 f \left (x \right )^{2} y+3 f \left (x \right ) g \left (x \right ) y^{2}-f \left (x \right ) y^{4}+2 y^{3} f^{\prime }\left (x \right )+\left (-6 f \left (x \right ) y^{2}+2 f^{\prime }\left (x \right )\right ) y^{\prime }+3 f \left (x \right ) {y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.803 |
|
| \(268\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a x +y {y^{\prime }}^{2}+y^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
0.350 |
|
| \(269\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y {y^{\prime }}^{2}+y^{2} y^{\prime \prime }&=b x +a \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.379 |
|
| \(270\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x +y^{2}\right ) y^{\prime \prime }&=2 \left (x -y^{2}\right ) {y^{\prime }}^{3}-y^{\prime } \left (1+4 y y^{\prime }\right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
0.992 |
|
| \(271\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+y^{2}\right ) y^{\prime \prime }&=\left (1+y^{2}\right ) \left (-y+y^{\prime } x \right ) \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.634 |
|
| \(272\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+y^{2}\right ) y^{\prime \prime }&=2 \left (1+y^{2}\right ) \left (-y+y^{\prime } x \right ) \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.625 |
|
| \(273\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (1-y\right ) y y^{\prime \prime }&=f \left (x \right ) \left (1-y\right ) y y^{\prime }+\left (1-2 y\right ) {y^{\prime }}^{2} \end {array} \]
|
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.066 |
|
| \(274\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (1-y\right ) y y^{\prime \prime }&=4 y \left (f \left (x \right )+g \left (x \right ) y\right ) y^{\prime }+\left (1-3 y\right ) {y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.718 |
|
| \(275\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (1-y\right ) y y^{\prime \prime }&=-\left (1-y\right )^{3} \left (\operatorname {F0} \left (x \right )^{2}-\operatorname {G0} \left (x \right )^{2} y^{2}\right )-4 \left (1-y\right ) y^{2} \left (f \left (x \right )^{2}-g \left (x \right )^{2}+f^{\prime }\left (x \right )+g^{\prime }\left (x \right )\right )-4 y \left (f \left (x \right )+g \left (x \right ) y\right ) y^{\prime }+\left (1-3 y\right ) {y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.493 |
|
| \(276\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y^{2} y^{\prime \prime }&=a \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
0.362 |
|
| \(277\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y^{2} y^{\prime \prime }&=\left (a -y^{2}\right ) y^{\prime }+x y {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.645 |
|
| \(278\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{2} y^{\prime \prime }&=\left (x^{2}+y^{2}\right ) \left (-y+y^{\prime } x \right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.988 |
|
| \(279\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a^{2}-x^{2}\right ) y {y^{\prime }}^{2}+\left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime }&=x \left (a^{2}-y^{2}\right ) y^{\prime } \end {array} \]
|
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.089 |
|
| \(280\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \operatorname {a2} x \left (1-y\right ) y^{2}+\operatorname {a3} \,x^{3} y^{2} \left (1+y\right )+\left (1-y\right )^{3} \left (\operatorname {a0} +\operatorname {a1} y^{2}\right )+2 x \left (1-y\right ) y y^{\prime }-x^{2} \left (1-3 y\right ) {y^{\prime }}^{2}+2 x^{2} \left (1-y\right ) y y^{\prime \prime }&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
2.023 |
|
| \(281\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x +y\right ) \left (-y+y^{\prime } x \right )^{3}+x^{3} y^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.606 |
|
| \(282\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (1-x \right ) x \left (1-y\right ) \left (x -y\right ) y y^{\prime \prime }&=-y^{2} \left (1-y^{2}\right )+2 \left (1-y\right ) y \left (x^{2}+y-2 y x \right ) y^{\prime }+\left (1-x \right ) x \left (x -2 y-2 y x +3 y^{2}\right ) {y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.728 |
|
| \(283\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (1-x \right ) x \left (1-y\right ) \left (x -y\right ) y y^{\prime \prime }&=f \left (x \right ) \left (\left (1-y\right ) \left (x -y\right ) y\right )^{{3}/{2}}-y^{2} \left (1-y^{2}\right )+2 \left (1-y\right ) y \left (x^{2}+y-2 y x \right ) y^{\prime }+\left (1-x \right ) x \left (x -2 y-2 y x +3 y^{2}\right ) {y^{\prime }}^{2} \end {array} \]
|
unknown |
✗ |
✗ |
✗ |
2.274 |
|
| \(284\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (1-x \right )^{2} x^{2} \left (1-y\right ) \left (x -y\right ) y y^{\prime \prime }&=\operatorname {a0} x \left (1-y\right )^{2} \left (x -y\right )^{2}+\left (-1+\operatorname {a2} \right ) \left (1-x \right ) x \left (1-y\right )^{2} y^{2}+\operatorname {a1} \left (1-x \right ) \left (x -y\right )^{2} y^{2}+\operatorname {a3} \left (1-y\right )^{2} \left (x -y\right )^{2} y^{2}+2 \left (1-x \right ) x \left (1-y\right )^{2} y \left (x^{2}+y-2 y x \right ) y^{\prime }+\left (1-x \right )^{2} x^{2} \left (x -2 y-2 y x +3 y^{2}\right ) {y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
4.243 |
|
| \(285\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a^{2} y+\left (x^{2}+y^{2}\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[NONE] |
✓ |
✗ |
✗ |
0.604 |
|
| \(286\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} A y+\left (a +2 b x +c \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }&=0 \end {array} \]
|
[NONE] |
✓ |
✓ |
✗ |
0.842 |
|
| \(287\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \operatorname {f3} \left (y\right )+\operatorname {f2} \left (y\right ) y^{\prime }+\operatorname {f1} \left (y\right ) {y^{\prime }}^{2}+\operatorname {f0} \left (y\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
6.279 |
|
| \(288\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sqrt {y}\, y^{\prime \prime }&=2 b x +2 a \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
0.376 |
|
| \(289\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} X \left (x , y\right )^{3} y^{\prime \prime }&=1 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.271 |
|
| \(290\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime } y^{\prime \prime }&=x y^{2}+x^{2} y y^{\prime } \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
✗ |
✗ |
6.694 |
|
| \(291\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a y^{2}+x^{3} y^{\prime } y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
1.196 |
|
| \(292\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left ({y^{\prime }}^{2}+a \left (-y+y^{\prime } x \right )\right ) y^{\prime \prime }&=b \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
0.890 |
|
| \(293\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} h \left (x \right )+g \left (y\right ) y^{\prime }+f \left (y^{\prime }\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
✗ |
✗ |
1.676 |
|
| \(294\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} {y^{\prime \prime }}^{2}&=a +b y \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✗ |
0.023 |
|
| \(295\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (x -y^{\prime }\right ) y^{\prime }-x \left (x +4 y^{\prime }\right ) y^{\prime \prime }+2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2}&=2 y \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.031 |
|
| \(296\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 {y^{\prime }}^{2}-2 \left (y+3 y^{\prime } x \right ) y^{\prime \prime }+3 x^{2} {y^{\prime \prime }}^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.551 |
|
| \(297\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 6 y y^{\prime \prime }-6 \left (1-6 x \right ) x y^{\prime } y^{\prime \prime }+\left (2-9 x \right ) x^{2} {y^{\prime \prime }}^{2}&=36 x {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
28.012 |
|
| \(298\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} h y^{2}+\operatorname {g1} y y^{\prime }+\operatorname {g0} {y^{\prime }}^{2}+\operatorname {f2} y y^{\prime \prime }+\operatorname {f1} y^{\prime } y^{\prime \prime }+\operatorname {f0} {y^{\prime \prime }}^{2}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✓ |
✗ |
✗ |
0.034 |
|
| \(299\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (y^{2}-x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime }\right )^{2}&=4 x y \left (-y+y^{\prime } x \right )^{3} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
0.032 |
|
| \(300\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} f \left (y^{\prime \prime }, y^{\prime }-y^{\prime \prime } x , y-y^{\prime } x +\frac {x^{2} y^{\prime \prime }}{2}\right )&=0 \end {array} \]
|
[NONE] |
✓ |
✗ |
✗ |
16.605 |
|
| \(301\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime } y^{\prime \prime }&=a x {y^{\prime }}^{5}+3 {y^{\prime \prime }}^{2} \end {array} \]
|
[[_2nd_order, _missing_y]] |
✗ |
✗ |
✗ |
126.022 |
|
| \(302\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (1+y\right ) y^{\prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.616 |
|
| \(303\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (1-x \right ) y^{\prime }+m y&=0 \end {array} \]
|
[_Laguerre] |
✓ |
✓ |
✗ |
4.847 |
|
| \(304\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (1-x \right ) y^{\prime \prime }-4 y^{\prime } x +5 y&=\cos \left (x \right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
8.727 |
|
| \(305\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} u^{\prime \prime }+u^{\prime }+u&=\cos \left (r +u\right ) \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.314 |
|
| \(306\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }-\left (1-\frac {{x^{\prime }}^{2}}{3}\right ) x^{\prime }+x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
53.600 |
|
| \(307\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+\cos \left (x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
2.716 |
|
| \(308\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+{\mathrm e}^{x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
3.718 |
|
| \(309\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y&=x +\frac {1}{x} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
75.263 |
|
| \(310\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\cos \left (x \right )-\sin \left (x \right )\right ) y^{\prime \prime }-2 \sin \left (x \right ) y^{\prime }+\left (\cos \left (x \right )+\sin \left (x \right )\right ) y&=\left (\cos \left (x \right )-\sin \left (x \right )\right )^{2} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
8.896 |
|
| \(311\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y y^{\prime \prime }&=-y^{2}+x^{2} {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.539 |
|
| \(312\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) z^{\prime \prime }+\left (1-3 x \right ) z^{\prime }+k z&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
80.551 |
|
| \(313\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) \eta ^{\prime \prime }-\left (x +1\right ) \eta ^{\prime }+\left (1+k \right ) \eta& =0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
76.619 |
|
| \(314\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-2 y^{\prime } x +2 \alpha y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.659 |
|
| \(315\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }&=y^{\prime }+{y^{\prime }}^{3} \end {array} \]
|
[NONE] |
✗ |
✓ |
✗ |
0.565 |
|
| \(316\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\frac {\left (t^{2}-1\right ) y^{\prime }}{t}+\frac {t^{2} y}{\left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{2}}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
16.158 |
|
| \(317\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=x \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.288 |
|
| \(318\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{2} y^{\prime \prime }&=x \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
0.421 |
|
| \(319\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 3 y y^{\prime \prime }&=\sin \left (x \right ) \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.373 |
|
| \(320\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-a x y^{\prime }-b x y-c x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.906 |
|
| \(321\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.687 |
|
| \(322\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{3}&=0 \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
4.724 |
|
| \(323\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
5.983 |
|
| \(324\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-x^{3} y^{\prime }-x^{3} y-x^{3}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
6.289 |
|
| \(325\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-x^{2} y^{\prime }-y x -x^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.697 |
|
| \(326\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{3}-x^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
5.413 |
|
| \(327\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-x^{2} y^{\prime }-x^{3} y-x^{4}-x^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
5.943 |
|
| \(328\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-x^{3} y^{\prime }-y x -x^{3}-x^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
5.112 |
|
| \(329\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.366 |
|
| \(330\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-x^{3} y^{\prime }-x^{3} y-x^{4}-x^{3}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
5.170 |
|
| \(331\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+1\right ) y^{\prime \prime }+y {y^{\prime }}^{2}&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.408 |
|
| \(332\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \frac {x y^{\prime \prime }}{1-x}+y&=\frac {1}{1-x} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
2.692 |
|
| \(333\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \frac {x y^{\prime \prime }}{1-x}+y&=\cos \left (x \right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
2.712 |
|
| \(334\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \frac {x y^{\prime \prime }}{-x^{2}+1}+y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
1.757 |
|
| \(335\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} {y^{\prime \prime }}^{2}+y^{\prime }+y&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
0.018 |
|
| \(336\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y^{2} {y^{\prime }}^{2}&=0 \end {array} \]
|
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.629 |
|
| \(337\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (\sin \left (x \right )+2 x \right ) y^{\prime }+\cos \left (y\right ) y {y^{\prime }}^{2}&=0 \end {array} \]
|
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.948 |
|
| \(338\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (x +3\right ) y^{\prime }+\left (3+y^{2}\right ) {y^{\prime }}^{2}&=0 \end {array} \]
|
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.671 |
|
| \(339\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 10 y^{\prime \prime }+\left ({\mathrm e}^{x}+3 x \right ) y^{\prime }+\frac {3 \,{\mathrm e}^{y} {y^{\prime }}^{2}}{\sin \left (y\right )}&=0 \end {array} \]
|
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.211 |
|
| \(340\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+1\right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }+y&=4 \cos \left (\ln \left (x +1\right )\right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
✗ |
30.878 |
|
| \(341\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (-y+y^{\prime } x \right )^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
0.722 |
|
| \(342\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (x^{2}+a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.165 |
|
| \(343\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (a^{2} x^{2 n}-1\right ) y&=0 \end {array} \]
|
[_Titchmarsh] |
✗ |
✗ |
✗ |
2.258 |
|
| \(344\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,x^{2 c}+b \,x^{c -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.664 |
|
| \(345\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (4 a^{2} b^{2} x^{2} {\mathrm e}^{2 b \,x^{2}}-1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
3.000 |
|
| \(346\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{x}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.310 |
|
| \(347\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \cosh \left (x \right )^{2}+b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.506 |
|
| \(348\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \cos \left (2 x \right )+b \right ) y&=0 \end {array} \]
|
[_ellipsoidal] |
✓ |
✓ |
✗ |
3.623 |
|
| \(349\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \cos \left (x \right )^{2}+b \right ) y&=0 \end {array} \]
|
[_ellipsoidal] |
✓ |
✓ |
✗ |
3.761 |
|
