| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
y^{\prime }&=y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
76.852 |
|
| \begin{align*}
y^{\prime }&=y^{2}-2 a b \cot \left (a x \right ) y+b^{2}-a^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✗ |
✗ |
56.072 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \cot \left (\beta x \right ) y+a b \cot \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
13.662 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \cot \left (b x \right )^{m} y+a \cot \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
16.096 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} \cot \left (x \right )^{m} y-a \cot \left (x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
38.533 |
|
| \begin{align*}
y^{\prime }&=a \cot \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] |
✓ |
✓ |
✓ |
✗ |
482.488 |
|
| \begin{align*}
y^{\prime } x&=a \cot \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cot \left (\lambda x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
110.695 |
|
| \begin{align*}
\left (a \cot \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
158.136 |
|
| \begin{align*}
y^{\prime }&=\sin \left (\lambda x \right ) a y^{2}+b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
15.113 |
|
| \begin{align*}
y^{\prime }&=a \cos \left (\lambda x \right ) y^{2}+b \cos \left (\lambda x \right ) \sin \left (\lambda x \right )^{n} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
41.582 |
|
| \begin{align*}
y^{\prime }&=\lambda \sin \left (\lambda x \right ) y^{2}+a \,x^{n} \cos \left (\lambda x \right ) y-x^{n} a \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
15.698 |
|
| \begin{align*}
\sin \left (2 x \right )^{n +1} y^{\prime }&=a y^{2} \sin \left (x \right )^{2 n}+b \cos \left (x \right )^{2 n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
45.582 |
|
| \begin{align*}
y^{\prime }&=y^{2}-\tan \left (x \right ) y+a \left (1-a \right ) \cot \left (x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.178 |
|
| \begin{align*}
y^{\prime }&=y^{2}-m y \tan \left (x \right )+b^{2} \cos \left (x \right )^{2 m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.976 |
|
| \begin{align*}
y^{\prime }&=y^{2}+m y \cot \left (x \right )+b^{2} \sin \left (x \right )^{2 m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.289 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \lambda +b \lambda +2 a b +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
15.506 |
|
| \begin{align*}
y^{\prime }&=y^{2}-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✗ |
✗ |
50.523 |
|
| \begin{align*}
y^{\prime }&=\lambda \sin \left (\lambda x \right ) y^{2}+a \sin \left (\lambda x \right ) y-a \tan \left (\lambda x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
15.533 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda \arcsin \left (x \right )^{n} y-a^{2}+a \lambda \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
12.151 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arcsin \left (x \right )^{n} y+\arcsin \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
24.216 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+\lambda \arcsin \left (x \right )^{n} \left (x^{1+k} y-1\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
66.367 |
|
| \begin{align*}
y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
17.129 |
|
| \begin{align*}
y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arcsin \left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
45.017 |
|
| \begin{align*}
y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
46.997 |
|
| \begin{align*}
y^{\prime }&=\lambda \arcsin \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
✓ |
✓ |
✗ |
72.161 |
|
| \begin{align*}
y^{\prime } x&=\lambda \arcsin \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
42.672 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda \arccos \left (x \right )^{n} y-a^{2}+a \lambda \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
13.289 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arccos \left (x \right )^{n} y+\arccos \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
33.856 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+\lambda \arccos \left (x \right )^{n} \left (x^{1+k} y-1\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
63.323 |
|
| \begin{align*}
y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
29.793 |
|
| \begin{align*}
y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arccos \left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
77.529 |
|
| \begin{align*}
y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
48.820 |
|
| \begin{align*}
y^{\prime }&=\lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
✓ |
✓ |
✗ |
86.013 |
|
| \begin{align*}
y^{\prime } x&=\lambda \arccos \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
69.704 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda \arctan \left (x \right )^{n} y-a^{2}+a \lambda \arctan \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
10.689 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arctan \left (x \right )^{n} y+\arctan \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
12.971 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+\lambda \arctan \left (x \right )^{n} \left (x^{1+k} y-1\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
