| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+c \left (a x +b \right )^{n -4} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
6.278 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+a x y^{\prime }-\left (b^{2} x^{n}+2 b \,x^{n -1}+a b x +a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
11.768 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }-a y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
✗ |
3.127 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (a \,x^{n -1}+b x \right ) y^{\prime }+\left (-1+a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
17.770 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (2 x^{n -1}+a \,x^{2}+b x \right ) y^{\prime }+b y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
63.690 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{n}+b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
11.385 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (a \,x^{n}-x^{n -1}+a b x +b \right ) y^{\prime }+a^{2} b x y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
48.273 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{n} y^{\prime \prime }+\left (a \,x^{n +m}+1\right ) y^{\prime }+a \,x^{m} \left (1+m \,x^{n -1}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
11.938 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b \right ) y^{\prime \prime }+\left (c \,x^{n}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{n}+d -b \lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
21.479 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b x +c \right ) y^{\prime \prime }&=a n \left (n -1\right ) x^{-2+n} y \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✗ |
✗ |
21.832 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (a -b \right ) x^{n}+a -n \right ) y^{\prime }+b \left (1-a \right ) x^{n -1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
54.361 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x \left (x^{2 n}+a \right ) y^{\prime \prime }+\left (x^{2 n}+a -a n \right ) y^{\prime }-b^{2} x^{2 n -1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
4.536 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a^{2} \left (n +1\right ) x^{2 n}+n -1\right ) y^{\prime }-\nu \left (\nu +1\right ) a^{2} n^{2} x^{2 n} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
140.036 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a p \,x^{n}+q \right ) y^{\prime }+\left (a r \,x^{n}+s \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
155.439 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{-2+n} \left (\left (b -1\right ) x^{n}+a \left (n -1\right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
4.943 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) \left (c \,x^{n}+d \right ) y^{\prime }+n \left (-a d +b c \right ) x^{n -1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
26.097 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{n}+a \right )^{2} y^{\prime \prime }+b \,x^{m} \left (x^{n}+a \right ) y^{\prime }-x^{-2+n} \left (b \,x^{m +1}+a n -a \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
66.439 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+c \,x^{m} \left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{m}-x^{n -1} a n -1\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
40.944 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (n +1\right ) x \left (a^{2} x^{2 n}-b^{2}\right ) y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
171.732 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n +1}+b \,x^{n}+c \right )^{2} y^{\prime \prime }+\left (\alpha \,x^{n}+\beta \,x^{n -1}+\gamma \right ) y^{\prime }+\left (n \left (-a n -a +\alpha \right ) x^{n -1}+\left (n -1\right ) \left (-n b +\beta \right ) x^{-2+n}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
227.337 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda -x \right ) y^{\prime }+y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
✗ |
2.254 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda ^{2}-x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
91.251 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+a n \,x^{n -1} b m \,x^{m -1} y^{\prime }+d y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
128.500 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,x^{n}+b \right )^{m +1} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }-a n m \,x^{n -1} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
9.470 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.241 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{x}-b \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.474 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \left (\lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{2 \lambda x}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
4.224 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
4.406 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
4.185 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+{\mathrm e}^{2 \lambda x} c -\frac {\lambda ^{2}}{4}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
4.481 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}-\frac {\lambda ^{2}}{4}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
4.621 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} b \,{\mathrm e}^{2 a x} y+a y^{\prime }+y^{\prime \prime }&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.936 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-a y^{\prime }+b \,{\mathrm e}^{2 a x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
3.628 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a y^{\prime }+\left (b \,{\mathrm e}^{\lambda x}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.987 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
9.025 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