| \(350\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (\frac {m \left (m -1\right )}{\cos \left (x \right )^{2}}+\frac {n \left (n -1\right )}{\sin \left (x \right )^{2}}+a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.445 |
|
| \(351\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (n \left (n +1\right ) k^{2} \operatorname {JacobiSN}\left (x , k\right )^{2}+b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.057 |
|
| \(352\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.130 |
|
| \(353\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.898 |
|
| \(354\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 a y^{\prime }+f \left (x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
4.356 |
|
| \(355\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+y^{\prime } x +\left (n +1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.149 |
|
| \(356\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+y^{\prime } x -n y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.793 |
|
| \(357\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -a y-y^{\prime } x +y^{\prime \prime }&=0 \end {array} \]
|
[_Hermite] |
✓ |
✓ |
✗ |
5.935 |
|
| \(358\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-2 y^{\prime } x +a y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.875 |
|
| \(359\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-4 y^{\prime } x +\left (3 x^{2}+2 n -1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.366 |
|
| \(360\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b y+a x y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.158 |
|
| \(361\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.806 |
|
| \(362\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.553 |
|
| \(363\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,x^{-1+q} y^{\prime }+b \,x^{q -2} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.416 |
|
| \(364\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 n y^{\prime } \cot \left (x \right )+\left (-a^{2}+n^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
7.279 |
|
| \(365\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\cot \left (x \right ) y^{\prime }+v \left (v +1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.578 |
|
| \(366\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b y+a \tan \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.737 |
|
| \(367\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+2 a \right ) y^{\prime }+\left (\frac {a f^{\prime }\left (x \right )}{f \left (x \right )}+a^{2}-b^{2} f \left (x \right )^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.379 |
|
| \(368\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right ) y^{\prime }+\left (\frac {f^{\prime }\left (x \right ) \left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right )}{f \left (x \right )}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {v^{2} {g^{\prime }\left (x \right )}^{2}}{g \left (x \right )^{2}}+{g^{\prime }\left (x \right )}^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.429 |
|
| \(369\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right ) y^{\prime }+\left (\frac {h^{\prime }\left (x \right ) \left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right )}{h \left (x \right )}-\frac {h^{\prime \prime }\left (x \right )}{h \left (x \right )}+{g^{\prime }\left (x \right )}^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.600 |
|
| \(370\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 y^{\prime \prime }-\left (x^{2}+a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.187 |
|
| \(371\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a y^{\prime \prime }-\left (a b +c +x \right ) y^{\prime }+\left (b \left (x +c \right )+d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.540 |
|
| \(372\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a +x \right ) y+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.886 |
|
| \(373\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +y^{\prime }+\left (a +x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.633 |
|
| \(374\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x -y^{\prime }+x^{3} \left ({\mathrm e}^{x^{2}}-v^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
7.145 |
|
| \(375\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (x +b \right ) y^{\prime }+a y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
7.148 |
|
| \(376\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (x +a +b \right ) y^{\prime }+a y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
7.202 |
|
| \(377\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x -y^{\prime } x -a y&=0 \end {array} \]
|
[_Laguerre] |
✓ |
✓ |
✗ |
4.997 |
|
| \(378\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (b -x \right ) y^{\prime }-a y&=0 \end {array} \]
|
[_Laguerre] |
✓ |
✓ |
✗ |
6.734 |
|
| \(379\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x -2 \left (-1+x \right ) y^{\prime }-y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.298 |
|
| \(380\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x -\left (3 x -2\right ) y^{\prime }-\left (2 x -3\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.799 |
|
| \(381\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a x +b +n \right ) y^{\prime }+n a y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.570 |
|
| \(382\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x -\left (a +b \right ) \left (x +1\right ) y^{\prime }+a b x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.831 |
|
| \(383\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a b x +a n +b m \right ) y+\left (m +n +\left (a +b \right ) x \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.476 |
|
| \(384\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.591 |
|
| \(385\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x -2 \left (x^{2}-a \right ) y^{\prime }+2 n x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
17.306 |
|
| \(386\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y^{\prime \prime } x -\left (-1+x \right ) y^{\prime }+a y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
7.143 |
|
| \(387\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y^{\prime \prime } x -\left (2 x -1\right ) y^{\prime }+a y&=0 \end {array} \]
|
[_Laguerre] |
✓ |
✓ |
✗ |
6.698 |
|
| \(388\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 y^{\prime \prime } x -\left (a +x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.026 |
|
| \(389\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 y^{\prime \prime } x +4 y-\left (2+x \right ) y+l y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
23.088 |
|
| \(390\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 y^{\prime \prime } x +4 m y^{\prime }-\left (x -2 m -4 n \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
7.982 |
|
| \(391\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 16 y^{\prime \prime } x +8 y^{\prime }-\left (a +x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.959 |
|
| \(392\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 5 \left (a x +b \right ) y^{\prime \prime }+8 a y^{\prime }+c \left (a x +b \right )^{{1}/{5}} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
24.852 |
|
| \(393\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 a x y^{\prime \prime }+\left (b x +a \right ) y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
29.812 |
|
| \(394\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
29.701 |
|
| \(395\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (\operatorname {a2} x +\operatorname {b2} \right ) y^{\prime \prime }+\left (\operatorname {a1} x +\operatorname {b1} \right ) y^{\prime }+\left (\operatorname {a0} x +\operatorname {b0} \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
28.187 |
|
| \(396\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.256 |
|
| \(397\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}-x \,{\mathrm e}^{x} \left (2+x \ln \left (x \right )\right )&=0 \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
3.181 |
|
| \(398\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+a y^{\prime }-y x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
9.452 |
|
| \(399\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+2 y^{\prime } x +\left (l \,x^{2}+a x -n \left (n +1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
33.686 |
|
| \(400\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+2 \left (-1+x \right ) y^{\prime }+a y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
15.260 |
|
| \(401\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+2 \left (a +x \right ) y^{\prime }-b \left (b -1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
15.604 |
|
| \(402\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
16.214 |
|
| \(403\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (a x +b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.679 |
|
| \(404\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -y+x \left (x +3\right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
16.965 |
|
| \(405\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }-\left (x^{2}-2 x \right ) y^{\prime }-\left (a +x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
18.096 |
|
| \(406\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-v \left (v -1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
7.007 |
|
| \(407\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+f \left (x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
6.238 |
|
| \(408\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (2 a x +b \right ) x y^{\prime }+\left (a b x +c \,x^{2}+d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
21.932 |
|
| \(409\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
24.309 |
|
| \(410\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
23.520 |
|
| \(411\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) x y^{\prime }+f \left (x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
16.760 |
|
| \(412\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (x^{3}+1\right ) x y^{\prime }-y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
26.894 |
|
| \(413\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (-x^{4}+\left (2 n +2 a +1\right ) x^{2}+\left (-1\right )^{n} a -a^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.847 |
|
| \(414\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2 n}+\operatorname {b1} \,x^{n}+\operatorname {c1} \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
18.352 |
|
| \(415\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }-\left (2 x^{2} \tan \left (x \right )-x \right ) y^{\prime }-\left (a +x \tan \left (x \right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.727 |
|
| \(416\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (2 x^{2} \cot \left (x \right )+x \right ) y^{\prime }+\left (x \cot \left (x \right )+a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
14.140 |
|
| \(417\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+2 x f \left (x \right ) y^{\prime }+\left (x f^{\prime }\left (x \right )+f \left (x \right )^{2}-f \left (x \right )+a \,x^{2}+b x +c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
18.997 |
|
| \(418\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+2 x^{2} f \left (x \right ) y^{\prime }+\left (x^{2} \left (f^{\prime }\left (x \right )+f \left (x \right )^{2}+a \right )-v \left (v -1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
15.840 |
|
| \(419\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (x -2 x^{2} f \left (x \right )\right ) y^{\prime }+\left (x^{2} \left (1+f \left (x \right )^{2}-f^{\prime }\left (x \right )\right )-f \left (x \right ) x -v^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
26.418 |
|
| \(420\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x -v \left (v -1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
26.621 |
|
| \(421\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }-v \left (v +1\right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✓ |
5.857 |
|
| \(422\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }-n \left (n +1\right ) y+\frac {\left (n +1\right ) \operatorname {LegendreP}\left (n +1, x\right )-\left (n +1\right ) x \operatorname {LegendreP}\left (n , x\right )}{x^{2}-1}&=0 \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
7.310 |
|
| \(423\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime } x +f \left (x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
85.904 |
|
| \(424\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }+2 y^{\prime } x -l y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
67.773 |
|
| \(425\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }+2 y^{\prime } x -v \left (v +1\right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
80.715 |
|
| \(426\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }-2 y^{\prime } x -\left (v +2\right ) \left (v -1\right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
702.639 |
|
| \(427\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (v +n +1\right ) \left (v -n \right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
66.958 |
|
| \(428\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (v -n +1\right ) \left (v +n \right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
66.132 |
|
| \(429\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }-2 \left (v -1\right ) x y^{\prime }-2 v y&=0 \end {array} \]
|
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✗ |
0.032 |
|
| \(430\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{2}+c x +d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
49.316 |
|
| \(431\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
119.335 |
|
| \(432\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x +1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
79.384 |
|
| \(433\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }-v \left (v +1\right ) y&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
65.512 |
|
| \(434\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (-1+x \right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
79.884 |
|
| \(435\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (-1+x \right ) y^{\prime \prime }+\left (\left (1+a \right ) x +b \right ) y^{\prime }-l y&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
97.276 |
|
| \(436\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (2+x \right ) y^{\prime \prime }+2 \left (n +1+\left (n +1-2 l \right ) x -l \,x^{2}\right ) y^{\prime }+\left (2 l \left (p -n -1\right ) x +2 p l +m \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
135.568 |
|
| \(437\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-1+x \right ) \left (x -2\right ) y^{\prime \prime }-\left (2 x -3\right ) y^{\prime }+y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
295.003 |
|
| \(438\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
65.829 |
|
| \(439\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x \left (-1+x \right ) y^{\prime \prime }+\left (\left (2 v +5\right ) x -2 v -3\right ) y^{\prime }+\left (v +1\right ) y&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
74.620 |
|
| \(440\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 x^{2} y^{\prime \prime }-\left (-4 k x +4 m^{2}+x^{2}-1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.602 |
|
| \(441\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 x^{2} y^{\prime \prime }+4 y^{\prime } x +\left (-x^{2}+2 \left (1-m +2 l \right ) x -m^{2}+1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
34.238 |
|
| \(442\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 x^{2} y^{\prime \prime }+4 y^{\prime } x +f \left (x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
8.597 |
|
| \(443\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (4 x -1\right ) y^{\prime \prime }+\left (\left (4 a +2\right ) x -a \right ) y^{\prime }+a \left (-1+a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
68.414 |
|
| \(444\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 48 x \left (-1+x \right ) y^{\prime \prime }+\left (152 x -40\right ) y^{\prime }+53 y&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
62.296 |
|
| \(445\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 144 x \left (-1+x \right ) y^{\prime \prime }+\left (120 x -48\right ) y^{\prime }+y&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
62.209 |
|
| \(446\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 144 x \left (-1+x \right ) y^{\prime \prime }+\left (168 x -96\right ) y^{\prime }+y&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
61.691 |
|
| \(447\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
19.120 |
|
| \(448\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \operatorname {a2} \,x^{2} y^{\prime \prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x \right ) y^{\prime }+\left (\operatorname {a0} \,x^{2}+\operatorname {b0} x +\operatorname {c0} \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
