| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (6 x -1\right ) y^{\prime }}{3 x \left (x -2\right )}+\frac {y}{3 x^{2} \left (x -2\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.441 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a \left (b +2\right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (a x +1\right ) x^{2}}-\frac {\left (a b x -c d \right ) y}{\left (a x +1\right ) x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
166.060 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {2 \left (a x +2 b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (2 a x +6 b \right ) y}{\left (a x +b \right ) x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.451 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}+A x \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
✓ |
✗ |
139.455 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {a y}{x^{4}} \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
0.701 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
5.415 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
3.741 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {y^{\prime }}{x^{3}}+\frac {2 y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.761 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (a +b \right ) y^{\prime }}{x^{2}}-\frac {\left (\left (a +b \right ) x +a b \right ) y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.293 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {y^{\prime }}{x}-\frac {y}{x^{4}} \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
0.661 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {y^{\prime }}{x}-\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
10.398 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
23.482 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 y^{\prime }}{x}-\frac {a^{2} y}{x^{4}} \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
8.889 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.654 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 \left (a +x \right ) y^{\prime }}{x^{2}}-\frac {b y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.648 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.730 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {2 y}{x^{4}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.210 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (x^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}+\frac {x y}{x^{3}+1} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.503 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
95.514 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
128.359 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (x^{2}-2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (x^{2}-2\right ) y}{x^{2} \left (x^{2}-1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.377 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2} \left (x^{2}-1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
35.931 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}+\frac {v \left (v +1\right ) y}{x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
45.496 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (a \left (1+a \right )-a \,x^{2} \left (a +3\right )\right ) y}{x^{2} \left (x^{2}-1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.813 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 a \,x^{2}+n \left (n +1\right ) \left (x^{2}-1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
143.737 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {b y}{x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
132.911 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (-1+a \right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (-1+a \right )-v \left (v +1\right )\right )-a \left (1+a \right )\right ) y}{x^{2} \left (x^{2}-1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
368.672 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {a y}{\left (x^{2}+1\right )^{2}} \end {array} \]
|
[_Halm] |
✓ |
✓ |
✓ |
✓ |
0.733 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}+1}-\frac {y}{\left (x^{2}+1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
1.460 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}+1}-\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (n +1\right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
110.524 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {a x y^{\prime }}{x^{2}+1}-\frac {b y}{\left (x^{2}+1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
90.465 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {a y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
0.997 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}+\frac {a^{2} y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
7.120 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
88.512 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (\left (x^{2}-1\right ) \left (a \,x^{2}+b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
106.790 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
103.963 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}-\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
151.182 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
188.963 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (2 x^{2}+a \right ) y^{\prime }}{x \left (x^{2}+a \right )}-\frac {b y}{x^{2} \left (x^{2}+a \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
0.947 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {b^{2} y}{\left (a^{2}+x^{2}\right )^{2}} \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.356 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 \left (x^{2}-1\right ) y^{\prime }}{x \left (-1+x \right )^{2}}-\frac {\left (-2 x^{2}+2 x +2\right ) y}{x^{2} \left (-1+x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.343 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {12 y}{\left (x +1\right )^{2} \left (x^{2}+2 x +3\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.947 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {b y}{x^{2} \left (x -a \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.207 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {b y}{x^{2} \left (x -a \right )^{2}}+c \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
1.632 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {c y}{\left (x -a \right )^{2} \left (x -b \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.804 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (\left (\alpha +\beta +1\right ) \left (x -a \right )^{2} \left (x -b \right )+\left (1-\alpha -\beta \right ) \left (x -b \right )^{2} \left (x -a \right )\right ) y^{\prime }}{\left (x -a \right )^{2} \left (x -b \right )^{2}}-\frac {\alpha \beta \left (a -b \right )^{2} y}{\left (x -a \right )^{2} \left (x -b \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.800 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
5.550 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a \,x^{2}+a -3\right ) y}{4 \left (x^{2}+1\right )^{2}} \end {array} \]
|
[_Halm] |
✓ |
✓ |
✓ |
✓ |
1.039 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
9.660 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {3 y}{4 \left (x^{2}+x +1\right )^{2}} \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✗ |
1.177 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (v \left (v +1\right ) \left (-1+x \right )-a^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
51.638 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (-v \left (v +1\right ) \left (-1+x \right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
50.933 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {3 y}{16 x^{2} \left (-1+x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
0.575 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (7 a \,x^{2}+5\right ) y^{\prime }}{x \left (a \,x^{2}+1\right )}-\frac {\left (15 a \,x^{2}+5\right ) y}{x^{2} \left (a \,x^{2}+1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.454 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
122.282 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (b \,x^{2}+c x +d \right ) y}{a \,x^{2} \left (-1+x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
