| # |
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] |
✓ |
✓ |
✓ |
✗ |
3.996 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b \,{\mathrm e}^{-\lambda x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.902 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
6.365 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
55.463 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.394 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.898 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
45.667 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.968 |
|
| \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] |
✗ |
✗ |
✓ |
✗ |
12.270 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.382 |
|
| \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] |
✗ |
✗ |
✗ |
✗ |
13.459 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
3.957 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.815 |
|
| \begin{align*}
x y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+k y+a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.922 |
|
| \begin{align*}
x y^{\prime }&=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] |
✓ |
✓ |
✓ |
✗ |
13.052 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
68.014 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}+\lambda x y+b^{2} a \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.696 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+\lambda x y+a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.736 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
2.605 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.145 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
8.877 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
13.343 |
|
| \begin{align*}
y^{\prime }&=\lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
6.254 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
39.382 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
93.860 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.774 |
|
| \begin{align*}
y^{\prime }&=\alpha y^{2}+\beta +\gamma \cosh \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
13.408 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.965 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.261 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
38.192 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
33.864 |
|
| \begin{align*}
y^{\prime }&=\sinh \left (\lambda x \right ) y^{2} a +b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
9.408 |
|
| \begin{align*}
y^{\prime }&=\cosh \left (\lambda x \right ) a y^{2}+b \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )^{n} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
36.176 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
89.907 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
18.967 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda a -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
7.733 |
|
| \begin{align*}
y^{\prime }&=y^{2}+3 \lambda a -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
29.556 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.541 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
95.595 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda a -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
7.776 |
|
| \begin{align*}
y^{\prime }&=y^{2}-\lambda ^{2}+3 \lambda a -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
35.279 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.583 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
106.514 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
3.068 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda a +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] |
✓ |
✓ |
✓ |
✗ |
18.550 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
7.734 |
|
| \begin{align*}
x y^{\prime }&=a y^{2}+b \ln \left (x \right )+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
39.824 |
|
| \begin{align*}
x y^{\prime }&=a y^{2}+b \ln \left (x \right )^{k}+c \ln \left (x \right )^{2 k +2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.061 |
|
| \begin{align*}
x y^{\prime }&=x y^{2}-a^{2} x \ln \left (\beta x \right )^{2}+a \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
96.599 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{n} y^{2}+b -a \,b^{2} x^{n} \ln \left (x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
32.735 |
|
| \begin{align*}
x^{2} y^{\prime }&=x^{2} y^{2}+a \ln \left (x \right )^{2}+b \ln \left (x \right )+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.663 |
|
| \begin{align*}
x^{2} \ln \left (a x \right ) \left (y^{\prime }-y^{2}\right )&=1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
16.290 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
5.038 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
4.550 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
3.090 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.226 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
3.191 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
4.173 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.586 |
|
| \begin{align*}
x y^{\prime }&=\left (a y+b \ln \left (x \right )\right )^{2} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
31.234 |
|
| \begin{align*}
x y^{\prime }&=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] |
✓ |
✓ |
✓ |
✗ |
35.612 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{n} \left (y+b \ln \left (x \right )\right )^{2}-b \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
38.474 |
|
| \begin{align*}
x y^{\prime }&=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] |
✓ |
✓ |
✓ |
✗ |
8.761 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2} a^{2} x^{2}-y x +b^{2} \ln \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
78.070 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.179 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
17.178 |
|
| \begin{align*}
y^{\prime }&=\alpha y^{2}+\beta +\gamma \sin \left (\lambda x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
15.743 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.800 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.940 |
|
| \begin{align*}
y^{\prime }&=\lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
18.177 |
|
| \begin{align*}
2 y^{\prime }&=\left (\lambda +a -\sin \left (\lambda x \right ) a \right ) y^{2}+\lambda -a -\sin \left (\lambda x \right ) a \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✗ |
✗ |
107.712 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
25.522 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
47.472 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
135.652 |
|
| \begin{align*}
x y^{\prime }&=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] |
✓ |
✓ |
✓ |
✗ |
67.467 |
|
| \begin{align*}
\left (\sin \left (\lambda x \right ) a +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] |
✓ |
✓ |
✓ |
✗ |
1.709 |
|
| \begin{align*}
\left (\sin \left (\lambda x \right ) a +b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \sin \left (\lambda x \right )&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
37.050 |
|
| \begin{align*}
y^{\prime }&=\alpha y^{2}+\beta +\gamma \cos \left (\lambda x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
15.321 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.030 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
8.061 |
|
| \begin{align*}
y^{\prime }&=\lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
37.366 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
49.003 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
24.298 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
32.185 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
175.640 |
|
| \begin{align*}
x y^{\prime }&=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] |
✓ |
✓ |
✓ |
✗ |
68.044 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
1.710 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
52.185 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda a +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
6.098 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda ^{2}+3 \lambda a +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
47.328 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \tan \left (x \right ) y+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
10.350 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
1.642 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
41.803 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.190 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
23.705 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
27.784 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
391.796 |
|
| \begin{align*}
x y^{\prime }&=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] |
✓ |
✓ |
✓ |
✗ |
90.255 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
79.035 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda a +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✗ |
✗ |
6.049 |
|