25.172 |
|
| \(449\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \operatorname {A2} \left (a x +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (a x +b \right ) y^{\prime }+\operatorname {A0} \left (a x +b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
84.069 |
|
| \(450\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +f \right ) y^{\prime }+g y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
126.912 |
|
| \(451\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -y+2 y^{\prime } x +x^{3} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
15.806 |
|
| \(452\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }+x^{2} y^{\prime }+\left (a \,x^{2}+b x +a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
24.513 |
|
| \(453\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+y x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
43.961 |
|
| \(454\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 x^{2}+1\right ) y^{\prime }-v \left (v +1\right ) x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
97.138 |
|
| \(455\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 \left (n +1\right ) x^{2}+2 n +1\right ) y^{\prime }-\left (v -n \right ) \left (v +n +1\right ) x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
139.096 |
|
| \(456\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{2}+1\right ) y^{\prime \prime }-\left (2 \left (n -1\right ) x^{2}+2 n -1\right ) y^{\prime }+\left (v +n \right ) \left (-v +n -1\right ) x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
138.898 |
|
| \(457\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{2}-1\right ) y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-y x&=0 \end {array} \]
|
[[_elliptic, _class_II]] |
✓ |
✓ |
✗ |
204.388 |
|
| \(458\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{2}-1\right ) y^{\prime \prime }+\left (3 x^{2}-1\right ) y^{\prime }+y x&=0 \end {array} \]
|
[[_elliptic, _class_I]] |
✓ |
✓ |
✗ |
55.043 |
|
| \(459\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{2}-1\right ) y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
119.052 |
|
| \(460\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (\left (a +b +1\right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (-1+x \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
132.551 |
|
| \(461\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {2 y^{\prime }}{x \left (x -2\right )}-\frac {y}{x^{2} \left (x -2\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
50.600 |
|
| \(462\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (-1+x \right ) \left (x -a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{x \left (-1+x \right ) \left (x -a \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
305.266 |
|
| \(463\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}-\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
317.525 |
|
| \(464\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}+\frac {v \left (v +1\right ) y}{4 x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
26.742 |
|
| \(465\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (\left (1+a \right ) x -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (-1+x \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
76.212 |
|
| \(466\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (a x +b \right ) y}{4 x \left (-1+x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
22.992 |
|
| \(467\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a \left (b +2\right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (a x +1\right ) x^{2}}-\frac {\left (a b x -c d \right ) y}{\left (a x +1\right ) x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
166.060 |
|
| \(468\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}+A x \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
139.455 |
|
| \(469\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.415 |
|
| \(470\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.741 |
|
| \(471\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {y^{\prime }}{x}-\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.398 |
|
| \(472\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
23.482 |
|
| \(473\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
95.514 |
|
| \(474\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
128.359 |
|
| \(475\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2} \left (x^{2}-1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
35.931 |
|
| \(476\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}+\frac {v \left (v +1\right ) y}{x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
45.496 |
|
| \(477\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 a \,x^{2}+n \left (n +1\right ) \left (x^{2}-1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
143.737 |
|
| \(478\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {b y}{x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
132.911 |
|
| \(479\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (-1+a \right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (-1+a \right )-v \left (v +1\right )\right )-a \left (1+a \right )\right ) y}{x^{2} \left (x^{2}-1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
368.672 |
|
| \(480\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}+1}-\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (n +1\right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
110.524 |
|
| \(481\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {a x y^{\prime }}{x^{2}+1}-\frac {b y}{\left (x^{2}+1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
90.465 |
|
| \(482\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
88.512 |
|
| \(483\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (\left (x^{2}-1\right ) \left (a \,x^{2}+b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
106.790 |
|
| \(484\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
103.963 |
|
| \(485\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}-\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
151.182 |
|
| \(486\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
188.963 |
|
| \(487\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.550 |
|
| \(488\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (v \left (v +1\right ) \left (-1+x \right )-a^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
51.638 |
|
| \(489\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (-v \left (v +1\right ) \left (-1+x \right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
50.933 |
|
| \(490\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
122.282 |
|
| \(491\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (b \,x^{2}+c x +d \right ) y}{a \,x^{2} \left (-1+x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.165 |
|
| \(492\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
136.334 |
|
| \(493\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (1+a \right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
275.737 |
|
| \(494\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\left (\frac {1-\operatorname {a1} -\operatorname {b1}}{x -\operatorname {c1}}+\frac {1-\operatorname {a2} -\operatorname {b2}}{x -\operatorname {c2}}+\frac {1-\operatorname {a3} -\operatorname {b3}}{x -\operatorname {c3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {a1} \operatorname {b1} \left (\operatorname {c1} -\operatorname {c3} \right ) \left (\operatorname {c1} -\operatorname {c2} \right )}{x -\operatorname {c1}}+\frac {\operatorname {a2} \operatorname {b2} \left (\operatorname {c2} -\operatorname {c1} \right ) \left (\operatorname {c2} -\operatorname {c3} \right )}{x -\operatorname {c2}}+\frac {\operatorname {a3} \operatorname {b3} \left (\operatorname {c3} -\operatorname {c2} \right ) \left (\operatorname {c3} -\operatorname {c1} \right )}{x -\operatorname {c3}}\right ) y}{\left (x -\operatorname {c1} \right ) \left (x -\operatorname {c2} \right ) \left (x -\operatorname {c3} \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1064.491 |
|
| \(495\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1311.950 |
|
| \(496\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (x^{2} \left (\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right )+\left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )+\left (x^{2}-\operatorname {a3} \right ) \left (x^{2}-\operatorname {a1} \right )\right )-\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}-\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
1913.327 |
|
| \(497\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}-\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
38.004 |
|
| \(498\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {y}{{\mathrm e}^{x}+1} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✗ |
2.855 |
|
| \(499\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.198 |
|
| \(500\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}-\left (-a^{2}+n^{2}\right ) y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
25.432 |
|
| \(501\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (2 n +1\right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\left (v +n +1\right ) \left (v -n \right ) y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.555 |
|
| \(502\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.947 |
|
| \(503\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {a y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.024 |
|
| \(504\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sin \left (x \right )^{2} y^{\prime \prime }-\left (a \sin \left (x \right )^{2}+n \left (n -1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.745 |
|
| \(505\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.834 |
|
| \(506\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sin \left (x \right )^{2} y^{\prime \prime }-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
16.651 |
|
| \(507\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (-\left (a^{2} b^{2}-\left (1+a \right )^{2}\right ) \sin \left (x \right )^{2}-a \left (1+a \right ) b \sin \left (2 x \right )-a \left (-1+a \right )\right ) y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.151 |
|
| \(508\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.620 |
|
| \(509\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (v \left (v +1\right ) \sin \left (x \right )^{2}-n^{2}\right ) y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.868 |
|
| \(510\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\sin \left (x \right ) y^{\prime }}{\cos \left (x \right )}-\frac {\left (2 x^{2}+x^{2} \sin \left (x \right )^{2}-24 \cos \left (x \right )^{2}\right ) y}{4 x^{2} \cos \left (x \right )^{2}}+\sqrt {\cos \left (x \right )} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
20.151 |
|
| \(511\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}-\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
13.549 |
|
| \(512\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.991 |
|
| \(513\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.465 |
|
| \(514\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (-a \cos \left (x \right )^{2} \sin \left (x \right )^{2}-m \left (m -1\right ) \sin \left (x \right )^{2}-n \left (n -1\right ) \cos \left (x \right )^{2}\right ) y}{\cos \left (x \right )^{2} \sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
8.230 |
|
| \(515\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {f^{\prime }\left (x \right ) y^{\prime }}{2 f \left (x \right )}-\frac {g \left (x \right ) y}{f \left (x \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
3.872 |
|
| \(516\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (2 f \left (x \right ) {g^{\prime }\left (x \right )}^{2} g \left (x \right )-\left (g \left (x \right )^{2}-1\right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )\right ) y^{\prime }}{f \left (x \right ) g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}-\frac {\left (\left (g \left (x \right )^{2}-1\right ) \left (f^{\prime }\left (x \right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )-f \left (x \right ) f^{\prime \prime }\left (x \right ) g^{\prime }\left (x \right )\right )-\left (2 f^{\prime }\left (x \right ) g \left (x \right )+v \left (v +1\right ) f \left (x \right ) g^{\prime }\left (x \right )\right ) f \left (x \right ) {g^{\prime }\left (x \right )}^{2}\right ) y}{f \left (x \right )^{2} g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
7.609 |
|
| \(517\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-6 y^{2}-x&=0 \end {array} \]
|
[[_Painleve, ‘1st‘]] |
✗ |
✗ |
✗ |
0.566 |
|
| \(518\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a y^{2}+b x +c&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.644 |
|
| \(519\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-2 y^{3}-y x +a&=0 \end {array} \]
|
[[_Painleve, ‘2nd‘]] |
✗ |
✗ |
✗ |
0.640 |
|
| \(520\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-2 a^{2} y^{3}+2 a b x y-b&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.721 |
|
| \(521\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+d +b x y+c y+a y^{3}&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.735 |
|
| \(522\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,x^{r} y^{2}&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.626 |
|
| \(523\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\frac {1}{\left (a y^{2}+b x y+c \,x^{2}+\alpha y+\beta x +\gamma \right )^{{3}/{2}}}&=0 \end {array} \]
|
[NONE] |
✓ |
✗ |
✗ |
1.313 |
|
| \(524\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,{\mathrm e}^{x} \sqrt {y}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.648 |
|
| \(525\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+{\mathrm e}^{x} \sin \left (y\right )&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.061 |
|
| \(526\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a^{2} \sin \left (y\right )-\beta \sin \left (x \right )&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.627 |
|
| \(527\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a^{2} \sin \left (y\right )-\beta f \left (x \right )&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.183 |
|
| \(528\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {f \left (\frac {y}{\sqrt {x}}\right )}{x^{{3}/{2}}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
15.336 |
|
| \(529\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-3 y^{\prime }-y^{2}-2 y&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
672.325 |
|
| \(530\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-7 y^{\prime }-y^{{3}/{2}}+12 y&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
125.997 |
|
| \(531\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+5 a y^{\prime }-6 y^{2}+6 a^{2} y&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✗ |
121.720 |
|
| \(532\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+3 a y^{\prime }-2 y^{3}+2 a^{2} y&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✗ |
45.677 |
|
| \(533\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a y^{\prime }+b \,x^{v} y^{n}&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.772 |
|
| \(534\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{y}-2 a&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
22.431 |
|
| \(535\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a y^{\prime }+f \left (x \right ) \sin \left (y\right )&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.789 |
|
| \(536\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (3 a +y\right ) y^{\prime }-y^{3}+a y^{2}+2 a^{2} y&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✗ |
52.214 |
|
| \(537\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (y+3 f \left (x \right )\right ) y^{\prime }-y^{3}+f \left (x \right ) y^{2}+y \left (2 f \left (x \right )^{2}+f^{\prime }\left (x \right )\right )&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.318 |
|
| \(538\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+f \left (x \right ) y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_potential_symmetries]] |
✓ |
✓ |
✗ |
2.240 |
|
| \(539\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-3 y y^{\prime }-3 a y^{2}-4 a^{2} y-b&=0 \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
56.821 |
|
| \(540\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+f \left (x \right ) y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_potential_symmetries]] |
✓ |
✓ |
✗ |
2.261 |
|
| \(541\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} c y+b y^{\prime }+a {y^{\prime }}^{2}+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
32.006 |
|
| \(542\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a y^{\prime } {| y^{\prime }|}+b \sin \left (y\right )&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
12.985 |
|
| \(543\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-a \left (-y+y^{\prime } x \right )^{v}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.211 |