9.165 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 y^{\prime }}{x}-\frac {c y}{x^{2} \left (a x +b \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.768 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {y}{\left (a x +b \right )^{4}} \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✗ |
1.036 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {A y}{\left (a \,x^{2}+b x +c \right )^{2}} \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✗ |
2.580 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {y^{\prime }}{x^{4}}+\frac {y}{x^{5}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.647 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
136.334 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (1+3 x \right ) y^{\prime }}{\left (-1+x \right ) \left (x +1\right )}-\frac {36 \left (x +1\right )^{2} y}{\left (-1+x \right )^{2} \left (3 x +5\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
7.463 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {y^{\prime }}{x}-\frac {a y}{x^{6}} \end {array} \]
|
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
0.753 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (3 x^{2}+a \right ) y^{\prime }}{x^{3}}-\frac {b y}{x^{6}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.557 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (1+a \right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
275.737 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\left (\frac {1-\operatorname {a1} -\operatorname {b1}}{x -\operatorname {c1}}+\frac {1-\operatorname {a2} -\operatorname {b2}}{x -\operatorname {c2}}+\frac {1-\operatorname {a3} -\operatorname {b3}}{x -\operatorname {c3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {a1} \operatorname {b1} \left (\operatorname {c1} -\operatorname {c3} \right ) \left (\operatorname {c1} -\operatorname {c2} \right )}{x -\operatorname {c1}}+\frac {\operatorname {a2} \operatorname {b2} \left (\operatorname {c2} -\operatorname {c1} \right ) \left (\operatorname {c2} -\operatorname {c3} \right )}{x -\operatorname {c2}}+\frac {\operatorname {a3} \operatorname {b3} \left (\operatorname {c3} -\operatorname {c2} \right ) \left (\operatorname {c3} -\operatorname {c1} \right )}{x -\operatorname {c3}}\right ) y}{\left (x -\operatorname {c1} \right ) \left (x -\operatorname {c2} \right ) \left (x -\operatorname {c3} \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1064.491 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (-2 x^{2}+1\right ) y}{4 x^{6}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.535 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.636 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {27 x y}{16 \left (x^{3}-1\right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.745 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1311.950 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (x^{2} \left (\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right )+\left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )+\left (x^{2}-\operatorname {a3} \right ) \left (x^{2}-\operatorname {a1} \right )\right )-\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}-\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1913.327 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-a \,x^{2 a -1} x^{-2 a} y^{\prime }-b^{2} x^{-2 a} y \end {array} \]
|
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
1.800 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}-\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
38.004 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {y}{{\mathrm e}^{x}+1} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
✓ |
✓ |
✗ |
2.855 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {y^{\prime }}{x \ln \left (x \right )}+\ln \left (x \right )^{2} y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
1.106 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {y^{\prime }}{x \left (\ln \left (x \right )-1\right )}-\frac {y}{x^{2} \left (\ln \left (x \right )-1\right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.267 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
11.198 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}-\left (-a^{2}+n^{2}\right ) y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
25.432 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (2 n +1\right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\left (v +n +1\right ) \left (v -n \right ) y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
10.555 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (\sin \left (x \right )^{2}-\cos \left (x \right )\right ) y^{\prime }}{\sin \left (x \right )}-y \sin \left (x \right )^{2} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.579 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {x \sin \left (x \right ) y^{\prime }}{\cos \left (x \right ) x -\sin \left (x \right )}+\frac {\sin \left (x \right ) y}{\cos \left (x \right ) x -\sin \left (x \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
10.622 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (x^{2} \sin \left (x \right )-2 \cos \left (x \right ) x \right ) y^{\prime }}{x^{2} \cos \left (x \right )}-\frac {\left (2 \cos \left (x \right )-x \sin \left (x \right )\right ) y}{x^{2} \cos \left (x \right )} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
✗ |
1.464 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
6.947 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {a \left (n -1\right ) \sin \left (2 a x \right ) y^{\prime }}{\cos \left (a x \right )^{2}}-\frac {n \,a^{2} \left (\left (n -1\right ) \sin \left (a x \right )^{2}+\cos \left (a x \right )^{2}\right ) y}{\cos \left (a x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.441 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {2 y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.676 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {a y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
6.024 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sin \left (x \right )^{2} y^{\prime \prime }-\left (a \sin \left (x \right )^{2}+n \left (n -1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
6.745 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
8.834 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sin \left (x \right )^{2} y^{\prime \prime }-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
16.651 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (-\left (a^{2} b^{2}-\left (1+a \right )^{2}\right ) \sin \left (x \right )^{2}-a \left (1+a \right ) b \sin \left (2 x \right )-a \left (-1+a \right )\right ) y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
13.151 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
8.620 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}+\frac {y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
58.350 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (v \left (v +1\right ) \sin \left (x \right )^{2}-n^{2}\right ) y}{\sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
10.868 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\cos \left (2 x \right ) y^{\prime }}{\sin \left (2 x \right )}-2 y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
5.214 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (-17 \sin \left (x \right )^{2}-1\right ) y}{4 \sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.829 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\sin \left (x \right ) y^{\prime }}{\cos \left (x \right )}-\frac {\left (2 x^{2}+x^{2} \sin \left (x \right )^{2}-24 \cos \left (x \right )^{2}\right ) y}{4 x^{2} \cos \left (x \right )^{2}}+\sqrt {\cos \left (x \right )} \end {array} \]
|
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
✓ |
✗ |
20.151 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}-\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
13.549 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
6.991 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=-\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
9.465 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }&=\frac {\left (3 \sin \left (x \right )^{2}+1\right ) y^{\prime }}{\cos \left (x \right ) \sin \left (x \right )}+\frac {\sin \left (x \right )^{2} y}{\cos \left (x \right )^{2}} \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.227 |
|