| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
y^{\prime }&=y^{2}+a x \,{\mathrm e}^{\lambda x} y+a \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.595 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b \,{\mathrm e}^{-\lambda x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
9.050 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b n \,x^{n -1}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
11.092 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
61.310 |
|
| \begin{align*}
y^{\prime }&=-\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,x^{n} {\mathrm e}^{\lambda x} y-a \,x^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
12.523 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b n \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
15.892 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
67.905 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.496 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b \lambda \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✓ |
✗ |
16.106 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} {\mathrm e}^{\lambda x} y-a \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
12.038 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}-a \,x^{n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) y+c \,x^{n} \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
45.203 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.868 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
12.785 |
|
| \begin{align*}
y^{\prime } x&=a \,{\mathrm e}^{\lambda x} y^{2}+k y+a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
14.562 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
19.964 |
|
| \begin{align*}
y^{\prime }&=y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
61.855 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}+\lambda y x +b^{2} a \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.744 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+\lambda y x +a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.433 |
|
| \begin{align*}
x^{4} \left (y^{\prime }-y^{2}\right )&=a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.272 |
|
| \begin{align*}
y^{\prime }&=y^{2}-a^{2}+a \lambda \sinh \left (\lambda x \right )-a^{2} \sinh \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
12.373 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \sinh \left (\beta x \right ) y+a b \sinh \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
12.316 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \sinh \left (b x \right )^{m} y+a \sinh \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
18.145 |
|
| \begin{align*}
y^{\prime }&=\lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
9.621 |
|
| \begin{align*}
y^{\prime }&=\left (a \sinh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}-a \sinh \left (\lambda x \right )^{2}+\lambda -a \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
59.243 |
|
| \begin{align*}
\left (\sinh \left (\lambda x \right ) a +b \right ) y^{\prime }&=y^{2}+c \sinh \left (\mu x \right ) y-d^{2}+c d \sinh \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
106.419 |
|
| \begin{align*}
\left (\sinh \left (\lambda x \right ) a +b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \sinh \left (\lambda x \right )&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
14.450 |
|
| \begin{align*}
y^{\prime }&=\alpha y^{2}+\beta +\gamma \cosh \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
18.865 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
14.526 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \cosh \left (b x \right )^{m} y+a \cosh \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
16.165 |
|
| \begin{align*}
y^{\prime }&=\left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
55.012 |
|
| \begin{align*}
2 y^{\prime }&=\left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
50.584 |
|
| \begin{align*}
y^{\prime }&=y^{2} \sinh \left (\lambda x \right ) a +b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
15.984 |
|
| \begin{align*}
y^{\prime }&=a \cosh \left (\lambda x \right ) y^{2}+b \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )^{n} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
147.332 |
|
| \begin{align*}
\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 ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
110.761 |
|
| \begin{align*}
\left (a \cosh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \cosh \left (\lambda x \right )&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
26.610 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
36.894 |
|
| \begin{align*}
y^{\prime }&=y^{2}+3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
131.520 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \tanh \left (b x \right )^{m} y+a \tanh \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
19.313 |
|
| \begin{align*}
\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 ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
106.724 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
37.337 |
|
| \begin{align*}
y^{\prime }&=y^{2}-\lambda ^{2}+3 a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
68.757 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \coth \left (b x \right )^{m} y+a \coth \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
18.973 |
|
| \begin{align*}
\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 ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
129.829 |
|
| \begin{align*}
y^{\prime }&=y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.092 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
30.946 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
16.514 |
|
| \begin{align*}
y^{\prime } x&=a y^{2}+b \ln \left (x \right )+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
48.716 |
|
| \begin{align*}
y^{\prime } x&=a y^{2}+b \ln \left (x \right )^{k}+c \ln \left (x \right )^{2 k +2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
13.397 |
|
| \begin{align*}
y^{\prime } x&=x y^{2}-a^{2} x \ln \left (\beta x \right )^{2}+a \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
119.389 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n} y^{2}+b -a \,b^{2} x^{n} \ln \left (x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
41.562 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2} x^{2}+a \ln \left (x \right )^{2}+b \ln \left (x \right )+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.339 |
|
| \begin{align*}
