| # |
ODE |
CAS classification |
Solved |
Maple |
Mma |
Sympy |
time(sec) |
| \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*}
\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 }&=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*}
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} \\
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*}
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*}
\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*}
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*}
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 }+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 }&=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 }&=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 }+\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*}
y^{\prime } x&=x^{2}+y \left (1+y\right ) \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.198 |
|
| \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&=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 +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&=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*}
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 }+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 y+c \,x^{2} y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.303 |
|
| \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 -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*}
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 (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*}
\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 -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^{3}+\frac {2 y}{x}-\frac {y^{2}}{x} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.756 |
|
| \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 }&=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 }&=\frac {\left (3+2 y\right )^{2}}{\left (5+4 x \right )^{2}} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.928 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}-1}{x^{2}-1} \\
y \left (2\right ) &= 2 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
3.093 |
|
| \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+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*}
\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 }+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 }+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 }-a \,x^{n} \left (1+y^{2}\right )&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.663 |
|
| \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 -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 +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 +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 +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 )+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*}
\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*}
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*}
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*}
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*}
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*}
\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 \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*}
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 {2 x \,{\mathrm e}^{x}-2 x -\ln \left (x \right )-1+x^{4} \ln \left (x \right )+x^{4}-2 y \ln \left (x \right ) x^{2}-2 x^{2} y+y^{2} \ln \left (x \right )+y^{2}}{{\mathrm e}^{x}-1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
21.471 |
|
| \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 {-\ln \left (x \right )+2 \ln \left (2 x \right ) x y+\ln \left (2 x \right )+\ln \left (2 x \right ) y^{2}+\ln \left (2 x \right ) x^{2}}{\ln \left (x \right )} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
16.125 |
|
| \begin{align*}
y^{\prime }&=\frac {2 x \sin \left (x \right )-\ln \left (2 x \right )+\ln \left (2 x \right ) x^{4}-2 \ln \left (2 x \right ) x^{2} y+\ln \left (2 x \right ) y^{2}}{\sin \left (x \right )} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.082 |
|
| \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 {-\sinh \left (x \right )+\ln \left (x \right ) x^{2}+2 x y \ln \left (x \right )+\ln \left (x \right )+y^{2} \ln \left (x \right )}{\sinh \left (x \right )} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
34.141 |
|
| \begin{align*}
y^{\prime }&=-\frac {\ln \left (x \right )-\sinh \left (x \right ) x^{2}-2 \sinh \left (x \right ) x y-\sinh \left (x \right )-\sinh \left (x \right ) y^{2}}{\ln \left (x \right )} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
75.561 |
|
| \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 {2 x \ln \left (\frac {1}{x -1}\right )-\coth \left (\frac {x +1}{x -1}\right )+\coth \left (\frac {x +1}{x -1}\right ) y^{2}-2 \coth \left (\frac {x +1}{x -1}\right ) x^{2} y+\coth \left (\frac {x +1}{x -1}\right ) x^{4}}{\ln \left (\frac {1}{x -1}\right )} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✗ |
✓ |
✗ |
124.306 |
|
| \begin{align*}
y^{\prime }&=\frac {2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-1+y^{2}-2 x^{2} y+x^{4}-x +x y^{2}-2 x^{3} y+x^{5}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
17.259 |
|
| \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 {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 }&=-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 (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 }&=-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 {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 }&=y^{2}-a^{2} x^{2}+3 a \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.990 |
|
| \begin{align*}
y^{\prime }&=y^{2}+x^{n -1} a n -a^{2} x^{2 n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
59.898 |
|
| \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 m \,x^{m -1}-a \,b^{2} x^{n +2 m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
64.591 |
|
| \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*}
\left (a \,x^{n}+b \right ) y^{\prime }&=b y^{2}+a \,x^{-2+n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
8.760 |
|
| \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-a b \,x^{n}-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.224 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} y+x^{m} b c -a \,c^{2} x^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.318 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} y+c k \,x^{-1+k}-b c \,x^{m +k}-a \,c^{2} x^{n +2 k} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
11.565 |
|
| \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}+\left (n +b \,x^{n}\right ) y+c \,x^{2 n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.819 |
|
| \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&=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 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*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.981 |
|
| \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 (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 (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 \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*}
\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }&=c y^{2}-b \,x^{m -1} y+a \,x^{-2+n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✗ |
✗ |
✗ |
1.888 |
|
| \begin{align*}
\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }&=a \,x^{-2+n} y^{2}+b \,x^{m -1} y+c \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✗ |
✗ |
✗ |
97.312 |
|
| \begin{align*}
\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }&=\alpha \,x^{k} y^{2}+\beta \,x^{s} y-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.169 |
|
| \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 }&=y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.431 |
|
| \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}^{\lambda x} y-a b \,{\mathrm e}^{\lambda x}-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.533 |
|
| \begin{align*}
y^{\prime }&=b \,{\mathrm e}^{\mu x} y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (2 \lambda +\mu \right ) x} \\
\end{align*} | [_Riccati] | ✓ | ✗ | ✓ | ✗ | 90.135 |
|
| \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*}
\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }&=y^{2}+k \,{\mathrm e}^{\nu x} y-m^{2}+k m \,{\mathrm e}^{\nu x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
28.092 |
|
| \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 }&=a \,{\mathrm e}^{\lambda x} y^{2}+b n \,x^{n -1}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
7.940 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
128.510 |
|
| \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}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
102.967 |
|