|
| \(544\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-k \,x^{a} y^{b} {y^{\prime }}^{r}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.922 |
|
| \(545\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=2 a \left (c +b x +y\right ) \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
1.213 |
|
| \(546\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +2 y^{\prime }-x y^{n}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.796 |
|
| \(547\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +2 y^{\prime }+a \,x^{v} y^{n}&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.825 |
|
| \(548\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +2 y^{\prime }+x \,{\mathrm e}^{y}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.807 |
|
| \(549\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b \,{\mathrm e}^{y} x +a y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.830 |
|
| \(550\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +a y^{\prime }+b \,x^{5-2 a} {\mathrm e}^{y}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
1.129 |
|
| \(551\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x -x^{2} {y^{\prime }}^{2}+2 y^{\prime }+y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.910 |
|
| \(552\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +a \left (-y+y^{\prime } x \right )^{2}-b&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.013 |
|
| \(553\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }&=a \left (y^{n}-y\right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.853 |
|
| \(554\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+a \left ({\mathrm e}^{y}-1\right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.828 |
|
| \(555\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+a \left (-y+y^{\prime } x \right )^{2}-b \,x^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.119 |
|
| \(556\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b x +a y {y^{\prime }}^{2}+x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.784 |
|
| \(557\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }-\sqrt {b y^{2}+a \,x^{2} {y^{\prime }}^{2}}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
1.304 |
|
| \(558\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 x^{2} y^{\prime \prime }-x^{4} {y^{\prime }}^{2}+4 y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.985 |
|
| \(559\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y+a y^{3}+9 x^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
0.694 |
|
| \(560\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 24+12 y x +x^{3} \left (-y^{3}+y y^{\prime }+y^{\prime \prime }\right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
1.593 |
|
| \(561\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }-a \left (-y+y^{\prime } x \right )^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.240 |
|
| \(562\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{3} y^{\prime \prime }+x^{2} \left (9+2 y x \right ) y^{\prime }+b +x y \left (a +3 y x -2 y^{2} x^{2}\right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
1.829 |
|
| \(563\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (-x^{k}+4 x^{3}\right ) \left (-y^{3}+y y^{\prime }+y^{\prime \prime }\right )-\left (-12 x^{2}+k \,x^{-1+k}\right ) \left (y^{2}+3 y^{\prime }\right )+a x y+b&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
3.989 |
|
| \(564\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }+a^{2} y^{n}&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.837 |
|
| \(565\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.560 |
|
| \(566\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }-x^{2} y^{\prime } \left (x +y^{\prime }\right )+4 y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.570 |
|
| \(567\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-y+y^{\prime } x \right )^{3}+x^{4} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.517 |
|
| \(568\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sqrt {x}\, y^{\prime \prime }-y^{{3}/{2}}&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.650 |
|
| \(569\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{2}+b x +c \right )^{{3}/{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {a \,x^{2}+b x +c}}\right )&=0 \end {array} \]
|
[NONE] |
✓ |
✓ |
✗ |
93.335 |
|
| \(570\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-a x&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.569 |
|
| \(571\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-a \,x^{2}&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.662 |
|
| \(572\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }+y^{2}-a x -b&=0 \end {array} \]
|
[NONE] |
✗ |
✓ |
✗ |
0.676 |
|
| \(573\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-{y^{\prime }}^{2}+{\mathrm e}^{x} y \left (c y^{2}+d \right )+{\mathrm e}^{2 x} \left (b +a y^{4}\right )&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.033 |
|
| \(574\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-{y^{\prime }}^{2}+f \left (x \right ) y^{\prime }-y f^{\prime }\left (x \right )-y^{3}&=0 \end {array} \]
|
[NONE] |
✗ |
✓ |
✗ |
1.320 |
|
| \(575\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-{y^{\prime }}^{2}+f^{\prime }\left (x \right ) y^{\prime }-y f^{\prime \prime }\left (x \right )+f \left (x \right ) y^{3}-y^{4}&=0 \end {array} \]
|
[NONE] |
✗ |
✓ |
✗ |
1.007 |
|
| \(576\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-{y^{\prime }}^{2}+a y y^{\prime }-2 a y^{2}+y^{3} b&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
31.598 |
|
| \(577\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-{y^{\prime }}^{2}-\left (a y-1\right ) y^{\prime }+2 y^{2} a^{2}-2 b^{2} y^{3}+a y&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✓ |
✗ |
62.529 |
|
| \(578\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-{y^{\prime }}^{2}+\left (a y-1\right ) y^{\prime }-y \left (1+y\right ) \left (b^{2} y^{2}-a^{2}\right )&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
131.457 |
|
| \(579\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-{y^{\prime }}^{2}+\left (\tan \left (x \right )+\cot \left (x \right )\right ) y y^{\prime }+\left (\cos \left (x \right )^{2}-n^{2} \cot \left (x \right )^{2}\right ) y^{2} \ln \left (y\right )&=0 \end {array} \]
|
[[_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
7.048 |
|
| \(580\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-{y^{\prime }}^{2}-f \left (x \right ) y y^{\prime }-g \left (x \right ) y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.218 |
|
| \(581\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-{y^{\prime }}^{2}+\left (g \left (x \right )+f \left (x \right ) y^{2}\right ) y^{\prime }-y \left (g^{\prime }\left (x \right )-f^{\prime }\left (x \right ) y^{2}\right )&=0 \end {array} \]
|
[[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
✗ |
1.824 |
|
| \(582\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }+a {y^{\prime }}^{2}+b y y^{\prime }+c y^{2}+d y^{1-a}&=0 \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
405.297 |
|
| \(583\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-\frac {\left (-1+a \right ) {y^{\prime }}^{2}}{a}-f \left (x \right ) y^{2} y^{\prime }+\frac {a f \left (x \right )^{2} y^{4}}{\left (2+a \right )^{2}}-\frac {a f^{\prime }\left (x \right ) y^{3}}{2+a}&=0 \end {array} \]
|
[NONE] |
✗ |
✓ |
✗ |
1.811 |
|
| \(584\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y^{\prime } \left (1+y^{\prime }\right )+\left (x -y\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
1.002 |
|
| \(585\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x -y\right ) y^{\prime \prime }-\left (1+y^{\prime }\right ) \left (1+{y^{\prime }}^{2}\right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
1.245 |
|
| \(586\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x -y\right ) y^{\prime \prime }-h \left (y^{\prime }\right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
4.866 |
|
| \(587\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }-{y^{\prime }}^{2}+f \left (x \right ) y^{2}+a&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.724 |
|
| \(588\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }-{y^{\prime }}^{2}-4 y^{2} \left (x +2 y\right )&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.663 |
|
| \(589\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }-{y^{\prime }}^{2}+1+2 x y^{2}+a y^{3}&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.717 |
|
| \(590\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }-{y^{\prime }}^{2}+\left (a y+b x \right ) y^{2}&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.691 |
|
| \(591\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }-{y^{\prime }}^{2}+b -4 \left (x^{2}+a \right ) y^{2}-8 x y^{3}-3 y^{4}&=0 \end {array} \]
|
[[_Painleve, ‘4th‘]] |
✗ |
✗ |
✗ |
0.881 |
|
| \(592\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y y^{\prime \prime }-3 {y^{\prime }}^{2}+f \left (x \right ) y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.659 |
|
| \(593\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 3 y y^{\prime \prime }-2 {y^{\prime }}^{2}-a \,x^{2}-b x -c&=0 \end {array} \]
|
[NONE] |
✓ |
✓ |
✗ |
0.741 |
|
| \(594\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} f \left (x \right )+a y y^{\prime }+x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.741 |
|
| \(595\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }-x {y^{\prime }}^{2}+y y^{\prime }+x \left (d +a y^{4}\right )+y \left (c +b y^{2}\right )&=0 \end {array} \]
|
[[_Painleve, ‘3rd‘]] |
✗ |
✗ |
✗ |
1.200 |
|
| \(596\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }-x {y^{\prime }}^{2}+a y y^{\prime }+y^{3} b x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.970 |
|
| \(597\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (1+y\right ) y^{\prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.080 |
|
| \(598\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) {y^{\prime }}^{2}-y y^{\prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
2.014 |
|
| \(599\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (-1+y\right ) y^{\prime \prime }-2 x^{2} {y^{\prime }}^{2}-2 x \left (-1+y\right ) y^{\prime }-2 y \left (-1+y\right )^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
4.032 |
|
| \(600\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (x +y\right ) y^{\prime \prime }-\left (-y+y^{\prime } x \right )^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.342 |
|
| \(601\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (x -y\right ) y^{\prime \prime }+a \left (-y+y^{\prime } x \right )^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.342 |
|
| \(602\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} y y^{\prime \prime }-x^{2} \left (1+{y^{\prime }}^{2}\right )+y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
0.809 |
|
| \(603\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.233 |
|
| \(604\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x +1\right )^{2} y y^{\prime \prime }-x \left (x +1\right )^{2} {y^{\prime }}^{2}+2 \left (x +1\right )^{2} y y^{\prime }-a \left (2+x \right ) y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.509 |
|
| \(605\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 x^{2} y y^{\prime }+3 x y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.656 |
|
| \(606\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a x +y {y^{\prime }}^{2}+y^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
0.560 |
|
| \(607\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}-a x -b&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.718 |
|
| \(608\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x +y^{2}\right ) y^{\prime \prime }-2 \left (x -y^{2}\right ) {y^{\prime }}^{3}+y^{\prime } \left (1+4 y y^{\prime }\right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
1.683 |
|
| \(609\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+y^{2}\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right ) \left (-y+y^{\prime } x \right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
1.460 |
|
| \(610\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+y^{2}\right ) y^{\prime \prime }-2 \left (1+{y^{\prime }}^{2}\right ) \left (-y+y^{\prime } x \right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
1.464 |
|
| \(611\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (1-y\right ) y y^{\prime \prime }-\left (1-2 y\right ) {y^{\prime }}^{2}+f \left (x \right ) \left (1-y\right ) y y^{\prime }&=0 \end {array} \]
|
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
3.376 |
|
| \(612\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y^{2} y^{\prime \prime }-a&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
0.504 |
|
| \(613\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime }+\left (a^{2}-x^{2}\right ) y {y^{\prime }}^{2}-x \left (a^{2}-y^{2}\right ) y^{\prime }&=0 \end {array} \]
|
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.775 |
|
| \(614\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} y \left (-1+y\right ) y^{\prime \prime }-x^{2} \left (3 y-1\right ) {y^{\prime }}^{2}+2 x y \left (-1+y\right ) y^{\prime }+\left (a y^{2}+b \right ) \left (-1+y\right )^{3}+c x y^{2} \left (-1+y\right )+d \,x^{2} y^{2} \left (1+y\right )&=0 \end {array} \]
|
[[_Painleve, ‘5th‘]] |
✗ |
✗ |
✗ |
3.643 |
|
| \(615\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x +y\right ) \left (-y+y^{\prime } x \right )^{3}+x^{3} y^{2} y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.848 |
|
| \(616\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y^{3} y^{\prime \prime }+y^{4}-a^{2} x y^{2}-1&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.677 |
|
| \(617\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y^{3} y^{\prime \prime }+y^{2} {y^{\prime }}^{2}-a \,x^{2}-b x -c&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.875 |
|
| \(618\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (c +2 b x +a \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }+d y&=0 \end {array} \]
|
[NONE] |
✓ |
✓ |
✗ |
1.162 |
|
| \(619\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sqrt {x^{2}+y^{2}}\, y^{\prime \prime }-a \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
1.537 |
|
| \(620\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2}&=0 \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
✗ |
✗ |
16.214 |
|
| \(621\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-y+y^{\prime } x \right ) y^{\prime \prime }+4 {y^{\prime }}^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.347 |
|
| \(622\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-y+y^{\prime } x \right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right )^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
7.571 |
|
| \(623\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \,x^{3} y^{\prime } y^{\prime \prime }+b y^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
2.475 |
|
| \(624\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left ({y^{\prime }}^{2}+a \left (-y+y^{\prime } x \right )\right ) y^{\prime \prime }-b&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.438 |
|
| \(625\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} {y^{\prime \prime }}^{2}-a y-b&=0 \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✗ |
0.051 |
|
| \(626\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2}-x \left (x +4 y^{\prime }\right ) y^{\prime \prime }+2 \left (x +y^{\prime }\right ) y^{\prime }-2 y&=0 \end {array} \]
|
[NONE] |
✓ |
✓ |
✗ |
0.056 |
|
| \(627\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 {y^{\prime }}^{2}-2 \left (y+3 y^{\prime } x \right ) y^{\prime \prime }+3 x^{2} {y^{\prime \prime }}^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
7.675 |
|
| \(628\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (2-9 x \right ) x^{2} {y^{\prime \prime }}^{2}-6 \left (1-6 x \right ) x y^{\prime } y^{\prime \prime }+6 y y^{\prime \prime }-36 x {y^{\prime }}^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
36.933 |
|
| \(629\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y {y^{\prime \prime }}^{2}-a \,{\mathrm e}^{2 x}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
2.258 |
|
| \(630\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (y^{2}-x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime }\right )^{2}-4 x y \left (-y+y^{\prime } x \right )^{3}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
0.049 |
|
| \(631\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (a \,x^{2}+b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
5.285 |
|
| \(632\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (a \,x^{2}+b x c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.619 |
|
| \(633\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.272 |
|
| \(634\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-a \,x^{-2+n} \left (a \,x^{n}+n +1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
