| # |
ODE |
CAS classification |
Solved |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
y^{\prime }&=x +\frac {y^{2}}{2} \\
y \left (-2\right ) &= 0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
2.842 |
|
| \begin{align*}
\left (x +1\right )^{2} y^{\prime }&=\left (1+y\right )^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.480 |
|
| \begin{align*}
x^{2} y^{\prime }&=1-x^{2}+y^{2}-y^{2} x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.529 |
|
| \begin{align*}
y^{\prime }&=3 x^{2} \left (1+y^{2}\right ) \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.790 |
|
| \begin{align*}
y^{\prime }&=\left (4 x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.024 |
|
| \begin{align*}
y^{\prime }+y^{2}&=x^{2}+1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.477 |
|
| \begin{align*}
y^{\prime }+2 y x&=1+x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.213 |
|
| \begin{align*}
y^{\prime }&=1+x^{2}+y^{2}+y^{2} x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.375 |
|
| \begin{align*}
y^{\prime }&=x^{2}-2 y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.075 |
|
| \begin{align*}
y^{\prime }&=y^{2}+x^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
39.950 |
|
| \begin{align*}
y^{\prime }&=x^{2}-y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.909 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }&=\left (1+y\right )^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.063 |
|
| \begin{align*}
x^{2} y^{\prime }&=1-x^{2}+y^{2}-y^{2} x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.131 |
|
| \begin{align*}
y^{\prime }&=3 x^{2} \left (1+y^{2}\right ) \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.048 |
|
| \begin{align*}
y^{\prime }&=\left (4 x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.057 |
|
| \begin{align*}
y^{\prime }&=1+x^{2}+y^{2}+y^{2} x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.441 |
|
| \begin{align*}
y^{\prime }&=x^{2}-2 y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.062 |
|
| \begin{align*}
y^{\prime }&=2 \left (x +1\right ) \left (1+y^{2}\right ) \\
y \left (0\right ) &= 0 \\
\end{align*} | [_separable] | ✓ | ✓ | ✓ | ✓ | 2.876 |
|
| \begin{align*}
y^{\prime }&=\frac {x^{2}+y x +y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.931 |
|
| \begin{align*}
x^{2}+3 y x +y^{2}-x^{2} y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.429 |
|
| \begin{align*}
y^{\prime }&=t -1-y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.661 |
|
| \begin{align*}
y^{\prime }&=1+2 x +y^{2}+2 x y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.346 |
|
| \begin{align*}
2 y^{\prime }+x \left (y^{2}-1\right )&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.073 |
|
| \begin{align*}
y^{\prime }&=x^{2} \left (1+y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.003 |
|
| \begin{align*}
y^{\prime }&=x \left (1+y^{2}\right ) \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.336 |
|
| \begin{align*}
\frac {y^{\prime }}{\left (1+y\right )^{2}}-\frac {1}{x \left (1+y\right )}&=-\frac {3}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.724 |
|
| \begin{align*}
y^{\prime }&=x^{2} \left (1+y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.532 |
|
| \begin{align*}
y^{\prime }&=\left (x -1\right ) \left (y-1\right ) \left (y-2\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.971 |
|
| \begin{align*}
\left (x^{2}+2\right ) y^{\prime }&=4 x \left (y^{2}+2 y+1\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.853 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.902 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2}+y x -x^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
0.148 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2}+y x -x^{2} \\
y \left (1\right ) &= 2 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.069 |
|
| \begin{align*}
x^{2} y^{\prime }&=x^{2}+y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.713 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}-3 y x -5 x^{2}}{x^{2}} \\
y \left (1\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.050 |
|
| \begin{align*}
x^{2} y^{\prime }&=2 x^{2}+y^{2}+4 y x \\
y \left (1\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.639 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2}+y x -4 x^{2} \\
y \left (-1\right ) &= 0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.775 |
|
| \begin{align*}
x^{3} y^{\prime }&=2 y^{2}+2 x^{2} y-2 x^{4} \\
\end{align*} | [[_homogeneous, ‘class G‘], _rational, _Riccati] | ✓ | ✓ | ✓ | ✓ | 3.089 |
|
| \begin{align*}
y^{\prime }&=y^{2} {\mathrm e}^{-x}+4 y+2 \,{\mathrm e}^{x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.103 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}+\tan \left (x \right ) y+\tan \left (x \right )^{2}}{\sin \left (x \right )^{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.279 |
|
| \begin{align*}
x \ln \left (x \right )^{2} y^{\prime }&=-4 \ln \left (x \right )^{2}+y \ln \left (x \right )+y^{2} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.070 |
|
| \begin{align*}
y^{\prime }&=1+x -\left (2 x +1\right ) y+x y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
2.575 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+y^{2}\right )-x \left (2+x \right ) y+x +2&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.095 |
|
| \begin{align*}
y^{\prime }+y^{2}+4 y x +4 x^{2}+2&=0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.601 |
|
| \begin{align*}
\left (2 x +1\right ) \left (y^{\prime }+y^{2}\right )-2 y-2 x -3&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.131 |
|
| \begin{align*}
\left (3 x -1\right ) \left (y^{\prime }+y^{2}\right )-\left (3 x +2\right ) y-6 x +8&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.498 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+y^{2}\right )+y x +x^{2}-\frac {1}{4}&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.033 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+y^{2}\right )-7 y x +7&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.625 |
|
| \begin{align*}
\left (t^{2}+1\right ) y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.372 |
|
| \begin{align*}
y^{\prime }&=1-t +y^{2}-t y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.578 |
|
| \begin{align*}
y^{\prime }&=t +y^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
23.086 |
|
| \begin{align*}
\left (t^{2}+1\right ) y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.707 |
|
| \begin{align*}
y^{\prime }&=1-t +y^{2}-t y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.780 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{t}+y^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.908 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\cos \left (t \right )^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
57.204 |
|
| \begin{align*}
y^{\prime }&=t +y^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} | [[_Riccati, _special]] | ✓ | ✓ | ✓ | ✗ | 18.077 |
|
| \begin{align*}
y^{\prime }&=1-t +y^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.754 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.830 |
|
| \begin{align*}
1+y^{2}&=\frac {y^{\prime }}{x^{3} \left (x -1\right )} \\
y \left (2\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.991 |
|
| \begin{align*}
\left (x^{2}+x +1\right ) y^{\prime }&=y^{2}+2 y+5 \\
y \left (1\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.496 |
|
| \begin{align*}
\left (x^{2}-2 x -8\right ) y^{\prime }&=y^{2}+y-2 \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.473 |
|
| \begin{align*}
y^{\prime }-\frac {y^{2}}{x^{2}}&={\frac {1}{4}} \\
y \left (1\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.211 |
|
| \begin{align*}
y^{\prime }-\frac {y^{2}}{x^{2}}&={\frac {1}{4}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.599 |
|
| \begin{align*}
y^{\prime }&=\frac {x \left (y^{2}-1\right )}{2 \left (x -2\right ) \left (x -1\right )} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.968 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }+y^{2}&=-1 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.389 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (x +y\right )^{2}}{2 x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.513 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2}+3 y x +x^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.583 |
|
| \begin{align*}
y^{\prime }&=\frac {x \left (y^{2}-1\right )}{2 \left (x -2\right ) \left (x -1\right )} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.405 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }+y^{2}&=-1 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.137 |
|
| \begin{align*}
y^{\prime }&=\frac {x^{2}+y x +y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.288 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (x +y\right )^{2}}{2 x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.228 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2}+3 y x +x^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.181 |
|
| \begin{align*}
y^{\prime }&=\left (9 x -y\right )^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.495 |
|
| \begin{align*}
y^{\prime }&=\left (4 x +y+2\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.997 |
|
| \begin{align*}
y^{\prime }&=2 x \left (x +y\right )^{2}-1 \\
y \left (0\right ) &= 1 \\
\end{align*} | [[_1st_order, _with_linear_symmetries], _Riccati] | ✓ | ✓ | ✓ | ✓ | 2.375 |
|
| \begin{align*}
y^{\prime }+\frac {2 y}{x}-y^{2}&=-\frac {2}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.939 |
|
| \begin{align*}
y^{\prime }+\frac {7 y}{x}-3 y^{2}&=\frac {3}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.066 |
|
| \begin{align*}
x^{2} y^{\prime }&=\left (y-1\right ) x +\left (y-1\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.295 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}+x^{2}}{2 x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.496 |
|
| \begin{align*}
2 y^{\prime } x&=1-y^{2} \\
y \left (1\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.535 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{x} \left (1+y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.424 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.133 |
|
| \begin{align*}
1&=\frac {y}{1-y^{2} x^{2}}+\frac {x y^{\prime }}{1-y^{2} x^{2}} \\
\end{align*} |
[_exact, _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.260 |
|
| \begin{align*}
y^{\prime } x&=x^{5}+x^{3} y^{2}+y \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.936 |