| \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 \,{\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 } x&=a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y+c \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
13.606 |
|
| \begin{align*}
y^{\prime }&=y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
163.724 |
|
| \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*}
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*}
\left (\sinh \left (\lambda x \right ) a +b \right ) y^{\prime }&=y^{2}+c \sinh \left (\mu x \right ) y-d^{2}+c d \sinh \left (\mu x \right ) \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 107.322 |
|
| \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*}
\left (a \cosh \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \cosh \left (\mu x \right ) y-d^{2}+c d \cosh \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
110.319 |
|
| \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*}
\left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \tanh \left (\mu x \right ) y-d^{2}+c d \tanh \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
233.962 |
|
| \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*}
\left (a \coth \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
458.147 |
|
| \begin{align*}
y^{\prime }&=a \ln \left (x \right )^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
11.097 |
|
| \begin{align*}
y^{\prime } x&=x y^{2}-a^{2} x \ln \left (\beta x \right )^{2}+a \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
241.592 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n} y^{2}+b -a \,b^{2} x^{n} \ln \left (x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
94.418 |
|
| \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 }&=-\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 )^{n} y^{2}+b \ln \left (x \right )^{m} y+b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.161 |
|
| \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*}
\left (a \ln \left (x \right )+b \right ) y^{\prime }&=y^{2}+c \ln \left (x \right )^{n} y-\lambda ^{2}+\lambda c \ln \left (x \right )^{n} \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 9.664 |
|
| \begin{align*}
\left (a \ln \left (x \right )+b \right ) y^{\prime }&=\ln \left (x \right )^{n} y^{2}+c y-\lambda ^{2} \ln \left (x \right )^{n}+\lambda c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
13.454 |
|
| \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 }&=-\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 } 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 ) y^{\prime }&=y^{2}+c \sin \left (\mu x \right ) y-d^{2}+c d \sin \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.542 |
|
| \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 }&=-\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 } 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 ) y^{\prime }&=y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.615 |
|
| \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 } 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*}
\left (a \tan \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+k \tan \left (\mu x \right ) y-d^{2}+k d \tan \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
94.793 |
|
| \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*}
\left (a \cot \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right ) \\
\end{align*} | [_Riccati] | ✓ | ✓ | ✓ | ✗ | 133.615 |
|
| \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 }&=\lambda \arcsin \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
13.540 |
|
| \begin{align*}
y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arcsin \left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
47.924 |
|
| \begin{align*}
y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
87.761 |
|
| \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 \arccos \left (x \right )^{n} y-a^{2}+a \lambda \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
10.638 |
|
| \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 }&=\lambda \arccos \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
18.486 |
|
| \begin{align*}
y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arccos \left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
84.398 |
|
| \begin{align*}
y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
76.574 |
|
| \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 }&=\lambda \arctan \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arctan \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.353 |
|
| \begin{align*}
y^{\prime }&=\lambda \arctan \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arctan \left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} | [_Riccati] | ✓ | ✗ | ✗ | ✗ | 78.757 |
|
| \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 }&=\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \operatorname {arccot}\left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
15.819 |
|
| \begin{align*}
y^{\prime }&=\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}-b \lambda \,x^{m} \operatorname {arccot}\left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
124.488 |
|
| \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 }&=f \left (x \right ) y^{2}+x^{n -1} a n -a^{2} x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
23.670 |
|
| \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 } x&=x^{2 n} f \left (x \right ) y^{2}+\left (a \,x^{n} f \left (x \right )-n \right ) y+b f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
24.010 |
|
| \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 }&=f \left (x \right ) y^{2}+g \left (x \right ) y+x^{n -1} a n -a \,x^{n} g \left (x \right )-a^{2} x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
88.035 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,x^{n} g \left (x \right ) y+x^{n -1} a n +a^{2} x^{2 n} \left (g \left (x \right )-f \left (x \right )\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
31.048 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \\
\end{align*} | [_Riccati] | ✓ | ✗ | ✗ | ✗ | 9.740 |
|
| \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 }&={\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 }&=f \left (x \right ) y^{2}+g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{\lambda x} g \left (x \right )-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
11.164 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,{\mathrm e}^{\lambda x} g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}+a^{2} {\mathrm e}^{2 \lambda x} \left (g \left (x \right )-f \left (x \right )\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
314.335 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} f \left (x \right ) {\mathrm e}^{2 \lambda \,x^{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
83.919 |
|
| \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*}
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 }&=\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*}
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 }&=\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 }&=\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 }+\left (8-x \right ) y-y^{2}&=-8 x \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.500 |
|
| \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 }&=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 }&=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*}
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*}
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 } 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*}
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*}
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 }&={\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 }&=-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*}
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*}
\left (-x^{2}+1\right ) y^{\prime }&=1-y^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.397 |
|
| \begin{align*}
y^{\prime } x +y^{2}&=1 \\
y \left (-2\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.692 |
|
| \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^{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 }&=\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 }&=\frac {y^{2}-4 t y+6 t^{2}}{t^{2}} \\
y \left (2\right ) &= 4 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.321 |
|
| \begin{align*}
y^{\prime }&=\left (t -y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
0.990 |
|
| \begin{align*}
y^{\prime }&=1+\left (t -y\right )^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.471 |
|
| \begin{align*}
y^{2}-1+y^{\prime } x&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.868 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (3+2 y\right )^{2}}{\left (5+4 x \right )^{2}} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.658 |
|