|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} y^{2}+\left (b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-\lambda \right ) y+c \,{\mathrm e}^{\mu x}
\] |
[_Riccati] |
✓ |
3.164 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{k n x}+d \,{\mathrm e}^{k \left (2 n +1\right ) x}
\] |
[_Riccati] |
✗ |
4.810 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\mu x} \left (y-b \,{\mathrm e}^{\lambda x}\right )^{2}+b \lambda \,{\mathrm e}^{\lambda x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.384 |
|
|
\[
{}\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime } = y^{2}+k \,{\mathrm e}^{\nu x} y-m^{2}+k m \,{\mathrm e}^{\nu x}
\] |
[_Riccati] |
✗ |
72.458 |
|
|
\[
{}\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} {\mathrm e}^{\lambda x}+b \,\mu ^{2} {\mathrm e}^{\mu x} = 0
\] |
[_Riccati] |
✓ |
4.282 |
|
|
\[
{}y^{\prime } = y^{2}+a x \,{\mathrm e}^{\lambda x} y+a \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
2.485 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b \,{\mathrm e}^{-\lambda x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
2.126 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b n \,x^{n -1}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n}
\] |
[_Riccati] |
✗ |
6.791 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x}
\] |
[_Riccati] |
✗ |
4.946 |
|
|
\[
{}y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,x^{n} {\mathrm e}^{\lambda x} y-a \,x^{n}
\] |
[_Riccati] |
✓ |
3.126 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b n \,x^{n -1}
\] |
[_Riccati] |
✓ |
4.137 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✗ |
5.550 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
2.687 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b \lambda \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✗ |
5.917 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} {\mathrm e}^{\lambda x} y-a \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
3.620 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) y+c \,x^{n}
\] |
[_Riccati] |
✗ |
6.682 |
|
|
\[
{}y^{\prime } = a \,x^{n} {\mathrm e}^{2 \lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-\lambda \right ) y+c \,x^{n}
\] |
[_Riccati] |
✓ |
5.325 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
4.144 |
|
|
\[
{}y^{\prime } x = a \,{\mathrm e}^{\lambda x} y^{2}+k y+a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
3.181 |
|
|
\[
{}y^{\prime } x = a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y+c \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✗ |
19.303 |
|
|
\[
{}y^{\prime } = y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}}
\] |
[_Riccati] |
✗ |
2.782 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}+\lambda x y+a \,b^{2}
\] |
[_Riccati] |
✓ |
2.098 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+\lambda x y+a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}}
\] |
[_Riccati] |
✓ |
3.385 |
|
|
\[
{}x^{4} \left (y^{\prime }-y^{2}\right ) = a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}}
\] |
[_Riccati] |
✓ |
3.823 |
|
|
\[
{}y^{\prime } = y^{2}-a^{2}+a \lambda \sinh \left (\lambda x \right )-a^{2} \sinh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
11.405 |
|
|
\[
{}y^{\prime } = y^{2}+a \sinh \left (\beta x \right ) y+a b \sinh \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
4.338 |
|
|
\[
{}y^{\prime } = y^{2}+a x \sinh \left (b x \right )^{m} y+a \sinh \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
8.797 |
|
|
\[
{}y^{\prime } = \lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
8.464 |
|
|
\[
{}y^{\prime } = \left (a \sinh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}-a \sinh \left (\lambda x \right )^{2}+\lambda -a
\] |
[_Riccati] |
✓ |
29.769 |
|
|
\[
{}\left (a \sinh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sinh \left (\mu x \right ) y-d^{2}+c d \sinh \left (\mu x \right )
\] |
[_Riccati] |
✗ |
117.309 |
|
|
\[
{}\left (a \sinh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \sinh \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
5.335 |
|
|
\[
{}y^{\prime } = \alpha y^{2}+\beta +\gamma \cosh \left (x \right )
\] |
[_Riccati] |
✓ |
2.560 |
|
|
\[
{}y^{\prime } = y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
3.942 |
|
|
\[