9.091 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 a \,{\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{\lambda x}+\lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.743 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
12.431 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{\mu x} \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+\mu \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
13.560 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+2 k \,{\mathrm e}^{\mu x} y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+k^{2} {\mathrm e}^{2 \mu x}+k \mu \,{\mathrm e}^{\mu x}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✓ |
✗ |
8.375 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }-\left (a +2 b \,{\mathrm e}^{a x}\right ) y^{\prime }+b^{2} {\mathrm e}^{2 a x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.559 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
11.980 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+b \,{\mathrm e}^{2 \lambda x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.222 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+c \left (a \,{\mathrm e}^{\lambda x}+b -c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
12.954 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a +b \,{\mathrm e}^{2 \lambda x}\right ) y^{\prime }+\lambda \left (a -\lambda -b \,{\mathrm e}^{2 \lambda x}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✓ |
✗ |
7.701 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (b -\lambda \right ) {\mathrm e}^{2 \lambda x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
14.642 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+c \,{\mathrm e}^{\mu x}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✓ |
✗ |
8.132 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a \left (b +\lambda \right ) {\mathrm e}^{\lambda x}+c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
3.483 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b -\lambda \right ) y^{\prime }+\left ({\mathrm e}^{2 \lambda x} c +a b \,{\mathrm e}^{\lambda x}+b^{2}-b \lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
14.726 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +b c -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
13.173 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
13.715 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{2 \mu x}+c \,{\mathrm e}^{\mu x}+k \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✓ |
✗ |
8.751 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b -\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \mu x}+d \,{\mathrm e}^{\mu x}+k \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✓ |
✗ |
9.768 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}\right ) y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\mu x}+\lambda \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
8.340 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{2 \mu x}+b \right ) y^{\prime }+\mu \left ({\mathrm e}^{\lambda x} \left (b -a \,{\mathrm e}^{2 \mu x}\right )-\mu \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
17.019 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a \lambda \,{\mathrm e}^{\lambda x}+{\mathrm e}^{\mu x} b \mu \right ) y&=0 \end {array} \]
|
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
2.698 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+{\mathrm e}^{\lambda x} a c +{\mathrm e}^{\mu x} b \mu \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
9.418 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b \,{\mathrm e}^{\mu x}-\lambda \right ) y^{\prime }+\left (a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+{\mathrm e}^{2 \lambda x} c +{\mathrm e}^{2 \mu x} b^{2}+b \left (\mu -\lambda \right ) {\mathrm e}^{\mu x}\right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✓ |
✗ |
11.053 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+\left (a \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+a \lambda \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}-2 \lambda \right ) y^{\prime }+a^{2} b \lambda \,{\mathrm e}^{\left (2 \lambda +\mu \right ) x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
12.339 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime \prime }+a \,{\mathrm e}^{b \,x^{n}} y^{\prime }+c \left (a \,{\mathrm e}^{b \,x^{n}}-c \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
✓ |
✗ |
✗ |
12.932 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime \prime }-a \,\lambda ^{2} {\mathrm e}^{\lambda x} y&=0 \end {array} \]
|
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✗ |
✗ |
1.183 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a^{2} {\mathrm e}^{2 \lambda x}+b \right ) y^{\prime \prime }-b \lambda y^{\prime }-a^{2} \lambda ^{2} k^{2} {\mathrm e}^{2 \lambda x} y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
4.993 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 \left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime \prime }+a \lambda \,{\mathrm e}^{\lambda x} y^{\prime }+c y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
18.889 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime \prime }+\left (c \,{\mathrm e}^{\lambda x}+d \right ) y^{\prime }+k \left (\left (-a k +c \right ) {\mathrm e}^{\lambda x}+d -b k \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
16.536 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime \prime }+\left (c \,{\mathrm e}^{\lambda x}+d \right ) y^{\prime }+\left (n \,{\mathrm e}^{\lambda x}+m \right ) y&=0 \end {array} \]
|
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
14.769 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \frac {2 y x +1}{y}+\frac {\left (y-x \right ) y^{\prime }}{y^{2}}&=0 \end {array} \]
|
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✗ |
