|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,x^{n} {\mathrm e}^{\lambda x} y-a \,x^{n}
\] |
[_Riccati] |
✓ |
3.082 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b n \,x^{n -1}
\] |
[_Riccati] |
✓ |
4.089 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✗ |
5.563 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
2.658 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b \lambda \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✗ |
5.719 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} {\mathrm e}^{\lambda x} y-a \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
3.426 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) y+c \,x^{n}
\] |
[_Riccati] |
✗ |
6.277 |
|
|
\[
{}y^{\prime } = a \,x^{n} {\mathrm e}^{2 \lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-\lambda \right ) y+c \,x^{n}
\] |
[_Riccati] |
✓ |
4.950 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
3.995 |
|
|
\[
{}y^{\prime } x = a \,{\mathrm e}^{\lambda x} y^{2}+k y+a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
3.039 |
|
|
\[
{}y^{\prime } x = a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y+c \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✗ |
19.380 |
|
|
\[
{}y^{\prime } = y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}}
\] |
[_Riccati] |
✗ |
2.799 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}+\lambda x y+a \,b^{2}
\] |
[_Riccati] |
✓ |
2.065 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+\lambda x y+a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}}
\] |
[_Riccati] |
✓ |
3.177 |
|
|
\[
{}x^{4} \left (y^{\prime }-y^{2}\right ) = a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}}
\] |
[_Riccati] |
✓ |
3.745 |
|
|
\[
{}y^{\prime } = y^{2}-a^{2}+a \lambda \sinh \left (\lambda x \right )-a^{2} \sinh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
10.855 |
|
|
\[
{}y^{\prime } = y^{2}+a \sinh \left (\beta x \right ) y+a b \sinh \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
4.224 |
|
|
\[
{}y^{\prime } = y^{2}+a x \sinh \left (b x \right )^{m} y+a \sinh \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
8.153 |
|
|
\[
{}y^{\prime } = \lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
8.057 |
|
|
\[
{}y^{\prime } = \left (a \sinh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}-a \sinh \left (\lambda x \right )^{2}+\lambda -a
\] |
[_Riccati] |
✓ |
26.957 |
|
|
\[
{}\left (a \sinh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sinh \left (\mu x \right ) y-d^{2}+c d \sinh \left (\mu x \right )
\] |
[_Riccati] |
✗ |
123.846 |
|
|
\[
{}\left (a \sinh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \sinh \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
5.190 |
|
|
\[
{}y^{\prime } = \alpha y^{2}+\beta +\gamma \cosh \left (x \right )
\] |
[_Riccati] |
✓ |
2.478 |
|
|
\[
{}y^{\prime } = y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
3.922 |
|
|
\[
{}y^{\prime } = y^{2}+a x \cosh \left (b x \right )^{m} y+a \cosh \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
7.052 |
|
|
\[
{}y^{\prime } = \left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
25.917 |
|
|
\[
{}2 y^{\prime } = \left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
26.191 |
|
|
\[
{}y^{\prime } = y^{2}-\lambda ^{2}+a \cosh \left (\lambda x \right )^{n} \sinh \left (\lambda x \right )^{-n -4}
\] |
[_Riccati] |
✗ |
27.464 |
|
|
\[
{}y^{\prime } = a \sinh \left (\lambda x \right ) y^{2}+b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
10.709 |
|
|
\[
{}y^{\prime } = a \cosh \left (\lambda x \right ) y^{2}+b \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
10.293 |
|
|
\[
{}\left (a \cosh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cosh \left (\mu x \right ) y-d^{2}+c d \cosh \left (\mu x \right )
\] |
[_Riccati] |
✗ |
174.952 |
|
|
\[
{}\left (a \cosh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \cosh \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
4.966 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
4.998 |
|
|
\[
{}y^{\prime } = y^{2}+3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
4.701 |
|
|
\[
{}y^{\prime } = y^{2}+a x \tanh \left (b x \right )^{m} y+a \tanh \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