48.808 |
|
| \begin{align*}
y^{\prime }&=\lambda \arctan \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arctan \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
12.114 |
|
| \begin{align*}
y^{\prime }&=\lambda \arctan \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arctan \left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
49.533 |
|
| \begin{align*}
y^{\prime }&=\lambda \arctan \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
✓ |
✓ |
✗ |
43.747 |
|
| \begin{align*}
y^{\prime } x&=\lambda \arctan \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arctan \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
41.243 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} y-a^{2}+a \lambda \operatorname {arccot}\left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
12.486 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \operatorname {arccot}\left (x \right )^{n} y+\operatorname {arccot}\left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
13.358 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} \left (x^{1+k} y-1\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
61.945 |
|
| \begin{align*}
y^{\prime }&=\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \operatorname {arccot}\left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
16.800 |
|
| \begin{align*}
y^{\prime }&=\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}-b \lambda \,x^{m} \operatorname {arccot}\left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
78.513 |
|
| \begin{align*}
y^{\prime }&=\lambda \operatorname {arccot}\left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
✓ |
✓ |
✗ |
89.800 |
|
| \begin{align*}
y^{\prime } x&=\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \operatorname {arccot}\left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
43.904 |
|
| \begin{align*}
y^{\prime }&=y^{2}+f \left (x \right ) y-a^{2}-a f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.499 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a y-a b -b^{2} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
8.016 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right )+x f \left (x \right ) y+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.850 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,x^{n} f \left (x \right ) y+x^{n -1} a n \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
7.490 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+x^{n -1} a n -a^{2} x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
8.352 |
|
| \begin{align*}
y^{\prime }&=-\left (n +1\right ) x^{n} y^{2}+x^{n +1} f \left (x \right ) y-f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
9.196 |
|
| \begin{align*}
y^{\prime } x&=f \left (x \right ) y^{2}+n y+a \,x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.651 |
|
| \begin{align*}
y^{\prime } x&=x^{2 n} f \left (x \right ) y^{2}+\left (a \,x^{n} f \left (x \right )-n \right ) y+f \left (x \right ) b \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.321 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+g \left (x \right ) y-a^{2} f \left (x \right )-a g \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.220 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+g \left (x \right ) y+x^{n -1} a n -a \,x^{n} g \left (x \right )-a^{2} x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
16.700 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,x^{n} g \left (x \right ) y+x^{n -1} a n +a^{2} x^{2 n} \left (g \left (x \right )-f \left (x \right )\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
11.678 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+\lambda f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
10.168 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✓ |
✗ |
11.280 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
9.563 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+\lambda y+a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.126 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-f \left (x \right ) \left (a \,{\mathrm e}^{\lambda x}+b \right ) y+a \lambda \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
13.201 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{\lambda x} f \left (x \right ) y^{2}+\left (a f \left (x \right )-\lambda \right ) y+b \,{\mathrm e}^{-\lambda x} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.991 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{\lambda x} g \left (x \right )-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
10.424 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,{\mathrm e}^{\lambda x} g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}+a^{2} {\mathrm e}^{2 \lambda x} \left (g \left (x \right )-f \left (x \right )\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
127.694 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} f \left (x \right ) {\mathrm e}^{2 \lambda \,x^{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
13.483 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \tanh \left (\lambda x \right )^{2} \left (a f \left (x \right )+\lambda \right )+a \lambda \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
82.390 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \coth \left (\lambda x \right )^{2} \left (a f \left (x \right )+\lambda \right )+a \lambda \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