|
| \begin{align*}
y^{\prime } x&=y+x^{2}+9 y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.122 |
|
| \begin{align*}
y^{\prime }&=\left (x +1\right )^{2}+\left (4 y+1\right )^{2}+8 y x +1 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
22.996 |
|
| \begin{align*}
x^{2}+y+y^{2}-y^{\prime } x&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.339 |
|
| \begin{align*}
y^{\prime }+y^{2}&=x^{2}+1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.415 |
|
| \begin{align*}
y^{\prime }&=x^{2}-y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.334 |
|
| \begin{align*}
y^{\prime }+f \left (x \right )^{2}&=f^{\prime }\left (x \right )+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
1.316 |
|
| \begin{align*}
y^{\prime }+1-x&=y \left (x +y\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.599 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.481 |
|
| \begin{align*}
y^{\prime }&=\left (x -y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.286 |
|
| \begin{align*}
y^{\prime }&=3 y-3 x +3+\left (x -y\right )^{2} \\
\end{align*} | [[_homogeneous, ‘class C‘], _Riccati] | ✓ | ✓ | ✓ | ✓ | 2.539 |
|
| \begin{align*}
y^{\prime }&=2 x -\left (x^{2}+1\right ) y+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.256 |
|
| \begin{align*}
y^{\prime }&=x \left (x^{3}+2\right )-\left (2 x^{2}-y\right ) y \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.447 |
|
| \begin{align*}
y^{\prime }&=1+x \left (-x^{3}+2\right )+\left (2 x^{2}-y\right ) y \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.720 |
|
| \begin{align*}
y^{\prime }&=\cos \left (x \right )-\left (-y+\sin \left (x \right )\right ) y \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.325 |
|
| \begin{align*}
y^{\prime }&=\cos \left (2 x \right )+\left (\sin \left (2 x \right )+y\right ) y \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.245 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right )+x f \left (x \right ) y+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.003 |
|
| \begin{align*}
y^{\prime }&=\left (3+x -4 y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.497 |
|
| \begin{align*}
y^{\prime }&=\left (1+4 x +9 y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
15.109 |
|
| \begin{align*}
y^{\prime }&=3 a +3 b x +3 b y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.205 |
|
| \begin{align*}
y^{\prime }&=a x +b y^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
35.309 |
|
| \begin{align*}
y^{\prime }&=a +b x +c y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.175 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n -1}+b \,x^{2 n}+c y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.514 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n}+b y^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
33.747 |
|
| \begin{align*}
y^{\prime }&=1+a \left (x -y\right ) y \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.743 |
|
| \begin{align*}
y^{\prime }&=1-x -x^{3}+\left (2 x^{2}+1\right ) y-x y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
4.540 |
|
| \begin{align*}
y^{\prime }&=x \left (2+x^{2} y-y^{2}\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
3.150 |
|
| \begin{align*}
y^{\prime }&=x +\left (1-2 x \right ) y-\left (1-x \right ) y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
4.359 |
|
| \begin{align*}
y^{\prime }&=x^{n} \left (a +b y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.339 |
|
| \begin{align*}
y^{\prime }&=a \,x^{m}+b \,x^{n} y^{2} \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 44.997 |
|
| \begin{align*}
y^{\prime }&=\sin \left (x \right ) \left (2 \sec \left (x \right )^{2}-y^{2}\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.703 |
|
| \begin{align*}
y^{\prime }+4 \csc \left (x \right )&=\left (3-\cot \left (x \right )\right ) y+\sin \left (x \right ) y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.648 |
|
| \begin{align*}
y^{\prime }+\tan \left (x \right ) \left (1-y^{2}\right )&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.808 |
|
| \begin{align*}
y^{\prime }&=\left (a +b y+c y^{2}\right ) f \left (x \right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
6.622 |
|
| \begin{align*}
2 y^{\prime }+2 \csc \left (x \right )^{2}&=y \csc \left (x \right ) \sec \left (x \right )-y^{2} \sec \left (x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.714 |
|
| \begin{align*}
y^{\prime } x +x^{2}+y^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.185 |
|
| \begin{align*}
y^{\prime } x&=x^{2}+y \left (1+y\right ) \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.198 |
|
| \begin{align*}
y^{\prime } x -y+y^{2}&=x^{{2}/{3}} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
50.119 |
|
| \begin{align*}
y^{\prime } x&=a +b y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.344 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{2}+y+b y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.803 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{2 n}+\left (n +b y\right ) y \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.368 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n}+b y+c y^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
28.151 |
|
| \begin{align*}
y^{\prime } x&=k +a \,x^{n}+b y+c y^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
7.213 |
|
| \begin{align*}
y^{\prime } x +a +x y^{2}&=0 \\
\end{align*} |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
2.513 |
|
| \begin{align*}
y^{\prime } x&=x^{3}+\left (2 x^{2}+1\right ) y+x y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.539 |
|
| \begin{align*}
y^{\prime } x +b x +\left (2+a x y\right ) y&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.650 |
|
| \begin{align*}
y^{\prime } x +a_{0} +a_{1} x +\left (a_{2} +a_{3} x y\right ) y&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
33.851 |
|
| \begin{align*}
y^{\prime } x +a \,x^{2} y^{2}+2 y&=b \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
28.418 |
|
| \begin{align*}
y^{\prime } x +x^{m}+\frac {\left (n -m \right ) y}{2}+x^{n} y^{2}&=0 \\
\end{align*} | [_rational, _Riccati] | ✓ | ✓ | ✓ | ✗ | 2.897 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{m}-b y-c \,x^{n} y^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.842 |
|
| \begin{align*}
y^{\prime } x&=2 x -y+a \,x^{n} \left (x -y\right )^{2} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.925 |
|
| \begin{align*}
y^{\prime } x&=y+\left (x^{2}-y^{2}\right ) f \left (x \right ) \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.040 |
|
| \begin{align*}
2 y^{\prime } x +1&=4 i x y+y^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
57.595 |
|
| \begin{align*}
3 y^{\prime } x&=3 x^{{2}/{3}}+\left (1-3 y\right ) y \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.288 |
|
| \begin{align*}
x^{2} y^{\prime }+x^{2}+y x +y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.111 |
|
| \begin{align*}
x^{2} y^{\prime }&=\left (1+2 x -y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.839 |
|
| \begin{align*}
x^{2} y^{\prime }&=a +b y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.404 |
|
| \begin{align*}
x^{2} y^{\prime }+a \,x^{2}+b x y+c y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
62.256 |
|
| \begin{align*}
x^{2} y^{\prime }&=a +b \,x^{n}+y^{2} x^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.389 |
|
| \begin{align*}
x^{2} y^{\prime }+2+x y \left (4+y x \right )&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.807 |
|
| \begin{align*}
x^{2} y^{\prime }+2+a x \left (-y x +1\right )-y^{2} x^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.713 |
|
| \begin{align*}
x^{2} y^{\prime }&=a +b \,x^{2} y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
6.015 |
|
| \begin{align*}
x^{2} y^{\prime }&=a +b \,x^{n}+c \,x^{2} y^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
88.596 |
|
| \begin{align*}
x^{2} y^{\prime }&=a +b x y+c \,x^{2} y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.303 |
|
| \begin{align*}
x^{2} y^{\prime }&=a +b x y+c \,x^{4} y^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.259 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.310 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }&=-1-y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.338 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime }&=1-y^{2} \\
\end{align*} | [_separable] | ✓ | ✓ | ✓ | ✓ | 3.707 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime }&=y^{2}-1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.687 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime }&=1-y \left (2 x -y\right ) \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.448 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime }&=n \left (y^{2}-2 y x +1\right ) \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.357 |
|
| \begin{align*}
\left (x -a \right )^{2} y^{\prime }+k \left (x +y-a \right )^{2}+y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
16.660 |
|
| \begin{align*}
\left (x -a \right ) \left (-b +x \right ) y^{\prime }+k \left (y-a \right ) \left (y-b \right )&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
10.528 |
|
| \begin{align*}
\left (x -a \right ) \left (-b +x \right ) y^{\prime }+k \left (x +y-a \right ) \left (x +y-b \right )+y^{2}&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
11.143 |
|
| \begin{align*}
2 x^{2} y^{\prime }+1+2 y x -y^{2} x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.902 |
|
| \begin{align*}
2 x^{2} y^{\prime }&=2 y x +\left (-x \cot \left (x \right )+1\right ) \left (x^{2}-y^{2}\right ) \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
99.829 |
|
| \begin{align*}
x \left (1-2 x \right ) y^{\prime }&=4 x -\left (4 x +1\right ) y+y^{2} \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.013 |
|
| \begin{align*}
2 x \left (1-x \right ) y^{\prime }+x +\left (1-x \right ) y^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
7.674 |
|
| \begin{align*}
a \,x^{2} y^{\prime }&=x^{2}+a x y+b^{2} y^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
27.248 |
|
| \begin{align*}
\left (b \,x^{2}+a \right ) y^{\prime }&=A +B y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.829 |
|
| \begin{align*}
\left (b \,x^{2}+a \right ) y^{\prime }&=-A -B y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.665 |
|
| \begin{align*}
x^{3} y^{\prime }&=x^{4}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.078 |
|
| \begin{align*}