5.938 |
|
|
\[
{}\left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \tanh \left (\mu x \right ) y-d^{2}+c d \tanh \left (\mu x \right )
\] |
[_Riccati] |
✗ |
239.618 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
4.409 |
|
|
\[
{}y^{\prime } = y^{2}-\lambda ^{2}+3 a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
4.183 |
|
|
\[
{}y^{\prime } = y^{2}+a x \coth \left (b x \right )^{m} y+a \coth \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
6.062 |
|
|
\[
{}\left (a \coth \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right )
\] |
[_Riccati] |
✗ |
339.923 |
|
|
\[
{}y^{\prime } = y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
22.943 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +b \lambda -2 a b -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
18.866 |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
6.721 |
|
|
\[
{}y^{\prime } x = a y^{2}+b \ln \left (x \right )+c
\] |
[_Riccati] |
✓ |
2.014 |
|
|
\[
{}y^{\prime } x = a y^{2}+b \ln \left (x \right )^{k}+c \ln \left (x \right )^{2 k +2}
\] |
[_Riccati] |
✓ |
31.272 |
|
|
\[
{}y^{\prime } x = x y^{2}-a^{2} x \ln \left (\beta x \right )^{2}+a
\] |
[_Riccati] |
✗ |
2.165 |
|
|
\[
{}y^{\prime } x = x y^{2}-a^{2} x \ln \left (\beta x \right )^{2 k}+a k \ln \left (\beta x \right )^{k -1}
\] |
[_Riccati] |
✗ |
4.507 |
|
|
\[
{}y^{\prime } x = a \,x^{n} y^{2}+b -a \,b^{2} x^{n} \ln \left (x \right )^{2}
\] |
[_Riccati] |
✗ |
3.295 |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+a \ln \left (x \right )^{2}+b \ln \left (x \right )+c
\] |
[_Riccati] |
✓ |
7.151 |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+a \left (b \ln \left (x \right )+c \right )^{n}+\frac {1}{4}
\] |
[_Riccati] |
✗ |
3.280 |
|
|
\[
{}x^{2} \ln \left (a x \right ) \left (y^{\prime }-y^{2}\right ) = 1
\] |
[_Riccati] |
✓ |
1.506 |
|
|
\[
{}y^{\prime } = y^{2}+a \ln \left (\beta x \right ) y-a b \ln \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
2.212 |
|
|
\[
{}y^{\prime } = y^{2}+a x \ln \left (b x \right )^{m} y+a \ln \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
2.145 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n +1} \ln \left (x \right ) y+b \ln \left (x \right )+b
\] |
[_Riccati] |
✓ |
3.308 |
|
|
\[
{}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+a \,x^{n +1} \ln \left (x \right )^{m} y-a \ln \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
3.519 |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{n} y-a b x \ln \left (x \right )^{n +1} y+b \ln \left (x \right )+b
\] |
[_linear] |
✓ |
1.825 |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
5.337 |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y+b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
5.349 |
|
|
\[
{}y^{\prime } x = \left (a y+b \ln \left (x \right )\right )^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
1.612 |
|
|
\[
{}y^{\prime } x = a \ln \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \ln \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
38.722 |
|
|
\[
{}y^{\prime } x = a \,x^{n} \left (y+b \ln \left (x \right )\right )^{2}-b
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.606 |
|
|
\[
{}y^{\prime } x = a \,x^{2 n} \ln \left (x \right ) y^{2}+\left (b \,x^{n} \ln \left (x \right )-n \right ) y+c \ln \left (x \right )
\] |
[_Riccati] |
✓ |
4.452 |
|
|
\[
{}x^{2} y^{\prime } = a^{2} x^{2} y^{2}-x y+b^{2} \ln \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
3.034 |
|
|
\[
{}\left (a \ln \left (x \right )+b \right ) y^{\prime } = y^{2}+c \ln \left (x \right )^{n} y-\lambda ^{2}+\lambda c \ln \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
38.337 |
|
|
\[
{}\left (a \ln \left (x \right )+b \right ) y^{\prime } = \ln \left (x \right )^{n} y^{2}+c y-\lambda ^{2} \ln \left (x \right )^{n}+\lambda c
\] |
[_Riccati] |
✗ |
44.360 |
|
|
\[
{}y^{\prime } = \alpha y^{2}+\beta +\gamma \sin \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
3.185 |
|
|
\[
{}y^{\prime } = y^{2}-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
7.402 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+c \sin \left (\lambda x +a \right )^{n} \sin \left (\lambda x +b \right )^{-n -4}
\] |
[_Riccati] |
✗ |