86.537 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a^{2} f \left (x \right )+a \lambda \sinh \left (\lambda x \right )-a^{2} f \left (x \right ) \sinh \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
34.543 |
|
| \begin{align*}
y^{\prime } x&=f \left (x \right ) y^{2}+a -a^{2} f \left (x \right ) \ln \left (x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
12.415 |
|
| \begin{align*}
y^{\prime } x&=f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}-a \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
✓ |
✓ |
✗ |
22.111 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a x \ln \left (x \right ) f \left (x \right ) y+a \ln \left (x \right )+a \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
25.814 |
|
| \begin{align*}
y^{\prime }&=-a \ln \left (x \right ) y^{2}+a f \left (x \right ) \left (x \ln \left (x \right )-x \right ) y-f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
24.460 |
|
| \begin{align*}
y^{\prime }&=\lambda \sin \left (\lambda x \right ) y^{2}+f \left (x \right ) \cos \left (\lambda x \right ) y-f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
64.043 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a^{2} f \left (x \right )+a \lambda \sin \left (\lambda x \right )+a^{2} f \left (x \right ) \sin \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
37.438 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a^{2} f \left (x \right )+a \lambda \cos \left (\lambda x \right )+a^{2} f \left (x \right ) \cos \left (\lambda x \right )^{2} \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
38.082 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \tan \left (\lambda x \right )^{2} \left (a f \left (x \right )-\lambda \right )+a \lambda \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
89.300 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \cot \left (\lambda x \right )^{2} \left (a f \left (x \right )-\lambda \right )+a \lambda \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
106.123 |
|
| \begin{align*}
y^{\prime }&=y^{2}-f \left (x \right )^{2}+f^{\prime }\left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
3.257 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-f \left (x \right ) g \left (x \right ) y+g^{\prime }\left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
11.344 |
|
| \begin{align*}
y^{\prime }&=-f^{\prime }\left (x \right ) y^{2}+f \left (x \right ) g \left (x \right ) y-g \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
12.158 |
|
| \begin{align*}
y^{\prime }&=g \left (x \right ) \left (y-f \left (x \right )\right )^{2}+f^{\prime }\left (x \right ) \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
30.481 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}-\frac {g^{\prime }\left (x \right )}{f \left (x \right )} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
8.971 |
|
| \begin{align*}
f \left (x \right )^{2} y^{\prime }-f^{\prime }\left (x \right ) y^{2}+g \left (x \right ) \left (y-f \left (x \right )\right )&=0 \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
29.293 |
|
| \begin{align*}
y^{\prime }&=f^{\prime }\left (x \right ) y^{2}+a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+a \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.322 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+g^{\prime }\left (x \right ) y+a f \left (x \right ) {\mathrm e}^{2 g \left (x \right )} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.216 |
|
| \begin{align*}
y^{\prime }&=y^{2}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.615 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a^{2} f \left (a x +b \right ) \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
5.818 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\frac {f \left (\frac {1}{x}\right )}{x^{4}} \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
6.931 |
|
| \begin{align*}
x^{2} y^{\prime }&=x^{4} f \left (x \right ) y^{2}+1 \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
8.598 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2} x^{4}+x^{2 n} f \left (x^{n} a +b \right )-\frac {n^{2}}{4}+\frac {1}{4} \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
44.309 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+g \left (x \right ) y+h \left (x \right ) \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
13.063 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2} x^{2}+f \left (a \ln \left (x \right )+b \right )+\frac {1}{4} \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
12.049 |
|
| \begin{align*}
y y^{\prime }-y&=A \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
1.692 |
|
| \begin{align*}
y y^{\prime }-y&=A x +B \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✗ |
63.523 |
|
| \begin{align*}
y y^{\prime }-y&=-\frac {2 x}{9}+A +\frac {B}{\sqrt {x}} \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✓ |
✗ |
178.790 |
|
| \begin{align*}
y y^{\prime }-y&=2 A \left (\sqrt {x}+4 A +\frac {3 A^{2}}{\sqrt {x}}\right ) \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✗ |
✗ |
117.497 |
|
| \begin{align*}
y y^{\prime }-y&=A x +\frac {B}{x}-\frac {B^{2}}{x^{3}} \\
\end{align*} |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✓ |
✗ |
✗ |
42.132 |
|