4.053 |
|
| \(635\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.220 |
|
| \(636\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a y^{\prime }-\left (b \,x^{2}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.085 |
|
| \(637\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}+a \,x^{n}+n \,x^{n -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✗ |
✗ |
8.796 |
|
| \(638\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}-a \,x^{n}+n \,x^{n -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
8.625 |
|
| \(639\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+y^{\prime } x +\left (n -1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.569 |
|
| \(640\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 n y-2 y^{\prime } x +y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.450 |
|
| \(641\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b y+a x y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.505 |
|
| \(642\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a x y^{\prime }+b x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.352 |
|
| \(643\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a x y^{\prime }+\left (b x +c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.758 |
|
| \(644\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 a x y^{\prime }+\left (b \,x^{4}+a^{2} x^{2}+c x +a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.448 |
|
| \(645\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.330 |
|
| \(646\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.806 |
|
| \(647\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
14.128 |
|
| \(648\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (-c \,x^{2 n}+a \,x^{n +1}+b \,x^{n}+n \,x^{n -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
11.927 |
|
| \(649\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
14.913 |
|
| \(650\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,x^{n} y^{\prime }+b \,x^{n -1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.365 |
|
| \(651\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.874 |
|
| \(652\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,x^{n} y^{\prime }-b \left (a \,x^{n +m}+b \,x^{2 m}+m \,x^{m -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
10.881 |
|
| \(653\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (a^{2} x^{2 n}+b \,x^{2 m}+x^{n -1} a n +c \,x^{m -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
12.626 |
|
| \(654\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}+b -c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
11.109 |
|
| \(655\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (x^{n} a b -a \,x^{n -1}+b^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
11.219 |
|
| \(656\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (x^{n} a b +b \,x^{n -1}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{n}+1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
13.871 |
|
| \(657\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (x^{n} a b +2 b \,x^{n -1}-a^{2} x \right ) y^{\prime }+a \left (x^{n} a b +b \,x^{n -1}-a^{2} x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
21.934 |
|
| \(658\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+x^{n} \left (a \,x^{2}+\left (a c +b \right ) x +b c \right ) y^{\prime }-x^{n} \left (a x +b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
39.302 |
|
| \(659\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+c \left (a \,x^{n}+b \,x^{m}-c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
12.990 |
|
| \(660\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (x^{n +m} a b +b \left (m +1\right ) x^{m -1}-a \,x^{n -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
10.958 |
|
| \(661\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (x^{n +m} a b +x^{m} b c +x^{n -1} a n \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
11.615 |
|
| \(662\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +a y^{\prime }+\left (b x +c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
12.227 |
|
| \(663\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y&=0 \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✗ |
11.799 |
|
| \(664\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +a y^{\prime }+b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
11.348 |
|
| \(665\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (b -x \right ) y^{\prime }-a y&=0 \end {array} \]
|
[_Laguerre] |
✓ |
✓ |
✗ |
11.247 |
|
| \(666\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.421 |
|
| \(667\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a b x +a n +b m \right ) y+\left (m +n +\left (a +b \right ) x \right ) y^{\prime }+y^{\prime \prime } x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
14.959 |
|
| \(668\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
12.400 |
|
| \(669\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x -\left (2 a x +1\right ) y^{\prime }+b \,x^{3} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
12.389 |
|
| \(670\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (b -2\right ) x^{3} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
42.425 |
|
| \(671\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{2}+B x +\operatorname {C0} \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
39.762 |
|
| \(672\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a \,x^{2}+b x +2\right ) y^{\prime }+\left (c \,x^{2}+d x +b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
38.293 |
|
| \(673\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +a \,x^{n} y^{\prime }+\left (x^{n} a b -a \,x^{n -1}-b^{2} x +2 b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
10.227 |
|
| \(674\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
16.453 |
|
| \(675\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}-c x +b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
13.597 |
|
| \(676\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (x^{n} a b +b -3 n +1\right ) y^{\prime }+a^{2} n \left (b -n \right ) x^{2 n -1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
11.263 |
|
| \(677\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{2 n -1}+d \,x^{n -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.256 |
|
| \(678\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a \,x^{n}+b \,x^{n -1}+2\right ) y^{\prime }+b \,x^{-2+n} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
23.880 |
|
| \(679\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (a \,x^{n}+b x \right ) y^{\prime }+\left (x^{n} a b +x^{n -1} a n -b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
32.238 |
|
| \(680\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (x^{n} a b +b \,x^{n -1}+a x -1\right ) y^{\prime }+a^{2} b \,x^{n} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
30.527 |
|
| \(681\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (x^{n +m} a b +a n \,x^{n}+b \,x^{m}+1-2 n \right ) y^{\prime }+a^{2} b n \,x^{2 n +m -1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
15.804 |
|
| \(682\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
35.821 |
|
| \(683\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a x +b \right ) y^{\prime \prime }+s \left (c x +d \right ) y^{\prime }-s^{2} \left (\left (a +c \right ) x +b +d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
39.095 |
|
| \(684\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a_{2} x +b_{2} \right ) y^{\prime \prime }+\left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x +b_{0} \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
41.766 |
|
| \(685\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.349 |
|
| \(686\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }-\left (a^{2} x^{2 n}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
6.794 |
|
| \(687\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
6.064 |
|
| \(688\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
5.622 |
|
| \(689\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{2 n} \left (b \,x^{n}+c \right )^{m}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
6.291 |
|
| \(690\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (a \,x^{2}+b x +c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
32.503 |
|
| \(691\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
22.621 |
|
| \(692\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
32.958 |
|
| \(693\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.864 |
|
| \(694\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
36.957 |
|
| \(695\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (b -n -1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
42.447 |
|
| \(696\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
42.099 |
|
| \(697\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
43.853 |
|
| \(698\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
59.827 |
|
| \(699\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (x^{n} a b +a c \,x^{n -1}+b^{2} x^{2}+2 b x c +c^{2}-c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
17.695 |
|
| \(700\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n +2 m}-b^{2} x^{4 m +2}+a m \,x^{n -1}-m^{2}-m \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
14.090 |
|
| \(701\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
34.704 |
|
| \(702\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (b +n -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
35.917 |
|
| \(703\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\left (a \,x^{n +2}+b \,x^{2}+c \right ) y^{\prime }+\left (a n \,x^{n +1}+a c \,x^{n}+b c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
27.017 |
|
| \(704\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) y^{\prime \prime }+n \left (n -1\right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✓ |
9.141 |
|
| \(705\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
23.171 |
|
| \(706\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n \left (n +1\right ) y-2 y^{\prime } x +\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
103.766 |
|
| \(707\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +\nu \left (\nu +1\right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
102.347 |
|
| \(708\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (\nu +n +1\right ) \left (\nu -n \right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
92.155 |
|
| \(709\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (\nu -n +1\right ) \left (\nu +n \right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
92.947 |
|
| \(710\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right ) y^{\prime \prime }+\left (2 a +1\right ) y^{\prime }-b \left (2 a +b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
104.248 |
|
| \(711\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) y^{\prime \prime }+\left (2 a -3\right ) x y^{\prime }+\left (n +1\right ) \left (n +2 a -1\right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
90.771 |
|
| \(712\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) y^{\prime \prime }+\left (\beta -\alpha -\left (\alpha +\beta +2\right ) x \right ) y^{\prime }+n \left (n +\alpha +\beta +1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
146.510 |
|
| \(713\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
135.358 |
|
| \(714\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (2 n +1\right ) a x y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
480.287 |
|
| \(715\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\left (2 a \,x^{2}+b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
55.661 |
|
| \(716\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
134.397 |
|
| \(717\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
30.177 |
|
| \(718\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
163.405 |
|
| \(719\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (-1+x \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
161.223 |
|
| \(720\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (a +x \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
114.745 |
|
| \(721\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y&=0 \end {array} \]
|
[_Jacobi] |
✓ |
✓ |
✗ |
63.365 |
|
| \(722\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
154.702 |
|
| \(723\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (k +x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
163.325 |
|
| \(724\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
163.308 |
|
| \(725\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
85.323 |
|
| \(726\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+b y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
84.246 |
|
| \(727\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
85.403 |
|
| \(728\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (c x +d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
91.427 |
|
| \(729\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
111.617 |
|
| \(730\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x -\left (a \,x^{n}-a b \,x^{n -1}+b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
91.803 |
|
| \(731\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) y^{\prime }+s x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
152.938 |
|
| \(732\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (c -2 a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
185.289 |
|
| \(733\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
152.330 |
|
| \(734\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
231.887 |
|
| \(735\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (x \alpha +2 b -\beta \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
227.031 |
|
| \(736\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
225.673 |
|
| \(737\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+x m +k \right ) y^{\prime }+\left (-1+k \right ) \left (\left (-a k +n \right ) x +m -b k \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
262.014 |
|
| \(738\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+x m +k \right ) y^{\prime }+\left (-2 \left (a +n \right ) x +1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
329.009 |
|
| \(739\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
176.292 |
|
| \(740\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (a \,x^{2}+b x +1\right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y^{\prime }+\left (n x +m \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
256.882 |
|
| \(741\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (-1+x \right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
337.888 |
|
| \(742\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }-\left (-\lambda ^{2}+x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
402.448 |
|
| \(743\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
380.704 |
|
| \(744\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\left (\alpha \gamma +\beta \right ) x +\beta \lambda \right ) y^{\prime }-\left (x \alpha +\beta \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
948.703 |
|
| \(745\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-\lambda x +x^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
947.481 |
|
| \(746\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
7.418 |
|
| \(747\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a \,x^{n -1}+a b \,x^{-2+n}+b^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
26.488 |
|
| \(748\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \,x^{2} \left (-1+x \right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.451 |
|
| \(749\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
154.238 |
|
| \(750\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }-\left (\nu \left (\nu +1\right ) \left (x^{2}-1\right )+n^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
116.653 |
|
| \(751\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right )^{2} y^{\prime \prime }-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (\nu \left (\nu +1\right ) \left (-x^{2}+1\right )-\mu ^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
114.550 |
|
| \(752\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a \left (x^{2}-1\right )^{2} y^{\prime \prime }+b x \left (x^{2}-1\right ) y^{\prime }+\left (c \,x^{2}+d x +e \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
135.901 |
|
| \(753\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+a \right )^{2} y^{\prime \prime }+b \,x^{n} \left (x^{2}+a \right ) y^{\prime }-\left (b \,x^{n +1}+a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
31.728 |
|
| \(754\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+a \right )^{2} y^{\prime \prime }+b \,x^{n} \left (x^{2}+a \right ) y^{\prime }-m \left (b \,x^{n +1}+\left (m -1\right ) x^{2}+a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
36.038 |
|