x^{3} y^{\prime }&=\left (y-1\right ) x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.849 |
|
| \begin{align*}
x^{3} y^{\prime }+20+x^{2} y \left (1-x^{2} y\right )&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.879 |
|
| \begin{align*}
x^{3} y^{\prime }+3+\left (3-2 x \right ) x^{2} y-x^{6} y^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.429 |
|
| \begin{align*}
x \left (-x^{2}+1\right ) y^{\prime }+x^{2}+\left (-x^{2}+1\right ) y^{2}&=0 \\
\end{align*} | [_rational, _Riccati] | ✓ | ✓ | ✓ | ✗ | 177.820 |
|
| \begin{align*}
x \left (c \,x^{2}+b x +a \right ) y^{\prime }+x^{2}-\left (c \,x^{2}+b x +a \right ) y&=y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.891 |
|
| \begin{align*}
x^{4} y^{\prime }+a^{2}+x^{4} y^{2}&=0 \\
\end{align*} |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
7.272 |
|
| \begin{align*}
\left (-x^{4}+1\right ) y^{\prime }&=2 x \left (1-y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.471 |
|
| \begin{align*}
x \left (-x^{3}+1\right ) y^{\prime }&=x^{2}+\left (1-2 y x \right ) y \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.502 |
|
| \begin{align*}
x \left (-x^{4}+1\right ) y^{\prime }&=2 x \left (x^{2}-y^{2}\right )+\left (-x^{4}+1\right ) y \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.876 |
|
| \begin{align*}
x^{n} y^{\prime }&=x^{2 n -1}-y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
70.642 |
|
| \begin{align*}
x^{n} y^{\prime }+x^{2 n -2}+y^{2}+\left (1-n \right ) x^{n -1} y&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.109 |
|
| \begin{align*}
x^{n} y^{\prime }&=a^{2} x^{2 n -2}+b^{2} y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
9.289 |
|
| \begin{align*}
x^{n} y^{\prime }&=x^{n -1} \left (a \,x^{2 n}+n y-b y^{2}\right ) \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.514 |
|
| \begin{align*}
y^{\prime } \sqrt {-x^{2}+1}&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.935 |
|
| \begin{align*}
y^{\prime } x -a y+y^{2}&=x^{-2 a} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.832 |
|
| \begin{align*}
y^{\prime } x -a y+y^{2}&=x^{-\frac {2 a}{3}} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.910 |
|
| \begin{align*}
u^{\prime }+u^{2}&=\frac {c}{x^{{4}/{3}}} \\
\end{align*} |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
0.310 |
|
| \begin{align*}
u^{\prime }+b u^{2}&=\frac {c}{x^{4}} \\
\end{align*} |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
0.273 |
|
| \begin{align*}
u^{\prime }-u^{2}&=\frac {2}{x^{{8}/{3}}} \\
\end{align*} |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
0.480 |
|
| \begin{align*}
y^{\prime }&=x^{3}+\frac {2 y}{x}-\frac {y^{2}}{x} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.756 |
|
| \begin{align*}
y^{\prime }&=2 \sec \left (x \right ) \tan \left (x \right )-\sin \left (x \right ) y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.743 |
|
| \begin{align*}
y^{\prime }&=\frac {1}{x^{2}}-\frac {y}{x}-y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.424 |
|
| \begin{align*}
y^{\prime }&=1+\frac {y}{x}-\frac {y^{2}}{x^{2}} \\
\end{align*} | [[_homogeneous, ‘class A‘], _rational, _Riccati] | ✓ | ✓ | ✓ | ✓ | 4.465 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.258 |
|
| \begin{align*}
y^{\prime } x -y^{2}+1&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.124 |
|
| \begin{align*}
x^{2} y^{\prime }+x^{2}+y x +y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.231 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.431 |
|
| \begin{align*}
y^{\prime }+y^{2}&=\frac {a^{2}}{x^{4}} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.635 |
|
| \begin{align*}
y^{\prime }&=x y^{2}-\frac {2 y}{x}-\frac {1}{x^{3}} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.880 |
|
| \begin{align*}
y^{\prime }&=\frac {2 y^{2}}{x}+\frac {y}{x}-2 x \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.733 |
|
| \begin{align*}
y^{\prime }&=y^{2} {\mathrm e}^{-x}+y-{\mathrm e}^{x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.138 |
|
| \begin{align*}
y^{\prime }&=\left (1+y^{2}\right ) \tan \left (x \right ) \\
y \left (0\right ) &= \sqrt {3} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
5.653 |
|
| \begin{align*}
y^{\prime }&=\sqrt {1+\sin \left (x \right )}\, \left (1+y^{2}\right ) \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
25.092 |
|
| \begin{align*}
\left (y-4 x -1\right )^{2}-y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
7.717 |
|
| \begin{align*}
y^{\prime }&=\left (x +y+2\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.077 |
|
| \begin{align*}
y^{\prime }&=\left (x -y+5\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.945 |
|
| \begin{align*}
y^{\prime }&=x^{3} \left (y-x \right )^{2}+\frac {y}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.069 |
|
| \begin{align*}
y^{\prime }&=\left (2 x +y-1\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.912 |
|
| \begin{align*}
\left (x +1\right )^{2} y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.200 |
|
| \begin{align*}
y^{\prime }+x +x y^{2}&=0 \\
y \left (1\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
4.434 |
|
| \begin{align*}
y^{\prime }&=-2 \left (2 x +3 y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
10.306 |
|
| \begin{align*}
1+y^{2}&=\left (x^{2}+x \right ) y^{\prime } \\
\end{align*} | [_separable] | ✓ | ✓ | ✓ | ✓ | 3.914 |
|
| \begin{align*}
y^{2}-1+y^{\prime } x&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.825 |
|
| \begin{align*}
y^{\prime }&=y^{2}+x^{2} \\
y \left (1\right ) &= -1 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
10.593 |
|
| \begin{align*}
y^{\prime }&=x^{2}-y^{2} \\
y \left (-2\right ) &= 1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.159 |
|
| \begin{align*}
y^{\prime }&=x^{2}-y^{2} \\
y \left (3\right ) &= 0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.687 |
|
| \begin{align*}
y^{\prime }&=y^{2}+x^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
36.454 |
|
| \begin{align*}
y^{\prime }&=x \left (y-4\right )^{2}-2 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
90.978 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (3+2 y\right )^{2}}{\left (5+4 x \right )^{2}} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.928 |
|
| \begin{align*}
\left (x^{4}+1\right ) y^{\prime }+x \left (1+4 y^{2}\right )&=0 \\
y \left (1\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
3.329 |
|
| \begin{align*}
y^{\prime }&=\left (1+y^{2}\right ) \sqrt {1+\cos \left (x^{3}\right )} \\
y \left (1\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
42.881 |
|
| \begin{align*}
y^{\prime }&=\left (x +y+1\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.864 |
|
| \begin{align*}
x^{2}+y x +y^{2}&=x^{2} y^{\prime } \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.427 |
|
| \begin{align*}
y^{\prime }&=x^{2} \left (1+y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.790 |
|
| \begin{align*}
y^{\prime }&=-\frac {2}{t}+\frac {y}{t}+\frac {y^{2}}{t} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.972 |
|
| \begin{align*}
y^{\prime }&=-\frac {y}{t}-1-y^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.985 |
|
| \begin{align*}
\left (\phi ^{\prime }-\frac {\phi ^{2}}{2}\right ) \sin \left (\theta \right )^{2}-\phi \sin \left (\theta \right ) \cos \left (\theta \right )&=\frac {\cos \left (2 \theta \right )}{2}+1 \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.980 |
|
| \begin{align*}
-y+y^{\prime } x&=y^{2}+x^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.335 |
|
| \begin{align*}
y^{\prime }&=y^{2} x^{2}-4 x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✗ |
3.261 |
|
| \begin{align*}
y^{\prime }&=\frac {x^{2}+y x +y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.786 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (x +y-1\right )^{2}}{2 \left (2+x \right )^{2}} \\
\end{align*} | [[_homogeneous, ‘class C‘], _rational, _Riccati] | ✓ | ✓ | ✓ | ✓ | 3.434 |
|
| \begin{align*}
y^{\prime } x&=y+y^{2}+x^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.079 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.400 |
|
| \begin{align*}
\frac {y}{1-y^{2} x^{2}}+\frac {x y^{\prime }}{1-y^{2} x^{2}}&=1 \\
\end{align*} |
[_exact, _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.141 |
|
| \begin{align*}
\frac {y^{\prime } x +y}{1-y^{2} x^{2}}+x&=0 \\
\end{align*} |
[_exact, _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.762 |
|
| \begin{align*}
y^{\prime }&=y^{2}-x \\
y \left (0\right ) &= 1 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
4.138 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }+y^{2}&=-1 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
3.612 |
|
| \begin{align*}
y^{\prime }&=\frac {5 x^{2}-y x +y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.792 |
|
| \begin{align*}
y^{\prime } x -2 y+b y^{2}&=c \,x^{4} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.266 |
|
| \begin{align*}
y^{\prime } x -y+y^{2}&=x^{{2}/{3}} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
61.681 |
|
| \begin{align*}
u^{\prime }+u^{2}&=\frac {1}{x^{{4}/{5}}} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.211 |
|
| \begin{align*}
y^{\prime }&=y^{2}+x^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
41.976 |
|
| \begin{align*}
y^{\prime }&=x^{2}+y^{2}-1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
82.563 |
|
| \begin{align*}
y^{\prime }-y^{2}-x -x^{2}&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.783 |
|
| \begin{align*}
y^{\prime }&=a x +b y^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
38.549 |
|
| \begin{align*}
c y^{\prime }&=a x +b y^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
35.672 |
|
| \begin{align*}
c y^{\prime }&=\frac {a x +b y^{2}}{r} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
3.052 |
|
| \begin{align*}
c y^{\prime }&=\frac {a x +b y^{2}}{r x} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.035 |
|
| \begin{align*}
c y^{\prime }&=\frac {a x +b y^{2}}{r \,x^{2}} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.700 |
|
| \begin{align*}
y^{\prime }&=\sin \left (x \right )+y^{2} \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 5.155 |
|
| \begin{align*}
y^{\prime }&=x +y+b y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
80.244 |
|
| \begin{align*}
y^{\prime }&=x -y^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
6.756 |
|
| \begin{align*}
y^{\prime }+y^{2}-a x -b&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.175 |