x^{2} \ln \left (a x \right ) \left (y^{\prime }-y^{2}\right )&=1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
34.829 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \ln \left (\beta x \right ) y-a b \ln \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
9.349 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \ln \left (b x \right )^{m} y+a \ln \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.416 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}-a b \,x^{n +1} \ln \left (x \right ) y+b \ln \left (x \right )+b \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
5.131 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.987 |
|
| \begin{align*}
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 \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✗ |
5.051 |
|
| \begin{align*}
y^{\prime }&=a \ln \left (x \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.694 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.950 |
|
| \begin{align*}
y^{\prime } x&=\left (a y+b \ln \left (x \right )\right )^{2} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
33.661 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
44.456 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n} \left (y+b \ln \left (x \right )\right )^{2}-b \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
35.047 |
|
| \begin{align*}
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 ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
14.279 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2} a^{2} x^{2}-y x +b^{2} \ln \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
95.326 |
|
| \begin{align*}
\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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
16.489 |
|
| \begin{align*}
\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 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
21.074 |
|
| \begin{align*}
y^{\prime }&=\alpha y^{2}+\beta +\gamma \sin \left (\lambda x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
31.510 |
|
| \begin{align*}
y^{\prime }&=y^{2}-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
10.163 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \sin \left (\beta x \right ) y+a b \sin \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
13.054 |
|
| \begin{align*}
y^{\prime }&=\lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
23.598 |
|
| \begin{align*}
2 y^{\prime }&=\left (\lambda +a -a \sin \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \sin \left (\lambda x \right ) \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✗ |
✗ |
211.539 |
|
| \begin{align*}
y^{\prime }&=\left (\lambda +a \sin \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \sin \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
30.825 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} \sin \left (x \right )^{m} y-a \sin \left (x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
52.288 |
|
| \begin{align*}
y^{\prime }&=a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
133.521 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
88.646 |
|
| \begin{align*}
\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 ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.408 |
|
| \begin{align*}
\left (a \sin \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \sin \left (\lambda x \right )&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
43.115 |
|
| \begin{align*}
y^{\prime }&=\alpha y^{2}+\beta +\gamma \cos \left (\lambda x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
20.250 |
|
| \begin{align*}
y^{\prime }&=y^{2}-a^{2}+a \lambda \cos \left (\lambda x \right )+a^{2} \cos \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.449 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \cos \left (\beta x \right ) y+a b \cos \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
12.540 |
|
| \begin{align*}
y^{\prime }&=\lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
57.205 |
|
| \begin{align*}
2 y^{\prime }&=\left (\lambda +a -\cos \left (\lambda x \right ) a \right ) y^{2}+\lambda -a -\cos \left (\lambda x \right ) a \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
57.625 |
|
| \begin{align*}
y^{\prime }&=\left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \cos \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
27.348 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} \cos \left (x \right )^{m} y-a \cos \left (x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
36.386 |
|
| \begin{align*}
y^{\prime }&=a \cos \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
127.757 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
90.497 |
|
| \begin{align*}
\left (\cos \left (\lambda x \right ) a +b \right ) y^{\prime }&=y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.371 |
|
| \begin{align*}
\left (\cos \left (\lambda x \right ) a +b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \cos \left (\lambda x \right )&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
57.007 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
19.966 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
136.383 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \tan \left (x \right ) y+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
15.805 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+2 a b \tan \left (x \right ) y+b \left (a b -1\right ) \tan \left (x \right )^{2} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.493 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \tan \left (\beta x \right ) y+a b \tan \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
47.799 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \tan \left (b x \right )^{m} y+a \tan \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
16.702 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} \tan \left (x \right )^{m} y-a \tan \left (x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
30.003 |
|
| \begin{align*}
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 \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
41.613 |
|
| \begin{align*}
y^{\prime }&=a \tan \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
415.996 |
|
| \begin{align*}
y^{\prime } x&=a \tan \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \tan \left (\lambda x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
127.730 |
|
| \begin{align*}
\left (a \tan \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+k \tan \left (\mu x \right ) y-d^{2}+k d \tan \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
85.579 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✗ |
✗ |
19.488 |
|