143.075 |
|
|
\[
{}y^{\prime } = y^{2}+a \sin \left (\beta x \right ) y+a b \sin \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
4.366 |
|
|
\[
{}y^{\prime } = y^{2}+a \sin \left (b x \right )^{m} y+a \sin \left (b x \right )^{m}
\] |
[_Riccati] |
✗ |
16.435 |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
7.676 |
|
|
\[
{}2 y^{\prime } = \left (\lambda +a -a \sin \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \sin \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
73.064 |
|
|
\[
{}y^{\prime } = \left (\lambda +a \sin \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \sin \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
31.304 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \sin \left (x \right )^{m} y-a \sin \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
20.398 |
|
|
\[
{}y^{\prime } = a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
63.119 |
|
|
\[
{}y^{\prime } x = a \sin \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \sin \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
21.331 |
|
|
\[
{}\left (a \sin \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sin \left (\mu x \right ) y-d^{2}+c d \sin \left (\mu x \right )
\] |
[_Riccati] |
✗ |
125.520 |
|
|
\[
{}\left (a \sin \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \sin \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
5.444 |
|
|
\[
{}y^{\prime } = \alpha y^{2}+\beta +\gamma \cos \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
2.682 |
|
|
\[
{}y^{\prime } = y^{2}-a^{2}+a \lambda \cos \left (\lambda x \right )+a^{2} \cos \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
5.963 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+c \cos \left (\lambda x +a \right )^{n} \cos \left (\lambda x +b \right )^{-n -4}
\] |
[_Riccati] |
✗ |
164.434 |
|
|
\[
{}y^{\prime } = y^{2}+a \cos \left (\beta x \right ) y+a b \cos \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
4.379 |
|
|
\[
{}y^{\prime } = y^{2}+a \cos \left (b x \right )^{m} y+a \cos \left (b x \right )^{m}
\] |
[_Riccati] |
✗ |
14.879 |
|
|
\[
{}y^{\prime } = \lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
17.296 |
|
|
\[
{}2 y^{\prime } = \left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \cos \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
41.687 |
|
|
\[
{}y^{\prime } = \left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \cos \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
27.921 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \cos \left (x \right )^{m} y-a \cos \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
16.590 |
|
|
\[
{}y^{\prime } = a \cos \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
37.133 |
|
|
\[
{}y^{\prime } x = a \cos \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cos \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
49.540 |
|
|
\[
{}\left (a \cos \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right )
\] |
[_Riccati] |
✗ |
128.445 |
|
|
\[
{}\left (a \cos \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \cos \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
5.172 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
5.616 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
5.722 |
|
|
\[
{}y^{\prime } = a y^{2}+b \tan \left (x \right ) y+c
\] |
[_Riccati] |
✓ |
5.126 |
|
|
\[
{}y^{\prime } = a y^{2}+2 a b \tan \left (x \right ) y+b \left (a b -1\right ) \tan \left (x \right )^{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
10.714 |
|
|
\[
{}y^{\prime } = y^{2}+a \tan \left (\beta x \right ) y+a b \tan \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
4.866 |
|
|
\[
{}y^{\prime } = y^{2}+a x \tan \left (b x \right )^{m} y+a \tan \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
6.153 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \tan \left (x \right )^{m} y-a \tan \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
13.329 |
|
|
\[
{}y^{\prime } = a \tan \left (\lambda x \right )^{n} y^{2}-a \,b^{2} \tan \left (\lambda x \right )^{n +2}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda
\] |
[_Riccati] |
✓ |
33.000 |
|
|
\[
{}y^{\prime } = a \tan \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
142.671 |
|