| \(755\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }+\left (\left (x^{2}-1\right ) \left (a^{2} x^{2}-\lambda \right )-m^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
115.211 |
|
| \(756\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+\left (\left (x^{2}+1\right ) \left (a^{2} x^{2}-\lambda \right )+m^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
115.885 |
|
| \(757\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+c \left (a x +b \right )^{n -4} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
6.278 |
|
| \(758\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+a x y^{\prime }-\left (b^{2} x^{n}+2 b \,x^{n -1}+a b x +a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
11.768 |
|
| \(759\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (a \,x^{n -1}+b x \right ) y^{\prime }+\left (-1+a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
17.770 |
|
| \(760\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (2 x^{n -1}+a \,x^{2}+b x \right ) y^{\prime }+b y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
63.690 |
|
| \(761\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{n}+b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
11.385 |
|
| \(762\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (a \,x^{n}-x^{n -1}+a b x +b \right ) y^{\prime }+a^{2} b x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
48.273 |
|
| \(763\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (a \,x^{n +m}+1\right ) y^{\prime }+a \,x^{m} \left (1+m \,x^{n -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
11.938 |
|
| \(764\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b \right ) y^{\prime \prime }+\left (c \,x^{n}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{n}+d -b \lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
21.479 |
|
| \(765\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (a -b \right ) x^{n}+a -n \right ) y^{\prime }+b \left (1-a \right ) x^{n -1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
54.361 |
|
| \(766\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a^{2} \left (n +1\right ) x^{2 n}+n -1\right ) y^{\prime }-\nu \left (\nu +1\right ) a^{2} n^{2} x^{2 n} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
140.036 |
|
| \(767\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a p \,x^{n}+q \right ) y^{\prime }+\left (a r \,x^{n}+s \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
155.439 |
|
| \(768\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{-2+n} \left (\left (b -1\right ) x^{n}+a \left (n -1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
4.943 |
|
| \(769\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) \left (c \,x^{n}+d \right ) y^{\prime }+n \left (-a d +b c \right ) x^{n -1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
26.097 |
|
| \(770\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{n}+a \right )^{2} y^{\prime \prime }+b \,x^{m} \left (x^{n}+a \right ) y^{\prime }-x^{-2+n} \left (b \,x^{m +1}+a n -a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
66.439 |
|
| \(771\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+c \,x^{m} \left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{m}-x^{n -1} a n -1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
40.944 |
|
| \(772\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (n +1\right ) x \left (a^{2} x^{2 n}-b^{2}\right ) y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
171.732 |
|
| \(773\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n +1}+b \,x^{n}+c \right )^{2} y^{\prime \prime }+\left (\alpha \,x^{n}+\beta \,x^{n -1}+\gamma \right ) y^{\prime }+\left (n \left (-a n -a +\alpha \right ) x^{n -1}+\left (n -1\right ) \left (-b n +\beta \right ) x^{-2+n}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
227.337 |
|
| \(774\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda ^{2}-x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
91.251 |
|
| \(775\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+a n \,x^{n -1} b m \,x^{m -1} y^{\prime }+d y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
128.500 |
|
| \(776\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \left (\lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{2 \lambda x}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.224 |
|
| \(777\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.406 |
|
| \(778\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.185 |
|
| \(779\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+{\mathrm e}^{2 \lambda x} c -\frac {\lambda ^{2}}{4}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
4.481 |
|
| \(780\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}-\frac {\lambda ^{2}}{4}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
4.621 |
|
| \(781\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.025 |
|
| \(782\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
9.091 |
|
| \(783\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
12.431 |
|
| \(784\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{\mu x} \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+\mu \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
13.560 |
|
| \(785\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 k \,{\mathrm e}^{\mu x} y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+k^{2} {\mathrm e}^{2 \mu x}+k \mu \,{\mathrm e}^{\mu x}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
8.375 |
|
| \(786\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.980 |
|
| \(787\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+c \left (a \,{\mathrm e}^{\lambda x}+b -c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
12.954 |
|
| \(788\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a +b \,{\mathrm e}^{2 \lambda x}\right ) y^{\prime }+\lambda \left (a -\lambda -b \,{\mathrm e}^{2 \lambda x}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
7.701 |
|
| \(789\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (b -\lambda \right ) {\mathrm e}^{2 \lambda x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
14.642 |
|
| \(790\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+c \,{\mathrm e}^{\mu x}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
8.132 |
|
| \(791\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b -\lambda \right ) y^{\prime }+\left ({\mathrm e}^{2 \lambda x} c +a b \,{\mathrm e}^{\lambda x}+b^{2}-b \lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
14.726 |
|
| \(792\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +b c -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.173 |
|
| \(793\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.715 |
|
| \(794\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{2 \mu x}+c \,{\mathrm e}^{\mu x}+k \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
8.751 |
|
| \(795\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b -\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \mu x}+d \,{\mathrm e}^{\mu x}+k \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
9.768 |
|
| \(796\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}\right ) y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\mu x}+\lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
8.340 |
|
| \(797\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{2 \mu x}+b \right ) y^{\prime }+\mu \left ({\mathrm e}^{\lambda x} \left (b -a \,{\mathrm e}^{2 \mu x}\right )-\mu \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
17.019 |
|
| \(798\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+{\mathrm e}^{\lambda x} a c +{\mathrm e}^{\mu x} b \mu \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
9.418 |
|
| \(799\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b \,{\mathrm e}^{\mu x}-\lambda \right ) y^{\prime }+\left (a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+{\mathrm e}^{2 \lambda x} c +{\mathrm e}^{2 \mu x} b^{2}+b \left (\mu -\lambda \right ) {\mathrm e}^{\mu x}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
11.053 |
|
| \(800\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+a \lambda \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}-2 \lambda \right ) y^{\prime }+a^{2} b \lambda \,{\mathrm e}^{\left (2 \lambda +\mu \right ) x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
12.339 |
|
| \(801\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,{\mathrm e}^{b \,x^{n}} y^{\prime }+c \left (a \,{\mathrm e}^{b \,x^{n}}-c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✗ |
✗ |
12.932 |
|
| \(802\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime \prime }+\left (c \,{\mathrm e}^{\lambda x}+d \right ) y^{\prime }+k \left (\left (-a k +c \right ) {\mathrm e}^{\lambda x}+d -b k \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
16.536 |
|
| \(803\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime \prime }+\left (c \,{\mathrm e}^{\lambda x}+d \right ) y^{\prime }+\left (n \,{\mathrm e}^{\lambda x}+m \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
14.769 |
|
| \(804\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
36.778 |
|
| \(805\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }-2 n x \left (x +1\right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
41.667 |
|
| \(806\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }+2 x^{3} \left (x +1\right ) y^{\prime }+n^{2} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
62.808 |
|
| \(807\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-y+y^{\prime } x \right )^{2}+x^{2} y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.321 |
|
| \(808\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }-\left (-y+y^{\prime } x \right )^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.703 |
|
| \(809\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }-{y^{\prime }}^{2}&=\ln \left (y\right ) y^{2}-y^{2} x^{2} \end {array} \]
|
[[_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.839 |
|
| \(810\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+y^{\prime } x +x^{2} y&=0\\ y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=0\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
12.413 |
|
| \(811\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (2 t +1\right ) x^{\prime \prime }+t^{3} x^{\prime }+x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
54.053 |
|
| \(812\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sin \left (t \right ) x^{\prime \prime }+\cos \left (t \right ) x^{\prime }+2 x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
7.344 |
|
| \(813\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (t^{4}+t^{2}\right ) x^{\prime \prime }+2 t^{3} x^{\prime }+3 x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
82.876 |
|
| \(814\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }-\tan \left (t \right ) x^{\prime }+x&=0 \end {array} \]
|
[_Lienard] |
✓ |
✓ |
✗ |
9.522 |
|
| \(815\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} f \left (t \right ) x^{\prime \prime }+x g \left (t \right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
3.154 |
|
| \(816\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+x^{4} x^{\prime }-x^{\prime }+x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
158.386 |
|
| \(817\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+x^{\prime }+{x^{\prime }}^{3}+x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
62.868 |
|
| \(818\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+\left (x^{4}+x^{2}\right ) x^{\prime }+x^{3}+x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
210.177 |
|
| \(819\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+\left (5 x^{4}-6 x^{2}\right ) x^{\prime }+x^{3}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
56.100 |
|
| \(820\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+\left (x^{2}+1\right ) x^{\prime }+x^{3}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
36.279 |
|
| \(821\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime }+y^{\prime \prime }&=1 \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
199.025 |
|
| \(822\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y^{\prime \prime }+3 y^{\prime }+4 x^{2} y&=1 \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
10.340 |
|
| \(823\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sinh \left (x \right ) {y^{\prime }}^{2}+y^{\prime \prime }&=y x \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.580 |
|
| \(824\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x -3\right ) y^{\prime \prime }+y \ln \left (x \right )&=x^{2}\\ y \left (1\right )&=1\\ y^{\prime }\left (1\right )&=2\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
26.991 |
|
| \(825\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cot \left (x \right ) y&=0\\ y \left (\frac {\pi }{4}\right )&=1\\ y^{\prime }\left (\frac {\pi }{4}\right )&=0\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
8.321 |
|
| \(826\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+1\right ) y^{\prime \prime }+\left (-1+x \right ) y^{\prime }+y&=0\\ y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
53.462 |
|
| \(827\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +2 x^{2} y^{\prime }+\sin \left (x \right ) y&=\sinh \left (x \right )\\ y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=1\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
35.705 |
|
| \(828\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sin \left (x \right ) y^{\prime \prime }+y^{\prime } x +7 y&=1\\ y \left (1\right )&=1\\ y^{\prime }\left (1\right )&=0\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
16.387 |
|
| \(829\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+x^{2} y&=\tan \left (x \right )\\ y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
12.686 |
|
| \(830\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (x^{2}+1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.226 |
|
| \(831\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\frac {k x}{y^{4}}&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
0.834 |
|
| \(832\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+x^{2} y^{\prime }+2 \left (1-x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
10.133 |
|
| \(833\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \ln \left (x^{2}+1\right ) y^{\prime \prime }+\frac {4 x y^{\prime }}{x^{2}+1}+\frac {\left (-x^{2}+1\right ) y}{\left (x^{2}+1\right )^{2}}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
232.536 |
|
| \(834\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +\left (6 x y^{2}+1\right ) y^{\prime }+2 y^{3}+1&=0 \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
✗ |
14.542 |
|
| \(835\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\frac {\left (-1+x \right ) y^{\prime }}{x}+\frac {y}{x^{3}}&=\frac {{\mathrm e}^{-\frac {1}{x}}}{x^{3}} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
9.507 |
|
| \(836\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} t^{2} y^{\prime \prime }-6 y^{\prime } t +\sin \left (2 t \right ) y&=\ln \left (t \right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
11.646 |
|
| \(837\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+3 y^{\prime }+\frac {y}{t}&=t \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
27.455 |
|
| \(838\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+y^{\prime } t -y \ln \left (t \right )&=\cos \left (2 t \right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
6.192 |
|
| \(839\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} t^{3} y^{\prime \prime }-2 y^{\prime } t +y&=t^{4} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
24.260 |
|
| \(840\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n \left (n +1\right ) y-2 y^{\prime } x +\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
109.434 |
|
| \(841\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+x^{\prime }+x-x^{3}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
37.677 |
|
| \(842\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+x^{\prime }+x+x^{3}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
37.119 |
|
| \(843\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sqrt {1-x}\, y^{\prime \prime }-4 y&=\sin \left (x \right )\\ y \left (-2\right )&=3\\ y^{\prime }\left (-2\right )&=-1\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
✗ |
7.135 |
|
| \(844\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-4\right ) y^{\prime \prime }+y \ln \left (x \right )&=x \,{\mathrm e}^{x}\\ y \left (1\right )&=1\\ y^{\prime }\left (1\right )&=2\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
5.263 |
|
| \(845\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{2} y^{\prime \prime }&=8 x^{2} \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.830 |
|
| \(846\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+x^{2} y^{\prime }-4 y&=x^{3} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
13.749 |
|
| \(847\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+x^{2} y^{\prime }-4 y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