|
| \begin{align*}
y^{\prime }+y^{2}+a \,x^{m}&=0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
31.501 |
|
| \begin{align*}
y^{\prime }+y^{2}-2 x^{2} y+x^{4}-2 x -1&=0 \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.243 |
|
| \begin{align*}
y^{\prime }+y^{2}+\left (y x -1\right ) f \left (x \right )&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.198 |
|
| \begin{align*}
y^{\prime }-y^{2}-y x -x +1&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.767 |
|
| \begin{align*}
y^{\prime }-\left (x +y\right )^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.996 |
|
| \begin{align*}
y^{\prime }-y^{2}+\left (x^{2}+1\right ) y-2 x&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.832 |
|
| \begin{align*}
y^{\prime }-y^{2}+y \sin \left (x \right )-\cos \left (x \right )&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.312 |
|
| \begin{align*}
y^{\prime }-y^{2}-y \sin \left (2 x \right )-\cos \left (2 x \right )&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.293 |
|
| \begin{align*}
y^{\prime }+a y^{2}-b \,x^{\nu }&=0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
32.036 |
|
| \begin{align*}
y^{\prime }+a y^{2}-b \,x^{2 \nu }-c \,x^{\nu -1}&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.374 |
|
| \begin{align*}
y^{\prime }+a y \left (y-x \right )-1&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.735 |
|
| \begin{align*}
y^{\prime }+x y^{2}-x^{3} y-2 x&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
4.074 |
|
| \begin{align*}
y^{\prime }+x^{-a -1} y^{2}-x^{a}&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
50.410 |
|
| \begin{align*}
y^{\prime }-a \,x^{n} \left (1+y^{2}\right )&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.663 |
|
| \begin{align*}
y^{\prime }+\sin \left (x \right ) y^{2}-\frac {2 \sin \left (x \right )}{\cos \left (x \right )^{2}}&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.554 |
|
| \begin{align*}
y^{\prime }+f \left (x \right ) \left (y^{2}+2 a y+b \right )&=0 \\
\end{align*} | [_separable] | ✓ | ✓ | ✓ | ✓ | 6.823 |
|
| \begin{align*}
y^{\prime } x +x^{2}+y^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.497 |
|
| \begin{align*}
y^{\prime } x -y^{2}+1&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.603 |
|
| \begin{align*}
y^{\prime } x +a y^{2}-y+b \,x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.427 |
|
| \begin{align*}
y^{\prime } x +a y^{2}-b y+c \,x^{2 b}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.283 |
|
| \begin{align*}
y^{\prime } x +a y^{2}-b y-c \,x^{\beta }&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
31.228 |
|
| \begin{align*}
y^{\prime } x +a +x y^{2}&=0 \\
\end{align*} |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
2.796 |
|
| \begin{align*}
y^{\prime } x +x y^{2}-y-a \,x^{3}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.356 |
|
| \begin{align*}
y^{\prime } x +x y^{2}-\left (2 x^{2}+1\right ) y-x^{3}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.073 |
|
| \begin{align*}
y^{\prime } x +a x y^{2}+2 y+b x&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.677 |
|
| \begin{align*}
y^{\prime } x +a x y^{2}+b y+c x +d&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
32.347 |
|
| \begin{align*}
y^{\prime } x +x^{a} y^{2}+\frac {\left (a -b \right ) y}{2}+x^{b}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.596 |
|
| \begin{align*}
y^{\prime } x +a \,x^{\alpha } y^{2}+b y-c \,x^{\beta }&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.178 |
|
| \begin{align*}
y^{\prime } x +f \left (x \right ) \left (y^{2}-x^{2}\right )-y&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.594 |
|
| \begin{align*}
x^{2} y^{\prime }+x^{2}+y x +y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.185 |
|
| \begin{align*}
x^{2} y^{\prime }-y^{2}-y x -x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.091 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+y^{2}\right )+a \,x^{k}-b \left (b -1\right )&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.469 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+y^{2}\right )+4 y x +2&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.411 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+y^{2}\right )+a x y+b&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.604 |
|
| \begin{align*}
x^{2} \left (y^{\prime }-y^{2}\right )-a \,x^{2} y+a x +2&=0 \\
\end{align*} | [_rational, _Riccati] | ✓ | ✓ | ✓ | ✗ | 2.696 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+a y^{2}\right )-b&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
3.454 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+a y^{2}\right )+b \,x^{\alpha }+c&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
77.568 |
|
| \begin{align*}
\left (x^{2}-1\right ) y^{\prime }+y^{2}-2 y x +1&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.937 |
|
| \begin{align*}
\left (x -a \right ) \left (-b +x \right ) y^{\prime }+k \left (x +y-a \right ) \left (x +y-b \right )+y^{2}&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
11.668 |
|
| \begin{align*}
2 x^{2} y^{\prime }-2 y^{2}-y x +2 a^{2} x&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.550 |
|
| \begin{align*}
2 x^{2} y^{\prime }-2 y^{2}-3 y x +2 a^{2} x&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.122 |
|
| \begin{align*}
x \left (2 x -1\right ) y^{\prime }+y^{2}-\left (4 x +1\right ) y+4 x&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.636 |
|
| \begin{align*}
2 x \left (x -1\right ) y^{\prime }+y^{2} \left (x -1\right )-x&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.728 |
|
| \begin{align*}
3 x^{2} y^{\prime }-7 y^{2}-3 y x -x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.063 |
|
| \begin{align*}
3 \left (x^{2}-4\right ) y^{\prime }+y^{2}-y x -3&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
170.149 |
|
| \begin{align*}
x^{3} y^{\prime }-y^{2}-x^{4}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.589 |
|
| \begin{align*}
x^{3} y^{\prime }-x^{4} y^{2}+x^{2} y+20&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.300 |
|
| \begin{align*}
x^{3} y^{\prime }-x^{6} y^{2}-\left (2 x -3\right ) x^{2} y+3&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.952 |
|
| \begin{align*}
x \left (x^{2}-1\right ) y^{\prime }+\left (x^{2}-1\right ) y^{2}-x^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
131.004 |
|
| \begin{align*}
2 x \left (x^{2}-1\right ) y^{\prime }+2 \left (x^{2}-1\right ) y^{2}-\left (3 x^{2}-5\right ) y+x^{2}-3&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.615 |
|
| \begin{align*}
3 x \left (x^{2}-1\right ) y^{\prime }+x y^{2}-\left (x^{2}+1\right ) y-3 x&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
82.796 |
|
| \begin{align*}
\left (a \,x^{2}+b x +c \right ) \left (-y+y^{\prime } x \right )-y^{2}+x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.649 |
|
| \begin{align*}
x^{4} \left (y^{\prime }+y^{2}\right )+a&=0 \\
\end{align*} | [_rational, [_Riccati, _special]] | ✓ | ✓ | ✓ | ✓ | 3.875 |
|
| \begin{align*}
x \left (x^{3}-1\right ) y^{\prime }-2 x y^{2}+y+x^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.627 |
|
| \begin{align*}
x^{n} y^{\prime }+y^{2}-\left (n -1\right ) x^{n -1} y+x^{2 n -2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.556 |
|
| \begin{align*}
x^{n} y^{\prime }-a y^{2}-b \,x^{2 n -2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.928 |
|
| \begin{align*}
x \ln \left (x \right ) y^{\prime }-y^{2} \ln \left (x \right )-\left (2 \ln \left (x \right )^{2}+1\right ) y-\ln \left (x \right )^{3}&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.417 |
|
| \begin{align*}
\sin \left (x \right ) y^{\prime }-y^{2} \sin \left (x \right )^{2}+\left (\cos \left (x \right )-3 \sin \left (x \right )\right ) y+4&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.429 |
|
| \begin{align*}
2 f \left (x \right ) y^{\prime }+2 f \left (x \right ) y^{2}-y f^{\prime }\left (x \right )-2 f \left (x \right )^{2}&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.691 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (2 y \ln \left (x \right )-1\right )^{2}}{x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
63.523 |
|
| \begin{align*}
y^{\prime }&=\frac {y+x^{3} a \,{\mathrm e}^{x}+a \,x^{4}+a \,x^{3}-x y^{2} {\mathrm e}^{x}-y^{2} x^{2}-x y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
7.027 |
|
| \begin{align*}
y^{\prime }&=\frac {y+x^{3} a \ln \left (x +1\right )+a \,x^{4}+a \,x^{3}-x y^{2} \ln \left (x +1\right )-y^{2} x^{2}-x y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
9.237 |
|
| \begin{align*}
y^{\prime }&=\frac {y+x^{3} \ln \left (x \right )+x^{4}+x^{3}+7 x y^{2} \ln \left (x \right )+7 y^{2} x^{2}+7 x y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
8.836 |
|
| \begin{align*}
y^{\prime }&=\frac {y+x^{3} b \ln \left (\frac {1}{x}\right )+b \,x^{4}+b \,x^{3}+x a y^{2} \ln \left (\frac {1}{x}\right )+a \,x^{2} y^{2}+a x y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
11.129 |
|
| \begin{align*}
y^{\prime }&=\frac {y+\ln \left (\left (x -1\right ) \left (x +1\right )\right ) x^{3}+7 \ln \left (\left (x -1\right ) \left (x +1\right )\right ) x y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
9.810 |
|
| \begin{align*}
y^{\prime }&=\frac {y-\ln \left (\frac {x +1}{x -1}\right ) x^{3}+\ln \left (\frac {x +1}{x -1}\right ) x y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
13.322 |
|
| \begin{align*}
y^{\prime }&=\frac {y+{\mathrm e}^{\frac {x +1}{x -1}} x^{3}+{\mathrm e}^{\frac {x +1}{x -1}} x y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
8.989 |
|
| \begin{align*}
y^{\prime }&=\frac {y x -y-{\mathrm e}^{x +1} x^{3}+{\mathrm e}^{x +1} x y^{2}}{\left (x -1\right ) x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
9.589 |
|
| \begin{align*}
y^{\prime }&=\frac {y \ln \left (x -1\right )+x^{4}+x^{3}+y^{2} x^{2}+x y^{2}}{\ln \left (x -1\right ) x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
9.882 |
|
| \begin{align*}
y^{\prime }&=\frac {y \ln \left (x -1\right )+{\mathrm e}^{x +1} x^{3}+7 \,{\mathrm e}^{x +1} x y^{2}}{\ln \left (x -1\right ) x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
10.148 |
|
| \begin{align*}
y^{\prime }&=\frac {-{\mathrm e}^{x} y+y x -x^{3} \ln \left (x \right )-x^{3}-x y^{2} \ln \left (x \right )-x y^{2}}{\left (x -{\mathrm e}^{x}\right ) x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
9.355 |
|
| \begin{align*}