70.948 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \frac {y^{2}-2 x^{2}}{-x^{3}+x y^{2}}+\frac {\left (2 y^{2}-x^{2}\right ) y^{\prime }}{y^{3}-x^{2} y}&=0 \end {array} \]
|
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
✓ |
✓ |
196.236 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \frac {1}{\sqrt {x^{2}+y^{2}}}+\left (\frac {1}{y}-\frac {x}{y \sqrt {x^{2}+y^{2}}}\right ) y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
✓ |
✓ |
77.605 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime } x +x +y&=0 \end {array} \]
|
[_linear] |
✓ |
✓ |
✓ |
✓ |
24.410 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 6 x -2 y+1+\left (2 y-2 x -3\right ) y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
36.941 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sec \left (x \right ) \cos \left (y\right )^{2}-\cos \left (x \right ) \sin \left (y\right ) y^{\prime }&=0 \end {array} \]
|
[_separable] |
✓ |
✓ |
✓ |
✓ |
360.832 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x +1\right ) y^{2}-x^{3} y^{\prime }&=0 \end {array} \]
|
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.221 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{2}+1\right ) \left (1+y^{2}\right ) y^{\prime }+2 x y \left (1-y^{2}\right )&=0 \end {array} \]
|
[_separable] |
✓ |
✓ |
✓ |
✓ |
27.225 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \sin \left (x \right ) \cos \left (y\right )^{2}+\cos \left (x \right )^{2} y^{\prime }&=0 \end {array} \]
|
[_separable] |
✓ |
✓ |
✓ |
✗ |
7.783 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} {\mathrm e}^{\frac {y}{x}} x +y-y^{\prime } x&=0 \end {array} \]
|
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
✓ |
✗ |
1001.870 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} y+3 y^{3}-\left (x^{3}+2 x y^{2}\right ) y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
✓ |
✓ |
91.020 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime }+y^{2}-y x&=0 \end {array} \]
|
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
✓ |
✓ |
22.852 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{2} y+y^{3}-x^{3} y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
✓ |
✓ |
73.819 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{3}+x^{3} y^{\prime }&=0 \end {array} \]
|
[_separable] |
✓ |
✓ |
✓ |
✓ |
48.609 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x +y \cos \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right ) y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
✓ |
✓ |
61.589 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x +y+1\right ) y^{\prime }+1+4 x +3 y&=0 \end {array} \]
|
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
302.869 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 x -y+2+\left (x +y+3\right ) y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
695.319 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x +y-\left (4 x +2 y-1\right ) y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✓ |
41.317 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y+2 x y^{2}-x^{2} y^{3}+2 x^{2} y y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.339 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 y+3 x y^{2}+\left (x +2 x^{2} y\right ) y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
✓ |
✓ |
98.677 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y+x y^{2}+\left (x -x^{2} y\right ) y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
✓ |
✓ |
46.528 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime }+\cot \left (x \right ) y&=\sec \left (x \right ) \end {array} \]
|
[_linear] |
✓ |
✓ |
✓ |
✓ |
410.188 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime } x +\left (x +1\right ) y&={\mathrm e}^{x} \end {array} \]
|
[_linear] |
✓ |
✓ |
✓ |
✓ |
5.612 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime }-\frac {2 y}{x +1}&=\left (x +1\right )^{3} \end {array} \]
|
[_linear] |
✓ |
✓ |
✓ |
✓ |
10.139 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (x^{3}+x \right ) y^{\prime }+4 x^{2} y&=2 \end {array} \]
|
[_linear] |
✓ |
✓ |
✓ |
✓ |
17.092 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime }+\left (1-2 x \right ) y&=x^{2} \end {array} \]
|
[_linear] |
✓ |
✓ |
✓ |
✓ |
8.178 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} \left (-x^{2}+1\right ) y^{\prime }-2 \left (x +1\right ) y&=y^{{5}/{2}} \end {array} \]
|
[_rational, _Bernoulli] |
✓ |
✓ |
✓ |
✗ |
11.044 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y y^{\prime }+x y^{2}&=x \end {array} \]
|
[_separable] |
✓ |
✓ |
✓ |
✓ |
26.298 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime } \sin \left (y\right )+\cos \left (y\right ) \sin \left (x \right )&=\sin \left (x \right ) \end {array} \]
|
[_separable] |
✓ |
✓ |
✓ |
✓ |
42.181 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 4 y^{\prime } x +3 y+{\mathrm e}^{x} x^{4} y^{5}&=0 \end {array} \]
|
[_Bernoulli] |
✓ |
✓ |
✓ |
✓ |
12.974 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{\prime }-\frac {1+y}{x +1}&=\sqrt {1+y} \end {array} \]
|
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
43.099 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{4} y \left (3 y+2 y^{\prime } x \right )+x^{2} \left (4 y+3 y^{\prime } x \right )&=0 \end {array} \]
|
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
✓ |
✗ |
205.211 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} y^{2} \left (3 y-6 y^{\prime } x \right )-x \left (y-2 y^{\prime } x \right )&=0 \end {array} \]
|
[_separable] |
✓ |
✓ |
✓ |
✓ |
0.581 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} 2 x^{3} y-y^{2}-\left (2 x^{4}+y x \right ) y^{\prime }&=0 \end {array} \]
|
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
✓ |
✓ |
87.480 |
|
| \[ \begin {array}{>{\displaystyle }r @{\;} >{\displaystyle }l} x^{2} y^{\prime }+y^{2}-y x&=0 \end {array} \]
|
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
✓ |
✓ |
29.738 |
|