12.172 |
|
| \(848\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+x^{2} y^{\prime }&=4 y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.396 |
|
| \(849\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+x^{2} y^{\prime }+4 y&=y^{3} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.204 |
|
| \(850\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x {y^{\prime \prime }}^{2}+2 y&=2 x \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
0.075 |
|
| \(851\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+2 \sin \left (x\right )&=\sin \left (2 t \right ) \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
2.389 |
|
| \(852\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} {y^{\prime \prime }}^{2}-5 y^{\prime \prime } y^{\prime }+4 y^{2}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✓ |
✗ |
✗ |
0.073 |
|
| \(853\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} {y^{\prime \prime }}^{2}-2 y^{\prime \prime } y^{\prime }+y^{2}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✓ |
✗ |
✗ |
0.055 |
|
| \(854\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} \left (2-\ln \left (x \right )\right ) y^{\prime \prime }+x \left (4-\ln \left (x \right )\right ) y^{\prime }-y&=\frac {\left (2-\ln \left (x \right )\right )^{2}}{\sqrt {x}}\\ y \left (\infty \right )&=0\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
✗ |
44.535 |
|
| \(855\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} \left (\ln \left (x \right )-1\right ) y^{\prime \prime }-x^{2} y^{\prime }+y x&=2 \ln \left (x \right )\\ y \left (\infty \right )&=0\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
100.944 |
|
| \(856\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }-2 {x^{\prime }}^{2}+x^{\prime }-2 x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
143.184 |
|
| \(857\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+{\mathrm e}^{-x^{\prime }}-x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
132.771 |
|
| \(858\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }-x^{\prime }+x-x^{2}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
136.913 |
|
| \(859\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+y^{\prime }+y+y^{3}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
93.957 |
|
| \(860\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +\alpha \left (\alpha +1\right ) y&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
104.460 |
|
| \(861\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\mu \left (1-y^{2}\right ) y^{\prime }+y&=0 \end {array} \]
|
[[_2nd_order, _missing_x], _Van_der_Pol] |
✗ |
✗ |
✗ |
108.648 |
|
| \(862\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (t -1\right ) y^{\prime \prime }-3 y^{\prime } t +4 y&=\sin \left (t \right )\\ y \left (-2\right )&=2\\ y^{\prime }\left (-2\right )&=1\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
24.595 |
|
| \(863\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} t \left (t -4\right ) y^{\prime \prime }+3 y^{\prime } t +4 y&=2\\ y \left (3\right )&=0\\ y^{\prime }\left (3\right )&=-1\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
114.015 |
|
| \(864\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\cos \left (t \right ) y^{\prime }+3 y \ln \left (t \right )&=0\\ y \left (2\right )&=3\\ y^{\prime }\left (2\right )&=1\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
11.628 |
|
| \(865\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x +3\right ) y^{\prime \prime }+y^{\prime } x +y \ln \left (x \right )&=0\\ y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=1\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
34.875 |
|
| \(866\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x -2\right ) y^{\prime \prime }+y^{\prime }+\left (x -2\right ) \tan \left (x \right ) y&=0\\ y \left (3\right )&=1\\ y^{\prime }\left (3\right )&=2\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
35.819 |
|
| \(867\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\frac {t}{y}&=\frac {1}{\pi }\\ y \left (0\right )&=y_{0}\\ y^{\prime }\left (0\right )&=y_{1}\\ \end {array} \]
|
[NONE] |
✗ |
✓ |
✗ |
1.055 |
|
| \(868\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+y+\frac {y^{3}}{5}&=\cos \left (w t \right )\\ y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
3.359 |
|
| \(869\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\frac {y^{\prime }}{5}+y+\frac {y^{3}}{5}&=\cos \left (w t \right )\\ y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
3.420 |
|
| \(870\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n \,x^{3} y^{\prime \prime }&=\left (y-y^{\prime } x \right )^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.055 |
|
| \(871\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{2} \left (x^{2} y^{\prime \prime }-y^{\prime } x +y\right )&=x^{3} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.597 |
|
| \(872\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{2} y^{\prime \prime }-3 y^{2} y^{\prime } x +4 y^{3}+x^{6}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.439 |
|
| \(873\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2}&=0 \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
✗ |
✗ |
101.730 |
|
| \(874\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{2} y^{\prime }+2 y x \right ) y^{\prime \prime }+4 x {y^{\prime }}^{2}+8 y y^{\prime } x +4 y^{2}-1&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
8.574 |
|
| \(875\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y y^{\prime \prime }+x^{2} {y^{\prime }}^{2}-5 y y^{\prime } x&=4 y^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.465 |
|
| \(876\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n \left (n +1\right ) y-2 y^{\prime } x +\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
109.759 |
|
| \(877\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\frac {y}{x^{2} \ln \left (x \right )}&={\mathrm e}^{x} \left (\frac {2}{x}+\ln \left (x \right )\right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
5.429 |
|
| \(878\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=x +y^{2}\\ y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1\\ \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.799 |
|
| \(879\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 y^{\prime }+y^{2}&=0\\ y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0\\ \end {array} \]
|
[[_2nd_order, _missing_x], [_Emden, _modified]] |
✗ |
✗ |
✗ |
90.609 |
|
| \(880\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x \,{\mathrm e}^{y}+y-x^{2}\right ) y^{\prime \prime }&=2 y x -{\mathrm e}^{y}-x \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.746 |
|
| \(881\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-f \left (x \right ) y^{\prime }+\left (f \left (x \right )-1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✗ |
✗ |
13.701 |
|
| \(882\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+3 y^{\prime } x +x^{2} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.683 |
|
| \(883\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+\left (5 x^{4}-9 x^{2}\right ) x^{\prime }+x^{5}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
197.221 |
|
| \(884\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }-\frac {x^{2} {y^{\prime }}^{2}}{2 y}+4 y^{\prime } x +4 y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.114 |
|
| \(885\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \cos \left (x \right ) y^{\prime }+\sin \left (x \right ) y^{\prime \prime }+n y \sin \left (x \right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.348 |
|
| \(886\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} v^{\prime \prime }&=\left (\frac {1}{v}+{v^{\prime }}^{4}\right )^{{1}/{3}} \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
82.149 |
|
| \(887\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sqrt {y^{\prime }+y}&=\left (y^{\prime \prime }+2 x \right )^{{1}/{4}} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
13.514 |
|
| \(888\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (1+y^{2}\right ) y^{\prime \prime }-2 y {y^{\prime }}^{2}-2 \left (1+y^{2}\right ) y^{\prime }&=y^{2} \left (1+y^{2}\right ) \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
127.372 |
|
| \(889\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (5+2 x \right )^{2} y^{\prime \prime }-6 \left (5-2 x \right ) y^{\prime }+8 y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
56.629 |
|
| \(890\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sqrt {x}\, y^{\prime \prime }+2 y^{\prime } x +3 y&=x \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.352 |
|
| \(891\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }&=x y^{2} \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
✗ |
✗ |
103.934 |
|
| \(892\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (y^{2}+2 x^{2} y^{\prime }\right ) y^{\prime \prime }+2 \left (x +y\right ) {y^{\prime }}^{2}+y^{\prime } x +y&=0 \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
✗ |
✗ |
147.934 |
|
| \(893\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\frac {y^{\prime }}{x^{{1}/{3}}}+\left (\frac {1}{4 x^{{2}/{3}}}-\frac {1}{6 x^{{1}/{3}}}-\frac {6}{x^{2}}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
14.352 |
|
| \(894\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-2 b y^{\prime }+b^{2} x^{2} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.621 |
|
| \(895\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-y+y^{\prime } x \right )^{2}+x^{2} y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.294 |
|
| \(896\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} y y^{\prime \prime }+y^{2}&=x^{2} {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.049 |
|
| \(897\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }&=\sqrt {m \,x^{2} {y^{\prime }}^{3}+n y^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
2.831 |
|
| \(898\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }&=\left (x^{3}+2 y x \right ) y^{\prime }-4 y^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.233 |
|
| \(899\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }-x^{3} y^{\prime }&=x^{2} {y^{\prime }}^{2}-4 y^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.845 |
|
| \(900\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+4 y^{2}-6 y&=x^{4} {y^{\prime }}^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.412 |
|
| \(901\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 x^{2} y^{\prime \prime }+4 x^{5} y^{\prime }+\left (x^{3}+6 x^{2}+4\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
73.651 |
|
| \(902\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-y+y^{\prime } x \right )^{2}+x^{2} y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.262 |
|
| \(903\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} y y^{\prime \prime }+4 y^{2}&=x^{2} {y^{\prime }}^{2}+2 y y^{\prime } x \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.484 |
|
| \(904\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sqrt {x}\, y^{\prime \prime }+2 y^{\prime } x +3 y&=x \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.339 |
|
| \(905\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} \cos \left (y\right ) y^{\prime \prime }-2 x^{2} \sin \left (y\right ) {y^{\prime }}^{2}+x \cos \left (y\right ) y^{\prime }-\sin \left (y\right )&=\ln \left (x \right ) \end {array} \]
|
[[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
12.747 |
|
| \(906\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y+3 y^{\prime } x +2 y {y^{\prime }}^{2}+\left (x^{2}+2 y^{2} y^{\prime }\right ) y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
3.319 |
|
| \(907\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (y^{2}+2 x^{2} y^{\prime }\right ) y^{\prime \prime }+2 \left (x +y\right ) {y^{\prime }}^{2}+y^{\prime } x +y&=0 \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
✗ |
✗ |
151.458 |
|
| \(908\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime }-y y^{\prime \prime }&=n \sqrt {{y^{\prime }}^{2}+a^{2} y^{\prime \prime }} \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
1167.633 |
|
| \(909\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }&=\left (y-y^{\prime } x \right )^{3} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.201 |
|
| \(910\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +2 y^{\prime }&=x^{2} y^{\prime }-y^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
2.792 |
|
| \(911\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\cot \left (x \right ) y^{\prime }-\left (1-\cot \left (x \right )\right ) y&={\mathrm e}^{x} \sin \left (x \right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
7.947 |
|
| \(912\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (1+\frac {2 \cot \left (x \right )}{x}-\frac {2}{x^{2}}\right ) y&=\cos \left (x \right ) x \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
✗ |
8.217 |
|
| \(913\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (1-\cot \left (x \right )\right ) y^{\prime }-\cot \left (x \right ) y&=\sin \left (x \right )^{2} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
8.252 |
|
| \(914\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+p \left (t \right ) x^{\prime }+q \left (t \right ) x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
35.382 |
|
| \(915\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+\frac {x^{\prime }}{t}+q \left (t \right ) x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
34.805 |
|
| \(916\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+\frac {\left (t^{5}+1\right ) x}{t^{4}+5}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
11.612 |
|
| \(917\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+\sqrt {t^{6}+3 t^{5}+1}\, x&=0 \end {array} \]
|
[[_Emden, _Fowler]] |
✗ |
✗ |
✗ |
7.489 |
|
| \(918\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }-p \left (t \right ) x&=q \left (t \right )\\ x \left (a \right )&=0\\ x \left (b \right )&=0\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
6.075 |
|
| \(919\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+p \left (t \right ) x^{\prime }+q \left (t \right ) x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
33.270 |
|
| \(920\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x x^{\prime \prime }-{x^{\prime }}^{2}+{\mathrm e}^{t} x^{2}&=0\\ x \left (0\right )&=1\\ x^{\prime }\left (0\right )&=-1\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.143 |
|
| \(921\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x x^{\prime \prime }-{x^{\prime }}^{2}+{\mathrm e}^{t} x^{2}&=0\\ x \left (0\right )&=-1\\ x^{\prime }\left (0\right )&=-1\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.818 |
|
| \(922\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} t^{2} x^{\prime \prime }+t x^{\prime }+t^{2} x&=0\\ x^{\prime }\left (0\right )&=a\\ \end {array} \]
|
[_Lienard] |
✗ |
✗ |
✓ |
39.970 |
|
| \(923\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} t^{2} x^{\prime \prime }+t x^{\prime }+\left (t^{2}-1\right ) x&=0\\ x^{\prime }\left (0\right )&=a\\ \end {array} \]
|
[_Bessel] |
✓ |
✓ |
✗ |
39.519 |
|
| \(924\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} t^{2} x^{\prime \prime }+t x^{\prime }+\left (-m^{2}+t^{2}\right ) x&=0\\ x \left (0\right )&=0\\ \end {array} \]
|
[_Bessel] |
✗ |
✓ |
✓ |
62.687 |
|
| \(925\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} t^{2} x^{\prime \prime }+t x^{\prime }+t^{2} x&=\lambda x\\ x \left (0\right )&=0\\ x^{\prime }\left (0\right )&=1\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
40.754 |
|
| \(926\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -x^{\prime \prime }&=1-x-x^{2}\\ x \left (a \right )&=0\\ x \left (b \right )&=0\\ \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✗ |
✗ |
✗ |
15.263 |
|
| \(927\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} -x^{\prime \prime }&=\arctan \left (x\right )\\ x \left (0\right )&=0\\ x \left (b \right )&=0\\ \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✗ |
✗ |
176.110 |
|
| \(928\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} a_{0} \left (x \right ) y^{\prime \prime }+a_{1} \left (x \right ) y^{\prime }+a_{2} \left (x \right ) y&=f \left (x \right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
49.966 |
|
| \(929\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} n \left (n +1\right ) y-2 y^{\prime } x +\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end {array} \]
|
[_Gegenbauer] |
✓ |
✓ |
✗ |
105.357 |
|
| \(930\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } \cos \left (y\right )+\left (\cos \left (y\right )-y^{\prime } \sin \left (y\right )\right ) y^{\prime }-2 y x&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
7.013 |
|
| \(931\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} s^{2} t^{\prime \prime }+s t t^{\prime }&=s \end {array} \]
|
[[_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.501 |
|
| \(932\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} {y^{\prime \prime }}^{2}-3 y y^{\prime }+y x&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