y^{\prime }&=\frac {x y \ln \left (x \right )-y+2 x^{5} b +2 a \,x^{3} y^{2}}{\left (x \ln \left (x \right )-1\right ) x} \\
\end{align*} | [[_homogeneous, ‘class D‘], _Riccati] | ✓ | ✓ | ✓ | ✗ | 10.147 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (18 x^{{3}/{2}}+36 y^{2}-12 x^{3} y+x^{6}\right ) \sqrt {x}}{36} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
10.546 |
|
| \begin{align*}
y^{\prime }&=\frac {2 x^{2}+2 x +x^{4}-2 x^{2} y-1+y^{2}}{x +1} \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
13.323 |
|
| \begin{align*}
y^{\prime }&=\frac {y \ln \left (x \right )+\cosh \left (x \right ) x a y^{2}+\cosh \left (x \right ) x^{3} b}{x \ln \left (x \right )} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
45.248 |
|
| \begin{align*}
y^{\prime }&=\frac {x +y+y^{2}-2 x y \ln \left (x \right )+\ln \left (x \right )^{2} x^{2}}{x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
11.627 |
|
| \begin{align*}
y^{\prime }&=\frac {x \left ({\mathrm e}^{-2 x^{2}} x^{4}-4 x^{2} {\mathrm e}^{-x^{2}} y-4 x^{2} {\mathrm e}^{-x^{2}}+4 y^{2}+4 \,{\mathrm e}^{-x^{2}}\right )}{4} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.590 |
|
| \begin{align*}
y^{\prime }&=\frac {30 x^{3}+25 \sqrt {x}+25 y^{2}-20 x^{3} y-100 \sqrt {x}\, y+4 x^{6}+40 x^{{7}/{2}}+100 x}{25 x} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.980 |
|
| \begin{align*}
y^{\prime }&=\frac {y+x^{2} \ln \left (x \right )^{3}+2 x^{2} \ln \left (x \right )^{2} y+x^{2} \ln \left (x \right ) y^{2}}{x \ln \left (x \right )} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
15.141 |
|
| \begin{align*}
y^{\prime }&=\frac {y+x^{3} \ln \left (x \right )^{3}+2 x^{3} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right ) y^{2}}{x \ln \left (x \right )} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
12.839 |
|
| \begin{align*}
y^{\prime }&=\frac {2 x y^{2}+4 y \ln \left (2 x +1\right ) x +2 \ln \left (2 x +1\right )^{2} x +y^{2}-2+\ln \left (2 x +1\right )^{2}+2 y \ln \left (2 x +1\right )}{2 x +1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
0.961 |
|
| \begin{align*}
y^{\prime }&=\frac {-2 \cos \left (x \right ) x +2 x^{2} \sin \left (x \right )+2 x +2 y^{2}+4 y \cos \left (x \right ) x -4 y x +\cos \left (2 x \right ) x^{2}+3 x^{2}-4 x^{2} \cos \left (x \right )}{2 x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.734 |
|
| \begin{align*}
y^{\prime }&=\frac {2 x^{2} \cos \left (x \right )+2 x^{3} \sin \left (x \right )-2 x \sin \left (x \right )+2 x +2 y^{2} x^{2}-4 x \sin \left (x \right ) y+4 y \cos \left (x \right ) x^{2}+4 y x +3-\cos \left (2 x \right )-2 \sin \left (2 x \right ) x -4 \sin \left (x \right )+\cos \left (2 x \right ) x^{2}+x^{2}+4 \cos \left (x \right ) x}{2 x^{3}} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
12.945 |
|
| \begin{align*}
y^{\prime }&=-F \left (x \right ) \left (-a \,x^{2}+y^{2}\right )+\frac {y}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
8.538 |
|
| \begin{align*}
y^{\prime }&=-F \left (x \right ) \left (y^{2}-2 y x -x^{2}\right )+\frac {y}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
10.454 |
|
| \begin{align*}
y^{\prime }&=-F \left (x \right ) \left (-a y^{2}-b \,x^{2}\right )+\frac {y}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
10.156 |
|
| \begin{align*}
y^{\prime }&=-F \left (x \right ) \left (-y^{2}+2 x^{2} y+1-x^{4}\right )+2 x \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
17.290 |
|
| \begin{align*}
y^{\prime }&=-F \left (x \right ) \left (x^{2}+2 y x -y^{2}\right )+\frac {y}{x} \\
\end{align*} | [[_homogeneous, ‘class D‘], _Riccati] | ✓ | ✓ | ✓ | ✗ | 8.858 |
|
| \begin{align*}
y^{\prime }&=-F \left (x \right ) \left (-7 x y^{2}-x^{3}\right )+\frac {y}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.974 |
|
| \begin{align*}
y^{\prime }&=-F \left (x \right ) \left (-y^{2}-2 y \ln \left (x \right )-\ln \left (x \right )^{2}\right )+\frac {y}{x \ln \left (x \right )} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
18.736 |
|
| \begin{align*}
y^{\prime }&=-x^{3} \left (-y^{2}-2 y \ln \left (x \right )-\ln \left (x \right )^{2}\right )+\frac {y}{x \ln \left (x \right )} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
13.730 |
|
| \begin{align*}
y^{\prime }&=\left (-{\mathrm e}^{x}+y\right )^{2}+{\mathrm e}^{x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
14.172 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (y-\operatorname {Si}\left (x \right )\right )^{2}+\sin \left (x \right )}{x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
14.185 |
|
| \begin{align*}
y^{\prime }&=\left (\cos \left (x \right )+y\right )^{2}+\sin \left (x \right ) \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
0.582 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (y-\ln \left (x \right )-\operatorname {Ci}\left (x \right )\right )^{2}+\cos \left (x \right )}{x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
70.760 |
|
| \begin{align*}
y^{\prime }&=\frac {2 x^{2} y+x^{3}+x y \ln \left (x \right )-y^{2}-y x}{x^{2} \left (\ln \left (x \right )+x \right )} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.231 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b x +c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.307 |
|
| \begin{align*}
y^{\prime }&=y^{2}-a^{2} x^{2}+3 a \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.990 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a^{2} x^{2}+b x +c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.208 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \,x^{n} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
40.608 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \,x^{2 n}+c \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.214 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{-n -2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.651 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
52.125 |
|
| \begin{align*}
y^{\prime }&=\left (a \,x^{2 n}+b \,x^{n -1}\right ) y^{2}+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
9.408 |
|
| \begin{align*}
\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} x +b_{0}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.622 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b \\
\end{align*} | [[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] | ✓ | ✓ | ✓ | ✓ | 4.262 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2} x^{2}-a^{2} x^{4}+a \left (1-2 b \right ) x^{2}-b \left (b +1\right ) \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.186 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b \,x^{n}+c \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
124.744 |
|
| \begin{align*}
\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
10.168 |
|
| \begin{align*}
x^{4} y^{\prime }&=-x^{4} y^{2}-a^{2} \\
\end{align*} |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
8.403 |
|
| \begin{align*}
a \,x^{2} \left (x -1\right )^{2} \left (y^{\prime }+\lambda y^{2}\right )+b \,x^{2}+c x +s&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✗ |
✗ |
3.232 |
|
| \begin{align*}
x^{n +1} y^{\prime }&=x^{2 n} a y^{2}+c \,x^{m}+d \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
106.598 |
|
| \begin{align*}
\left (a \,x^{n}+b \right ) y^{\prime }&=b y^{2}+a \,x^{-2+n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
8.760 |
|
| \begin{align*}
\left (a \,x^{n}+b \,x^{m}+c \right ) \left (y^{\prime }-y^{2}\right )+a n \left (n -1\right ) x^{-2+n}+b m \left (m -1\right ) x^{m -2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.135 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b y+c x +k \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
38.793 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,x^{n} y+a \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.980 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,x^{n} y+b \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.165 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\left (x \alpha +\beta \right ) y+a \,x^{2}+b x +c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.136 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}-a \,x^{n} \left (b \,x^{m}+c \right ) y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✓ |
✗ |
6.153 |
|
| \begin{align*}
y^{\prime }&=-a n \,x^{n -1} y^{2}+c \,x^{m} \left (a \,x^{n}+b \right ) y-c \,x^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.751 |
|
| \begin{align*}
y^{\prime } x&=a y^{2}+b y+c \,x^{2 b} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.905 |
|
| \begin{align*}
y^{\prime } x&=a y^{2}+b y+c \,x^{n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
32.911 |
|
| \begin{align*}
y^{\prime } x&=a y^{2}+\left (n +b \,x^{n}\right ) y+c \,x^{2 n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.819 |
|
| \begin{align*}
y^{\prime } x&=x y^{2}+a y+b \,x^{n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.731 |
|
| \begin{align*}
y^{\prime } x +a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0}&=0 \\
\end{align*} | [_rational, _Riccati] | ✓ | ✓ | ✓ | ✗ | 37.207 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n} y^{2}+b y+c \,x^{-n} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.484 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n} y^{2}+m y-a \,b^{2} x^{n +2 m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.184 |
|
| \begin{align*}
y^{\prime } x&=x^{2 n} y^{2}+\left (m -n \right ) y+x^{2 m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.376 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n} y^{2}+b y+c \,x^{m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.742 |
|
| \begin{align*}
y^{\prime } x&=x^{2 n} a y^{2}+\left (b \,x^{n}-n \right ) y+c \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.228 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{2 n +m} y^{2}+\left (b \,x^{n +m}-n \right ) y+c \,x^{m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
38.549 |
|
| \begin{align*}
\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (a_{1} x +b_{1} \right ) y+a_{0} x +b_{0}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
11.099 |
|
| \begin{align*}
\left (a x +c \right ) y^{\prime }&=\alpha \left (b x +a y\right )^{2}+\beta \left (b x +a y\right )-b x +\gamma \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.238 |
|
| \begin{align*}
2 x^{2} y^{\prime }&=2 y^{2}+y x -2 a^{2} x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.224 |
|
| \begin{align*}