{}y^{\prime } = y^{2}+a x \cosh \left (b x \right )^{m} y+a \cosh \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
7.430 |
|
|
\[
{}y^{\prime } = \left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
25.754 |
|
|
\[
{}2 y^{\prime } = \left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
26.316 |
|
|
\[
{}y^{\prime } = y^{2}-\lambda ^{2}+a \cosh \left (\lambda x \right )^{n} \sinh \left (\lambda x \right )^{-n -4}
\] |
[_Riccati] |
✗ |
27.323 |
|
|
\[
{}y^{\prime } = a \sinh \left (\lambda x \right ) y^{2}+b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
10.928 |
|
|
\[
{}y^{\prime } = a \cosh \left (\lambda x \right ) y^{2}+b \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
10.664 |
|
|
\[
{}\left (a \cosh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cosh \left (\mu x \right ) y-d^{2}+c d \cosh \left (\mu x \right )
\] |
[_Riccati] |
✗ |
170.129 |
|
|
\[
{}\left (a \cosh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \cosh \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
4.985 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
4.948 |
|
|
\[
{}y^{\prime } = y^{2}+3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
4.821 |
|
|
\[
{}y^{\prime } = y^{2}+a x \tanh \left (b x \right )^{m} y+a \tanh \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
5.999 |
|
|
\[
{}\left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \tanh \left (\mu x \right ) y-d^{2}+c d \tanh \left (\mu x \right )
\] |
[_Riccati] |
✗ |
264.523 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
4.583 |
|
|
\[
{}y^{\prime } = y^{2}-\lambda ^{2}+3 a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
4.068 |
|
|
\[
{}y^{\prime } = y^{2}+a x \coth \left (b x \right )^{m} y+a \coth \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
6.204 |
|
|
\[
{}\left (a \coth \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right )
\] |
[_Riccati] |
✗ |
302.645 |
|
|
\[
{}y^{\prime } = y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
22.861 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +b \lambda -2 a b -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
19.787 |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
6.921 |
|
|
\[
{}y^{\prime } x = a y^{2}+b \ln \left (x \right )+c
\] |
[_Riccati] |
✓ |
2.150 |
|
|
\[
{}y^{\prime } x = a y^{2}+b \ln \left (x \right )^{k}+c \ln \left (x \right )^{2 k +2}
\] |
[_Riccati] |
✓ |
31.676 |
|
|
\[
{}y^{\prime } x = x y^{2}-a^{2} x \ln \left (\beta x \right )^{2}+a
\] |
[_Riccati] |
✗ |
2.185 |
|
|
\[
{}y^{\prime } x = x y^{2}-a^{2} x \ln \left (\beta x \right )^{2 k}+a k \ln \left (\beta x \right )^{k -1}
\] |
[_Riccati] |
✗ |
4.641 |
|
|
\[
{}y^{\prime } x = a \,x^{n} y^{2}+b -a \,b^{2} x^{n} \ln \left (x \right )^{2}
\] |
[_Riccati] |
✗ |
3.402 |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+a \ln \left (x \right )^{2}+b \ln \left (x \right )+c
\] |
[_Riccati] |
✓ |
7.438 |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+a \left (b \ln \left (x \right )+c \right )^{n}+\frac {1}{4}
\] |
[_Riccati] |
✗ |
3.323 |
|
|
\[
{}x^{2} \ln \left (a x \right ) \left (y^{\prime }-y^{2}\right ) = 1
\] |
[_Riccati] |
✓ |
1.572 |
|
|
\[
{}y^{\prime } = y^{2}+a \ln \left (\beta x \right ) y-a b \ln \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
2.278 |
|
|
\[
{}y^{\prime } = y^{2}+a x \ln \left (b x \right )^{m} y+a \ln \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
2.220 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n +1} \ln \left (x \right ) y+b \ln \left (x \right )+b
\] |
[_Riccati] |
✓ |
3.364 |
|
|
\[
{}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+a \,x^{n +1} \ln \left (x \right )^{m} y-a \ln \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
3.487 |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{n} y-a b x \ln \left (x \right )^{n +1} y+b \ln \left (x \right )+b
\] |
[_linear] |
✓ |
2.115 |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
5.408 |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y+b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
5.295 |
|
|
\[
{}y^{\prime } x = \left (a y+b \ln \left (x \right )\right )^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
1.510 |
|
|
\[
{}y^{\prime } x = a \ln \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \ln \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