0.088 |
|
| \(933\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} {r^{\prime \prime }}^{2}+r^{\prime \prime }+y r^{\prime }&=0 \end {array} \]
|
[[_2nd_order, _missing_y]] |
✗ |
✗ |
✗ |
121.419 |
|
| \(934\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} {y^{\prime \prime }}^{{3}/{2}}+y&=x \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
0.047 |
|
| \(935\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y^{\prime \prime } x +x^{2} y^{\prime }-\sin \left (x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
44.340 |
|
| \(936\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+{\mathrm e}^{x} y^{\prime }+\left (x +1\right ) y&=0\\ y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=0\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
13.080 |
|
| \(937\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 y^{\prime } x +y&=4 x y^{2} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
1.659 |
|
| \(938\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime }+y^{\prime \prime }&=x^{2} \end {array} \]
|
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
✗ |
122.146 |
|
| \(939\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+y x&=\sin \left (y^{\prime \prime }\right ) \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
2.463 |
|
| \(940\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (r^{2}+r \right ) R^{\prime \prime }+r R^{\prime }-n \left (n +1\right ) R&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
74.461 |
|
| \(941\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\tan \left (x \right ) y^{\prime }-\frac {\tan \left (x \right ) y}{x}&=\frac {y^{3}}{x^{3}} \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
2.498 |
|
| \(942\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (1+a \cos \left (2 x \right )\right ) y^{\prime \prime }+\lambda y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
11.000 |
|
| \(943\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\cos \left (x \right ) y^{\prime }+{\mathrm e}^{x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
13.723 |
|
| \(944\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x +y&=0\\ y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0\\ \end {array} \]
|
[[_Emden, _Fowler]] |
✗ |
✗ |
✗ |
97.598 |
|
| \(945\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (1-x \right ) y^{\prime \prime }-y^{\prime } x +{\mathrm e}^{x} y&=0\\ y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=4\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
43.037 |
|
| \(946\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sin \left (x \right ) y^{\prime \prime }+y^{\prime } x +y&=2\\ y \left (\frac {3 \pi }{4}\right )&=1\\ y^{\prime }\left (\frac {3 \pi }{4}\right )&=1\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
38.040 |
|
| \(947\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \cos \left (x \right ) y^{\prime \prime }+3 y&=1\\ y \left (1\right )&=0\\ y^{\prime }\left (0\right )&=0\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
9.391 |
|
| \(948\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+{\mathrm e}^{x} y&=0\\ y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
9.936 |
|
| \(949\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y^{\prime \prime } x -7 \cos \left (x \right ) y^{\prime }+y&={\mathrm e}^{-x} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
19.010 |
|
| \(950\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+4 \tan \left (x \right ) y^{\prime }-y x&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
11.156 |
|
| \(951\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \cos \left (x \right ) y^{\prime \prime }+y&=\sin \left (x \right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
✗ |
7.271 |
|
| \(952\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}-4\right ) y^{\prime \prime }+3 x^{3} y^{\prime }+\frac {4 y}{-1+x}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
203.574 |
|
| \(953\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
34.186 |
|
| \(954\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-y y^{\prime }&=6 \end {array} \]
|
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
221.332 |
|
| \(955\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+y^{\prime } t +\left (t^{2}+1\right )^{2} y^{2}&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
2.136 |
|
| \(956\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\sqrt {y^{\prime }}+y&=t \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
1.980 |
|
| \(957\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\sqrt {t}\, y^{\prime }+y&=\sqrt {t} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
14.329 |
|
| \(958\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 y+t \sin \left (y\right )&=0 \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
2.265 |
|
| \(959\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 y^{\prime }+\sin \left (t \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
7.554 |
|
| \(960\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sin \left (t \right ) y^{\prime \prime }+y&=\cos \left (t \right )\\ y \left (\frac {\pi }{2}\right )&=y_{1}\\ y^{\prime }\left (\frac {\pi }{2}\right )&=y_{1}\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
10.338 |
|
| \(961\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (t^{2}+1\right ) y^{\prime \prime }-y^{\prime } t +t^{2} y&=\cos \left (t \right )\\ y \left (0\right )&=y_{1}\\ y^{\prime }\left (0\right )&=y_{1}\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
✗ |
46.897 |
|
| \(962\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\sqrt {t}\, y^{\prime }-\sqrt {t -3}\, y&=0\\ y \left (10\right )&=y_{1}\\ y^{\prime }\left (10\right )&=y_{1}\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
16.163 |
|
| \(963\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} t \left (t^{2}-4\right ) y^{\prime \prime }+y&={\mathrm e}^{t}\\ y \left (1\right )&=y_{1}\\ y^{\prime }\left (1\right )&=y_{1}\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
34.582 |
|
| \(964\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a_{1} \left (t \right ) y^{\prime }+a_{0} \left (t \right ) y&=f \left (t \right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
✗ |
35.482 |
|
| \(965\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (1-x \right ) y^{\prime \prime }-4 y^{\prime } x +5 y&=\cos \left (x \right ) \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
29.178 |
|
| \(966\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} u^{\prime \prime }+u^{\prime }+u&=\cos \left (r +u\right ) \end {array} \]
|
[NONE] |
✗ |
✗ |
✗ |
5.072 |
|
| \(967\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }-\left (1-\frac {{x^{\prime }}^{2}}{3}\right ) x^{\prime }+x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
181.805 |
|
| \(968\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+y \sec \left (x \right )&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
6.258 |
|
| \(969\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (1+\frac {1}{x^{2}+1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
8.465 |
|
| \(970\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y \left (1+\ln \left (y\right )\right ) y^{\prime \prime }+{y^{\prime }}^{2}&=2 x y \,{\mathrm e}^{x^{2}} \end {array} \]
|
[[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.913 |
|
| \(971\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y^{\prime } \left (y y^{\prime \prime }-{y^{\prime }}^{2}\right )-y {y^{\prime }}^{2}&=x^{4} y^{3} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
✗ |
2.516 |
|
| \(972\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y^{\prime \prime }&=\left (y-y^{\prime } x \right )^{3}\\ y \left (1\right )&=1\\ y^{\prime }\left (1\right )&=1\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
3.210 |
|
| \(973\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (1-x \ln \left (x \right )\right ) y^{\prime \prime }+\left (1+x^{2} \ln \left (x \right )\right ) y^{\prime }-\left (x +1\right ) y&=\left (1-x \ln \left (x \right )\right )^{2} {\mathrm e}^{x} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✗ |
166.049 |
|
| \(974\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} \left (2-\ln \left (x \right )\right ) y^{\prime \prime }+x \left (4-\ln \left (x \right )\right ) y^{\prime }-y&=\frac {\left (2-\ln \left (x \right )\right )^{2}}{\sqrt {x}}\\ y \left (\infty \right )&=0\\ \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
✗ |
43.326 |
|
| \(975\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} \left (\ln \left (x \right )-1\right ) y^{\prime \prime }-x^{2} y^{\prime }+y x&=2 \ln \left (x \right )\\ y \left (\infty \right )&=0\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
108.385 |
|
| \(976\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }-2 {x^{\prime }}^{2}+x^{\prime }-2 x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
33.557 |
|
| \(977\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+{\mathrm e}^{-x^{\prime }}-x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
39.995 |
|
| \(978\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }-x^{\prime }+x-x^{2}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
56.534 |
|
| \(979\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime } x -y^{\prime }&=x^{2} y y^{\prime } \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.625 |
|
| \(980\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&={y^{\prime }}^{2}+15 y^{2} \sqrt {x} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.990 |
|
| \(981\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y y^{\prime \prime }&=\left (y-y^{\prime } x \right )^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.633 |
|
| \(982\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\frac {y^{\prime }}{x}+\frac {y}{x^{2}}&=\frac {{y^{\prime }}^{2}}{y} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.374 |
|
| \(983\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y \left (y^{\prime \prime } x +y^{\prime }\right )&=x {y^{\prime }}^{2} \left (1-x \right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.527 |
|
| \(984\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y y^{\prime \prime }+{y^{\prime }}^{2}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
0.960 |
|
| \(985\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left ({y^{\prime }}^{2}-2 y y^{\prime \prime }\right )&=y^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.151 |
|
| \(986\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x y y^{\prime \prime }&=y^{\prime } \left (y^{\prime }+y\right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.395 |
|
| \(987\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 x^{2} y^{3} y^{\prime \prime }&=x^{2}-y^{4} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.203 |
|
| \(988\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{3} y^{\prime \prime }&=\left (y-y^{\prime } x \right ) \left (y-y^{\prime } x -x \right ) \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
3.256 |
|
| \(989\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \frac {y^{2}}{x^{2}}+{y^{\prime }}^{2}&=3 y^{\prime \prime } x +\frac {2 y y^{\prime }}{x} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.706 |
|
| \(990\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\left (2 y x -\frac {5}{x}\right ) y^{\prime }+4 y^{2}-\frac {4 y}{x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.645 |
|
| \(991\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (2 y y^{\prime \prime }-{y^{\prime }}^{2}\right )&=1-2 y y^{\prime } x \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✗ |
1.513 |
|
| \(992\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (y y^{\prime \prime }-{y^{\prime }}^{2}\right )+y y^{\prime } x&=\left (2 y^{\prime } x -3 y\right ) \sqrt {x^{3}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
2.159 |
|
| \(993\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} \left ({y^{\prime }}^{2}-2 y y^{\prime \prime }\right )&=4 y y^{\prime } x^{3}+1 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✗ |
2.411 |
|
| \(994\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime }+x y y^{\prime \prime }-x {y^{\prime }}^{2}&=x^{3} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
1.317 |
|
| \(995\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime }+2 x^{2} y^{\prime \prime }&=x {y^{\prime }}^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.874 |
|
| \(996\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} {y^{\prime }}^{2}+2 x y y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✗ |
0.778 |
|
| \(997\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x y^{2} \left (y^{\prime \prime } x +y^{\prime }\right )+1&=0 \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✗ |
1.255 |
|
| \(998\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y \left (2 y^{\prime \prime } x +y^{\prime }\right )&=x {y^{\prime }}^{2}+1 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✗ |
1.352 |
|
| \(999\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 y {y^{\prime }}^{2}&=\left (2 x +\frac {1}{x}\right ) y^{\prime } \end {array} \]
|
[_Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
1.615 |
|
| \(1000\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&={y^{\prime }}^{2}+2 x y^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✗ |
0.860 |
|
| \(1001\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y y^{\prime \prime }+1&=x \left (1-y\right ) y^{\prime } \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✗ |
1.630 |
|
| \(1002\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime \prime }&=2 x {y^{\prime }}^{2}\\ y \left (2\right )&=2\\ y^{\prime }\left (2\right )&={\frac {1}{2}}\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
0.906 |
|
| \(1003\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }-3 y^{\prime } x&=\frac {6 y^{2}}{x^{2}}-4 y\\ y \left (1\right )&=1\\ y^{\prime }\left (1\right )&=4\\ \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
1.622 |
|
| \(1004\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-2 y^{\prime } x +\left (x +1\right )^{2} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
12.393 |
|
| \(1005\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-2 \,{\mathrm e}^{x} y^{\prime }+{\mathrm e}^{2 x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.785 |
|
| \(1006\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-2 \left (-1+x \right ) y^{\prime }+x^{2} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
11.452 |
|
| \(1007\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (x^{4}+1\right ) y&=0 \end {array} \]
|
[_Titchmarsh] |
✓ |
✓ |
✗ |
9.308 |
|
| \(1008\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+1\right ) y^{\prime \prime }-y&=0 \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
9.990 |
|
| \(1009\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime \prime }+\ln \left (x \right )^{2} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
5.625 |
|
| \(1010\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+2 x^{\prime }+{x^{\prime }}^{2}+x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
45.705 |
|
| \(1011\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }-5 x^{\prime }-4 x+x^{2}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
93.691 |
|
| \(1012\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+x^{\prime }+2 x-x^{2}&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
70.230 |
|
| \(1013\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{\prime \prime }+{x^{\prime }}^{3}-x^{\prime }+x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
45.462 |
|
| \(1014\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+\left (x^{2}-1\right ) x^{\prime }+x&=0 \end {array} \]
|
[[_2nd_order, _missing_x], _Van_der_Pol] |
✗ |
✗ |
✗ |
17.171 |
|
| \(1015\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+x^{\prime }-2 \arctan \left (x^{\prime }\right )+x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
210.870 |
|
| \(1016\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{\prime \prime }+2^{x^{\prime }}-x^{\prime }+x&=0 \end {array} \]
|
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
71.738 |
|
| \(1017\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\tan \left (x \right ) y^{\prime }-\frac {2 y}{1+\sin \left (x \right )}&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
9.820 |
|
| \(1018\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (x +1\right ) y^{\prime \prime }+y^{\prime } x -\left (x +1\right )^{2} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
✗ |
100.163 |
|
| \(1019\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} w^{\prime \prime }-2 w^{\prime } z +2 k w&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
14.067 |
|
| \(1020\) |
\[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-2 y^{\prime } x +2 \lambda y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
13.430 |
|