2 x^{2} y^{\prime }&=2 y^{2}+3 y x -2 a^{2} x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.792 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.981 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \,x^{n}+s \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.941 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \,x^{2 n}+s \,x^{n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.087 |
|
| \begin{align*}
x^{2} y^{\prime }&=\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
18.927 |
|
| \begin{align*}
\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha }&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
437.849 |
|
| \begin{align*}
\left (a \,x^{2}+b \right ) y^{\prime }+y^{2}-2 y x +\left (-a +1\right ) x^{2}-b&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.616 |
|
| \begin{align*}
\left (a \,x^{2}+b x +c \right ) y^{\prime }&=y^{2}+\left (2 \lambda x +b \right ) y+\lambda \left (\lambda -a \right ) x^{2}+\mu \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
9.829 |
|
| \begin{align*}
\left (a \,x^{2}+b x +c \right ) y^{\prime }&=y^{2}+\left (a x +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +\lambda c \\
\end{align*} | [_rational, _Riccati] | ✓ | ✓ | ✓ | ✗ | 73.343 |
|
| \begin{align*}
\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime }&=y^{2}+\left (a_{1} x +b_{1} \right ) y-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
94.498 |
|
| \begin{align*}
\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime }&=y^{2}+\left (a_{1} x +b_{1} \right ) y+a_{0} x^{2}+b_{0} x +c_{0} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✗ |
✗ |
85.471 |
|
| \begin{align*}
\left (x -a \right ) \left (-b +x \right ) y^{\prime }+k \left (x +y-a \right ) \left (x +y-b \right )+y^{2}&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
16.976 |
|
| \begin{align*}
\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b_{1} x +a_{1} \right ) y+a_{0}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
30.819 |
|
| \begin{align*}
x^{3} y^{\prime }&=a \,x^{3} y^{2}+\left (b \,x^{2}+c \right ) y+s x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
76.926 |
|
| \begin{align*}
x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y+s x&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.523 |
|
| \begin{align*}
x^{2} \left (x +a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y+x \alpha +\beta &=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
7.981 |
|
| \begin{align*}
\left (a \,x^{2}+b x +e \right ) \left (-y+y^{\prime } x \right )-y^{2}+x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.539 |
|
| \begin{align*}
x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b \,x^{2}+c \right ) y+s&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
8.533 |
|
| \begin{align*}
x^{n +1} y^{\prime }&=x^{2 n} a y^{2}+b \,x^{n} y+c \,x^{m}+d \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.634 |
|
| \begin{align*}
x \left (a \,x^{k}+b \right ) y^{\prime }&=\alpha \,x^{n} y^{2}+\left (\beta -a n \,x^{k}\right ) y+\gamma \,x^{-n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
74.086 |
|
| \begin{align*}
x^{2} \left (a \,x^{n}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (p \,x^{n}+q \right ) x y+r \,x^{n}+s&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
10.117 |
|
| \begin{align*}
\left (a \,x^{n}+b \,x^{m}+c \right ) \left (-y+y^{\prime } x \right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right )&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
45.987 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
31.385 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.431 |
|
| \begin{align*}
y^{\prime }&=\sigma y^{2}+a y+b \,{\mathrm e}^{x}+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
33.744 |
|
| \begin{align*}
y^{\prime }&=y^{2}+b y+a \left (\lambda -b \right ) {\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.978 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.192 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{k x} y^{2}+b \,{\mathrm e}^{s x} \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 1.648 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b y+c \,{\mathrm e}^{-\lambda x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
92.642 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\mu x} y^{2}+\lambda y-a \,b^{2} {\mathrm e}^{\left (2 \lambda +\mu \right ) x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.636 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y+a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.678 |
|
| \begin{align*}
y^{\prime }&=-\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y-a \,{\mathrm e}^{\left (\mu -\lambda \right ) x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.191 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-k x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.435 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\left (2 \lambda +\mu \right ) x} y^{2}+\left (b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-\lambda \right ) y+c \,{\mathrm e}^{\mu x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.964 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{\mu x} \left (y-b \,{\mathrm e}^{\lambda x}\right )^{2}+b \lambda \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.011 |
|
| \begin{align*}
\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 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.894 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \,{\mathrm e}^{\lambda x} y+a \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.688 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b \,{\mathrm e}^{-\lambda x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.164 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
6.723 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.000 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.585 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.502 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
4.042 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.312 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.684 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}+\lambda x y+b^{2} a \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 4.931 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.033 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
2.331 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.074 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
8.477 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.816 |
|
| \begin{align*}
y^{\prime }&=\lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
6.521 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
55.186 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
12.508 |
|
| \begin{align*}
y^{\prime }&=\alpha y^{2}+\beta +\gamma \cosh \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
62.033 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.646 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.998 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
46.502 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
57.902 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.270 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
36.842 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.401 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.704 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
3.447 |
|
| \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] | ✓ | ✓ | ✓ | ✗ | 19.509 |
|
| \begin{align*}
y^{\prime } x&=a y^{2}+b \ln \left (x \right )+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
42.611 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
4.759 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
1.530 |
|
| \begin{align*}
x^{2} \ln \left (a x \right ) \left (y^{\prime }-y^{2}\right )&=1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
54.706 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
4.927 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.013 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
2.688 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
8.562 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.698 |
|
| \begin{align*}
y^{\prime } x&=\left (a y+b \ln \left (x \right )\right )^{2} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
36.909 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
40.046 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
102.704 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.308 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2} a^{2} x^{2}-y x +b^{2} \ln \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
127.203 |
|
| \begin{align*}
y^{\prime }&=\alpha y^{2}+\beta +\gamma \sin \left (\lambda x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
20.423 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.816 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.251 |
|
| \begin{align*}
y^{\prime }&=\lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3} \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✗ | ✗ | 15.309 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
83.845 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
26.421 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
62.165 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
62.431 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
89.113 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
14.253 |
|
| \begin{align*}
y^{\prime }&=\alpha y^{2}+\beta +\gamma \cos \left (\lambda x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
58.337 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.586 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.894 |
|
| \begin{align*}
y^{\prime }&=\lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
46.104 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
71.203 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
26.478 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
55.142 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
45.682 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
97.864 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
94.211 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \tan \left (x \right ) y+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.197 |
|
| \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] | ✓ | ✓ | ✓ | ✓ | 1.759 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
23.769 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
