| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
y^{\prime }&=y^{2}+\lambda ^{2}+3 \lambda a +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] |
✓ |
✓ |
✓ |
✗ |
12.240 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
14.111 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
30.016 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
321.929 |
|
| \begin{align*}
x y^{\prime }&=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] |
✓ |
✓ |
✓ |
✗ |
92.992 |
|
| \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*} |
unknown |
✓ |
✓ |
✓ |
✗ |
132.317 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
13.093 |
|
| \begin{align*}
y^{\prime }&=\cos \left (\lambda x \right ) a 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-a \,x^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
13.485 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
41.167 |
|
| \begin{align*}
y^{\prime }&=y^{2}-y \tan \left (x \right )+a \left (1-a \right ) \cot \left (x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.089 |
|
| \begin{align*}
y^{\prime }&=y^{2}-m y \tan \left (x \right )+b^{2} \cos \left (x \right )^{2 m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.759 |
|
| \begin{align*}
y^{\prime }&=y^{2}+m y \cot \left (x \right )+b^{2} \sin \left (x \right )^{2 m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.974 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda a +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] |
✓ |
✓ |
✗ |
✗ |
13.683 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
13.669 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
11.642 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arcsin \left (x \right )^{n} y+\arcsin \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
25.560 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
58.553 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
17.052 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
38.853 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
44.690 |
|
| \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*}
x y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
39.107 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
12.383 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arccos \left (x \right )^{n} y+\arccos \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
27.589 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
50.768 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
24.370 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
69.658 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
46.966 |
|
| \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*}
x y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
63.324 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
9.517 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arctan \left (x \right )^{n} y+\arctan \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.381 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
46.244 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.900 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
44.935 |
|
| \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*}
x y^{\prime }&=\lambda \arctan \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arctan \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
35.202 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.296 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.941 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
57.400 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
14.887 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
57.690 |
|
| \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*}
x y^{\prime }&=\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] |
✓ |
✓ |
✓ |
✗ |
40.376 |
|
| \begin{align*}
y^{\prime }&=y^{2}+f \left (x \right ) y-a^{2}-a f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.877 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a y-a b -b^{2} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
7.430 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right )+x f \left (x \right ) y+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.190 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,x^{n} f \left (x \right ) y+a n \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
6.932 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
7.693 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
8.298 |
|
| \begin{align*}
x y^{\prime }&=f \left (x \right ) y^{2}+n y+a \,x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.838 |
|
| \begin{align*}
x y^{\prime }&=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] |
✓ |
✓ |
✓ |
✗ |
10.366 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.588 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+g \left (x \right ) y+a n \,x^{n -1}-a \,x^{n} g \left (x \right )-a^{2} x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
18.026 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,x^{n} g \left (x \right ) y+a n \,x^{n -1}+a^{2} x^{2 n} \left (g \left (x \right )-f \left (x \right )\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
10.881 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.173 |
|
| \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] |
✓ |
✗ |
✓ |
✗ |
10.508 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
9.124 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
6.479 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
12.316 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.276 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
10.001 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
117.582 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
16.354 |
|
| \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 )+\lambda a \\
\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 )+\lambda a \\
\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*}
x y^{\prime }&=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*}
x y^{\prime }&=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] |
✓ |
✓ |
✗ |
✗ |
21.584 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
62.062 |
|
| \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 )+\lambda a \\
\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 )+\lambda a \\
\end{align*} |
[_Riccati] |
✗ |
✗ |
✗ |
✗ |
106.123 |
|
| \begin{align*}
y^{\prime }&=y^{2}-f \left (x \right )^{2}+f^{\prime }\left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
3.633 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
14.788 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
15.482 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
32.072 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
15.751 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
1.338 |
|
| \begin{align*}
y^{\prime }&=y^{2}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.927 |
|
| \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 (a \,x^{n}+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 }&=x^{2} y^{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] |
✓ |
✓ |
✓ |
✓ |
1.676 |
|
| \begin{align*}
y y^{\prime }-y&=A x +B \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
✓ |
✗ |
53.872 |
|
| \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 |
|