39.325 |
|
|
\[
{}y^{\prime } x = a \,x^{n} \left (y+b \ln \left (x \right )\right )^{2}-b
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.691 |
|
|
\[
{}y^{\prime } x = a \,x^{2 n} \ln \left (x \right ) y^{2}+\left (b \,x^{n} \ln \left (x \right )-n \right ) y+c \ln \left (x \right )
\] |
[_Riccati] |
✓ |
4.583 |
|
|
\[
{}x^{2} y^{\prime } = a^{2} x^{2} y^{2}-x y+b^{2} \ln \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
3.199 |
|
|
\[
{}\left (a \ln \left (x \right )+b \right ) y^{\prime } = y^{2}+c \ln \left (x \right )^{n} y-\lambda ^{2}+\lambda c \ln \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
38.407 |
|
|
\[
{}\left (a \ln \left (x \right )+b \right ) y^{\prime } = \ln \left (x \right )^{n} y^{2}+c y-\lambda ^{2} \ln \left (x \right )^{n}+\lambda c
\] |
[_Riccati] |
✗ |
44.779 |
|
|
\[
{}y^{\prime } = \alpha y^{2}+\beta +\gamma \sin \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
3.242 |
|
|
\[
{}y^{\prime } = y^{2}-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
7.671 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+c \sin \left (\lambda x +a \right )^{n} \sin \left (\lambda x +b \right )^{-n -4}
\] |
[_Riccati] |
✗ |
160.381 |
|
|
\[
{}y^{\prime } = y^{2}+a \sin \left (\beta x \right ) y+a b \sin \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
4.418 |
|
|
\[
{}y^{\prime } = y^{2}+a \sin \left (b x \right )^{m} y+a \sin \left (b x \right )^{m}
\] |
[_Riccati] |
✗ |
17.338 |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
7.970 |
|
|
\[
{}2 y^{\prime } = \left (\lambda +a -a \sin \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \sin \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
79.783 |
|
|
\[
{}y^{\prime } = \left (\lambda +a \sin \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \sin \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
31.134 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \sin \left (x \right )^{m} y-a \sin \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
20.654 |
|
|
\[
{}y^{\prime } = a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
73.462 |
|
|
\[
{}y^{\prime } x = a \sin \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \sin \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
23.507 |
|
|
\[
{}\left (a \sin \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sin \left (\mu x \right ) y-d^{2}+c d \sin \left (\mu x \right )
\] |
[_Riccati] |
✗ |
124.536 |
|
|
\[
{}\left (a \sin \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \sin \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
5.554 |
|
|
\[
{}y^{\prime } = \alpha y^{2}+\beta +\gamma \cos \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
2.980 |
|
|
\[
{}y^{\prime } = y^{2}-a^{2}+a \lambda \cos \left (\lambda x \right )+a^{2} \cos \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
6.308 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+c \cos \left (\lambda x +a \right )^{n} \cos \left (\lambda x +b \right )^{-n -4}
\] |
[_Riccati] |
✗ |
166.290 |
|
|
\[
{}y^{\prime } = y^{2}+a \cos \left (\beta x \right ) y+a b \cos \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
4.454 |
|
|
\[
{}y^{\prime } = y^{2}+a \cos \left (b x \right )^{m} y+a \cos \left (b x \right )^{m}
\] |
[_Riccati] |
✗ |
14.457 |
|
|
\[
{}y^{\prime } = \lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
18.184 |
|
|
\[
{}2 y^{\prime } = \left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \cos \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
44.348 |
|
|
\[
{}y^{\prime } = \left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \cos \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
29.055 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \cos \left (x \right )^{m} y-a \cos \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
17.005 |
|
|
\[
{}y^{\prime } = a \cos \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
40.573 |
|
|
\[
{}y^{\prime } x = a \cos \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cos \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
49.212 |
|
|
\[
{}\left (a \cos \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right )
\] |
[_Riccati] |
✗ |
119.171 |
|
|
\[
{}\left (a \cos \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \cos \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
5.197 |
|