33.868 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
48.358 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
55.569 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
158.418 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
13.837 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
35.927 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
67.869 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
159.813 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
17.801 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
18.370 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
62.855 |
|
| \begin{align*}
y^{\prime }&=y^{2}-\tan \left (x \right ) y+a \left (-a +1\right ) \cot \left (x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.298 |
|
| \begin{align*}
y^{\prime }&=y^{2}-m y \tan \left (x \right )+b^{2} \cos \left (x \right )^{2 m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.984 |
|
| \begin{align*}
y^{\prime }&=y^{2}+m y \cot \left (x \right )+b^{2} \sin \left (x \right )^{2 m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.979 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
17.040 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
20.542 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
9.769 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arcsin \left (x \right )^{n} y+\arcsin \left (x \right )^{n} \lambda \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 49.269 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
72.003 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
68.152 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arccos \left (x \right )^{n} y+\arccos \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
47.253 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
86.053 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
133.343 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
10.141 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arctan \left (x \right )^{n} y+\arctan \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
10.454 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
71.620 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
62.167 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.562 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.826 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
92.391 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
70.153 |
|
| \begin{align*}
y^{\prime }&=y^{2}+f \left (x \right ) y-a^{2}-a f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.205 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a y-a b -b^{2} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
8.105 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right )+x f \left (x \right ) y+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.423 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
8.852 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
25.589 |
|
| \begin{align*}
y^{\prime } x&=f \left (x \right ) y^{2}+n y+a \,x^{2 n} f \left (x \right ) \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 7.282 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
8.088 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
9.547 |
|
| \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] |
✓ |
✗ |
✓ |
✗ |
10.848 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.147 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
24.825 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
25.233 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
51.138 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
96.381 |
|
| \begin{align*}
y^{\prime }&=y^{2}-f \left (x \right )^{2}+f^{\prime }\left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
6.226 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
46.905 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
47.124 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
68.117 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.884 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
1.474 |
|
| \begin{align*}
y^{\prime }&=y^{2}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.525 |
|
| \begin{align*}
-y+y^{\prime } x&=y^{2}+x^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.050 |
|
| \begin{align*}
y^{\prime } x -a y+b y^{2}&=c \,x^{2 a} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.734 |
|
| \begin{align*}
y^{\prime }+2 y x&=y^{2}+x^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.283 |
|
| \begin{align*}
x^{\prime }&=t^{2}+x^{2} \\
\end{align*} | [[_Riccati, _special]] | ✓ | ✓ | ✓ | ✗ | 39.881 |
|
| \begin{align*}
R^{\prime }&=\left (t +1\right ) \left (1+R^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.244 |
|
| \begin{align*}
x^{\prime }&=\left (4 t -x\right )^{2} \\
x \left (0\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.206 |
|
| \begin{align*}
T^{\prime }&=2 a t \left (T^{2}-a^{2}\right ) \\
T \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✗ |
9.231 |
|
| \begin{align*}
x^{\prime }&=t -x^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
4.858 |
|
| \begin{align*}
x^{\prime }&=\left (t +x\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.063 |
|
| \begin{align*}
2 r \left (s^{2}+1\right )+\left (r^{4}+1\right ) s^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.829 |
|
| \begin{align*}
y^{\prime }&=\left (1-x \right ) y^{2}+\left (2 x -1\right ) y-x \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
2.967 |
|
| \begin{align*}
y^{\prime }&=-y^{2}+y x +1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.678 |
|
| \begin{align*}
y^{\prime }&=-8 x y^{2}+4 x \left (4 x +1\right ) y-8 x^{3}-4 x^{2}+1 \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.174 |
|
| \begin{align*}
2 x^{2}+y x +y^{2}+2 x^{2} y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
10.933 |
|
| \begin{align*}
y^{\prime }&=\left (1+y^{2}\right ) \tan \left (x \right ) \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.947 |
|
| \begin{align*}
y x +y^{2}+x^{2}-x^{2} y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.685 |
|
| \begin{align*}
y^{\prime }&=x +y^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
17.819 |
|
| \begin{align*}
y^{\prime }&=x^{2}-y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.105 |
|
| \begin{align*}
y^{\prime }&=x -y^{2} \\
y \left (1\right ) &= 0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
20.332 |
|
| \begin{align*}
x^{2} y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
1.861 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.228 |
|
| \begin{align*}
1+s^{2}-\sqrt {t}\, s^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.490 |
|
| \begin{align*}
y^{\prime }&=x +y^{2} \\
y \left (0\right ) &= 1 \\
\end{align*} | [[_Riccati, _special]] | ✓ | ✓ | ✓ | ✗ | 3.086 |
|
| \begin{align*}
y^{\prime }&=x^{2}-y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.023 |
|
| \begin{align*}
y^{\prime }&=y^{2}-x^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.477 |
|
| \begin{align*}
y^{\prime }&=\left (1+y^{2}\right ) t \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
3.266 |
|
| \begin{align*}
y^{\prime }&=\left (y+\frac {1}{2}\right ) \left (t +y\right ) \\
y \left (0\right ) &= {\frac {1}{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.734 |
|
| \begin{align*}
y^{\prime }&=t -y^{2} \\
y \left (0\right ) &= 1 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
5.294 |
|
| \begin{align*}
y^{\prime }&=y^{2}-4 t \\
y \left (0\right ) &= {\frac {1}{2}} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
5.847 |
|
| \begin{align*}
y^{\prime }&=\left (y-2\right ) \left (y+1-\cos \left (t \right )\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
6.017 |
|
| \begin{align*}
x^{2} y^{\prime }+x y^{2}&=x \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.881 |
|
| \begin{align*}
y^{\prime }-y^{2}&=x \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
2.532 |
|
| \begin{align*}
y^{\prime }+\left (8-x \right ) y-y^{2}&=-8 x \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.500 |
|
| \begin{align*}
y^{\prime } x&=\left (x -y\right )^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
72.385 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.051 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.786 |
|
| \begin{align*}
y^{\prime }-3 y^{2} x^{2}&=-3 x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.448 |
|
| \begin{align*}
y^{\prime }-3 y^{2} x^{2}&=3 x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.313 |
|
| \begin{align*}
y^{\prime }-x y^{2}&=\sqrt {x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.250 |
|
| \begin{align*}
y^{\prime }&=1+\left (y x +3 y\right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.977 |
|
| \begin{align*}
y^{\prime }&=1+\left (y-x \right )^{2} \\
y \left (0\right ) &= {\frac {1}{4}} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.362 |
|
| \begin{align*}
y^{\prime }&=\left (x -y+3\right )^{2} \\
\end{align*} | [[_homogeneous, ‘class C‘], _Riccati] | ✓ | ✓ | ✓ | ✓ | 2.419 |
|
| \begin{align*}
y^{\prime }&=x \left (1+\frac {2 y}{x^{2}}+\frac {y^{2}}{x^{4}}\right ) \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.352 |
|
| \begin{align*}
y^{\prime }&=x^{2}-2 y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.224 |
|
| \begin{align*}
y^{\prime }&=x y^{2}+3 y^{2}+x +3 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.694 |
|
| \begin{align*}
y^{\prime }&=4 t^{2}-t y^{2} \\
y \left (2\right ) &= 1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.529 |
|
| \begin{align*}
y^{\prime }&=t^{2} y^{2}+y^{2}-t^{2}-1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✗ |
3.358 |
|
| \begin{align*}
4 \left (x -1\right )^{2} y^{\prime }-3 \left (3+y\right )^{2}&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.398 |
|
| \begin{align*}
y^{\prime }&=\left (x +y-4\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.576 |
|
| \begin{align*}
y^{\prime }&=y^{2}-x \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
22.806 |
|
| \begin{align*}
y^{\prime }&=y^{2}+x^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
45.764 |
|
| \begin{align*}
y^{\prime }&=x^{2}-y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.887 |
|
| \begin{align*}
y^{\prime }&=x^{2}-y^{2} \\
y \left (-1\right ) &= 0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.106 |
|
| \begin{align*}
y^{\prime }&=x +y^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
128.487 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.691 |
|
| \begin{align*}
1+y^{2}&=y^{\prime } x \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.924 |
|
| \begin{align*}
x^{2} y^{\prime }&=x^{2}-y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.802 |
|
| \begin{align*}
2 x^{2} y^{\prime }&=y^{2}+x^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.651 |
|
| \begin{align*}
{\mathrm e}^{-x} y^{\prime }+y^{2}-2 \,{\mathrm e}^{x} y&=1-{\mathrm e}^{2 x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
155.714 |
|
| \begin{align*}
y^{\prime }+y^{2}-2 y \sin \left (x \right )+\sin \left (x \right )^{2}-\cos \left (x \right )&=0 \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
0.447 |
|
| \begin{align*}
y^{\prime } x -y^{2}+\left (2 x +1\right ) y&=x^{2}+2 x \\
\end{align*} | [[_1st_order, _with_linear_symmetries], _rational, _Riccati] | ✓ | ✓ | ✓ | ✓ | 1.987 |
|
| \begin{align*}
x^{2} y^{\prime }&=1+y x +y^{2} x^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.953 |
|
| \begin{align*}
y^{\prime }&=\left (x -y\right )^{2}+1 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.154 |
|
| \begin{align*}
y^{\prime }&=\left (1+y^{2}\right ) \tan \left (2 x \right ) \\
y \left (0\right ) &= -\sqrt {3} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
5.576 |
|
| \begin{align*}
y^{\prime }&=2 \left (x +1\right ) \left (1+y^{2}\right ) \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
2.936 |
|
| \begin{align*}
y^{\prime }+3 t y&=4-4 t^{2}+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.541 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}}{3}+\frac {2}{3 x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
2.378 |
|
| \begin{align*}
y^{\prime }+y^{2}+\frac {y}{x}-\frac {4}{x^{2}}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.900 |
|
| \begin{align*}
y^{\prime } x -3 y+y^{2}&=4 x^{2}-4 x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.848 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\frac {1}{x^{4}} \\
\end{align*} |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
3.029 |
|
| \begin{align*}
y^{\prime }&=y^{2}-x^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.468 |
|
| \begin{align*}
y^{\prime } x&=y+y^{2}+x^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.388 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.800 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.930 |
|
| \begin{align*}
1&=\frac {y}{1-y^{2} x^{2}}+\frac {x y^{\prime }}{1-y^{2} x^{2}} \\
\end{align*} |
[_exact, _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
25.913 |
|
| \begin{align*}
\frac {y^{\prime } x +y}{1-y^{2} x^{2}}+x&=0 \\
\end{align*} |
[_exact, _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
26.764 |
|
| \begin{align*}
y^{\prime } x&=x^{5}+x^{3} y^{2}+y \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.220 |
|
| \begin{align*}
y^{\prime } x&=y+x^{2}+9 y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.247 |
|
| \begin{align*}
y^{\prime }&=1+\frac {y}{x}-\frac {y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
23.794 |
|
| \begin{align*}
x^{2} y^{\prime }&=x^{2}+y x +y^{2} \\
\end{align*} | [[_homogeneous, ‘class A‘], _rational, _Riccati] | ✓ | ✓ | ✓ | ✓ | 9.085 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.274 |
|
| \begin{align*}
y^{\prime }&=x \left (a y^{2}+b \right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
9.432 |
|
| \begin{align*}
n^{\prime }&=\left (n^{2}+1\right ) x \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.446 |
|
| \begin{align*}
3 x^{2} y^{\prime }+2 x^{2}-3 y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
10.801 |
|
| \begin{align*}
y^{\prime }+2 y x&=y^{2}+x^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.429 |
|
| \begin{align*}
y^{\prime }&=\left (4 x +y+1\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.140 |
|
| \begin{align*}
y^{\prime } x -y^{2}+\left (2 x +1\right ) y&=x^{2}+2 x \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
9.926 |
|
| \begin{align*}
{\mathrm e}^{-x} y^{\prime }+y^{2}-2 \,{\mathrm e}^{x} y&=1-{\mathrm e}^{2 x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
279.825 |
|
| \begin{align*}
y^{\prime }&=\left (x -y+3\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
9.120 |
|
| \begin{align*}
y^{\prime }-y+y^{2} {\mathrm e}^{x}+5 \,{\mathrm e}^{-x}&=0 \\
y \left (0\right ) &= \eta \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
14.710 |
|
| \begin{align*}
x^{\prime }&=t +x^{2} \\
x \left (0\right ) &= 0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
24.842 |
|
| \begin{align*}
x^{\prime }&={\mathrm e}^{t} \left (1+x^{2}\right ) \\
x \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.720 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
8.797 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
10.889 |
|
| \begin{align*}
y^{\prime }&=-\frac {2+x}{x \left (x +1\right )^{2}}-\frac {\left (-x^{2}+x +2\right ) y}{x \left (x +1\right )}+\left (x +1\right ) y^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
177.970 |
|
| \begin{align*}
y^{\prime }&=-\frac {2+x}{x \left (x +1\right )^{2}}-\frac {\left (-x^{2}+x +2\right ) y}{x \left (x +1\right )}+\left (x +1\right ) y^{2} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
130.049 |
|
| \begin{align*}
y^{\prime }&=1-y+y^{2} {\mathrm e}^{2 x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.840 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{2 x}+\left (2+\frac {5 \,{\mathrm e}^{x}}{2}\right ) y+y^{2} \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 118.375 |
|
| \begin{align*}
y^{\prime }&=-x^{2}-x -1-\left (2 x +1\right ) y-y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
12.020 |
|
| \begin{align*}
y^{\prime }&=\frac {y+y^{2}+x^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
22.816 |
|
| \begin{align*}
y^{\prime }+x \left (y-x \right )+x^{3} \left (y-x \right )^{2}&=1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
13.724 |
|
| \begin{align*}
1+y+y^{2}+x \left (x^{2}-4\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
38.114 |
|
| \begin{align*}
x^{2} y^{\prime }-y x&=x^{2}-y^{2} \\
y \left (1\right ) &= 0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
36.465 |
|
| \begin{align*}
y^{\prime }&=x +y^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
2.126 |
|
| \begin{align*}
x^{2}+y+y^{2}-y^{\prime } x&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.062 |
|
| \begin{align*}
y+x^{3}+x y^{2}-y^{\prime } x&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.902 |
|
| \begin{align*}
y^{\prime } x&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.413 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.276 |
|
| \begin{align*}
y+x^{3}+x y^{2}-y^{\prime } x&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.337 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (3+y\right )^{2}}{4 x^{2}} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.558 |
|
| \begin{align*}
y^{\prime }&=1-\left (x -y\right )^{2} \\
y \left (0\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.174 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
2.491 |
|
| \begin{align*}
y^{\prime }&=x y^{2}-2 y+4-4 x \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
1.800 |
|
| \begin{align*}
y^{\prime }+y^{2}&=x^{2}+1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.461 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}}{x -1}-\frac {x y}{x -1}+1 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.826 |
|
| \begin{align*}
y^{\prime } x&=y^{2} x^{2}-y+1 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.041 |
|
| \begin{align*}
1+y^{2}&=\left (x^{2}+1\right ) y^{\prime } \\
\end{align*} | [_separable] | ✓ | ✓ | ✓ | ✓ | 2.084 |
|
| \begin{align*}
y^{\prime }&=\left (1-y\right ) \left (\frac {1}{t}-\frac {1}{10}+\frac {y}{10}\right ) \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.976 |
|
| \begin{align*}
y^{\prime }&=\left (1-y\right ) \left (-\frac {1}{t \ln \left (t \right )}-\frac {3}{100}+\frac {3 y}{100}\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
2.988 |
|
| \begin{align*}
y^{\prime }&=y^{2}+x^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
43.588 |
|
| \begin{align*}
y^{\prime }&=1-\frac {y^{2}}{x} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.365 |
|
| \begin{align*}
y^{2}+7 y x +16 x^{2}+x^{2} y^{\prime }&=0 \\
y \left (1\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.745 |
|
| \begin{align*}
a^{2} \left (y^{\prime }-1\right )&=x^{2} y^{\prime }+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.073 |
|
| \begin{align*}
y^{\prime }&=\left (9 x +4 y+1\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
32.637 |
|
| \begin{align*}
y^{\prime }&=2 \left (3 x +y\right )^{2}-1 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
31.937 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}-4 t y+6 t^{2}}{t^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.996 |
|
| \begin{align*}
y^{\prime }&=t y^{2}-y^{2}+t -1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.387 |
|
| \begin{align*}
y^{\prime }&=t^{2}+y^{2} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
37.593 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{t} \\
y \left (1\right ) &= \sqrt {3} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.496 |
|
| \begin{align*}
t^{2} y^{\prime }&=y^{2}+t y+t^{2} \\
y \left (1\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.209 |
|
| \begin{align*}
y^{\prime }&=\left (t -y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
0.990 |
|
| \begin{align*}
y^{\prime }&=t +y^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
25.266 |
|
| \begin{align*}
y^{\prime }&=y^{2}-t \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
2.280 |
|
| \begin{align*}
y^{2}-1+y^{\prime } x&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.868 |
|
| \begin{align*}
y^{\prime }&=y^{2}+x^{2} \\
y \left (1\right ) &= -1 \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
13.135 |
|
| \begin{align*}
y^{\prime }&=x^{2}-y^{2} \\
y \left (-2\right ) &= 1 \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 7.864 |
|
| \begin{align*}
y^{\prime }&=x^{2}-y^{2} \\
y \left (3\right ) &= 0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.389 |
|
| \begin{align*}
y^{\prime }&=x \left (y-4\right )^{2}-2 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
85.678 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (3+2 y\right )^{2}}{\left (5+4 x \right )^{2}} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.658 |
|