| # |
ODE |
CAS classification |
Solved |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
\left (x +1\right )^{2} y^{\prime }&=\left (y+1\right )^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.763 |
|
| \begin{align*}
x^{2} y^{\prime }&=1-x^{2}+y^{2}-x^{2} y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.664 |
|
| \begin{align*}
y^{\prime }&=3 x^{2} \left (1+y^{2}\right ) \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.512 |
|
| \begin{align*}
y^{\prime }&=\left (4 x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.897 |
|
| \begin{align*}
y^{\prime }+y^{2}&=x^{2}+1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.556 |
|
| \begin{align*}
y^{\prime }+2 y x&=1+x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.316 |
|
| \begin{align*}
y^{\prime }&=1+x^{2}+y^{2}+x^{2} y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.524 |
|
| \begin{align*}
y^{\prime }&=x^{2}-2 y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.809 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }&=\left (y+1\right )^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.369 |
|
| \begin{align*}
x^{2} y^{\prime }&=1-x^{2}+y^{2}-x^{2} y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.792 |
|
| \begin{align*}
y^{\prime }&=3 x^{2} \left (1+y^{2}\right ) \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.726 |
|
| \begin{align*}
y^{\prime }&=\left (4 x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.930 |
|
| \begin{align*}
y^{\prime }&=1+x^{2}+y^{2}+x^{2} y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.093 |
|
| \begin{align*}
y^{\prime }&=x^{2}-2 y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.887 |
|
| \begin{align*}
y^{\prime }&=2 \left (x +1\right ) \left (1+y^{2}\right ) \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.848 |
|
| \begin{align*}
y^{\prime }&=\frac {x^{2}+y x +y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.217 |
|
| \begin{align*}
x^{2}+3 y x +y^{2}-x^{2} y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.106 |
|
| \begin{align*}
y^{\prime }&=1+2 x +y^{2}+2 x y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.701 |
|
| \begin{align*}
2 y^{\prime }+x \left (y^{2}-1\right )&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.840 |
|
| \begin{align*}
y^{\prime }&=x^{2} \left (1+y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.237 |
|
| \begin{align*}
y^{\prime }&=x \left (1+y^{2}\right ) \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.647 |
|
| \begin{align*}
y^{\prime }&=x^{2} \left (1+y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.198 |
|
| \begin{align*}
y^{\prime }&=\left (x -1\right ) \left (-1+y\right ) \left (-2+y\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.507 |
|
| \begin{align*}
y^{\prime } \left (x^{2}+2\right )&=4 x \left (y^{2}+2 y+1\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.357 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.433 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2}+y x -x^{2} \\
y \left (1\right ) &= 2 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
10.716 |
|
| \begin{align*}
x^{2} y^{\prime }&=x^{2}+y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.100 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
11.773 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
6.141 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
12.665 |
|
| \begin{align*}
x^{3} y^{\prime }&=2 y^{2}+2 x^{2} y-2 x^{4} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.246 |
|
| \begin{align*}
y^{\prime }&=y^{2} {\mathrm e}^{-x}+4 y+2 \,{\mathrm e}^{x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.252 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
4.638 |
|
| \begin{align*}
y^{\prime }&=1+x -\left (2 x +1\right ) y+x y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
4.279 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+y^{2}\right )-x \left (x +2\right ) y+x +2&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.093 |
|
| \begin{align*}
y^{\prime }+y^{2}+4 y x +4 x^{2}+2&=0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.225 |
|
| \begin{align*}
\left (2 x +1\right ) \left (y^{\prime }+y^{2}\right )-2 y-2 x -3&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.320 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
7.031 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
3.131 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+y^{2}\right )-7 y x +7&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
9.172 |
|
| \begin{align*}
\left (t^{2}+1\right ) y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.992 |
|
| \begin{align*}
y^{\prime }&=1-t +y^{2}-t y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.468 |
|
| \begin{align*}
\left (t^{2}+1\right ) y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.345 |
|
| \begin{align*}
y^{\prime }&=1-t +y^{2}-t y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.622 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.543 |
|
| \begin{align*}
1+y^{2}&=\frac {y^{\prime }}{x^{3} \left (x -1\right )} \\
y \left (2\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
9.173 |
|
| \begin{align*}
\left (x^{2}+x +1\right ) y^{\prime }&=y^{2}+2 y+5 \\
y \left (1\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
12.212 |
|
| \begin{align*}
\left (x^{2}-2 x -8\right ) y^{\prime }&=y^{2}+y-2 \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
11.210 |
|
| \begin{align*}
y^{\prime }-\frac {y^{2}}{x^{2}}&={\frac {1}{4}} \\
y \left (1\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.435 |
|
| \begin{align*}
y^{\prime }-\frac {y^{2}}{x^{2}}&={\frac {1}{4}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.554 |
|
| \begin{align*}
y^{\prime }&=\frac {x \left (y^{2}-1\right )}{2 \left (x -2\right ) \left (x -1\right )} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.362 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }+y^{2}&=-1 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.089 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (x +y\right )^{2}}{2 x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.314 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2}+3 y x +x^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.512 |
|
| \begin{align*}
y^{\prime }&=\frac {x \left (y^{2}-1\right )}{2 \left (x -2\right ) \left (x -1\right )} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.800 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }+y^{2}&=-1 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.407 |
|
| \begin{align*}
y^{\prime }&=\frac {x^{2}+y x +y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.723 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (x +y\right )^{2}}{2 x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.334 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2}+3 y x +x^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.263 |
|
| \begin{align*}
y^{\prime }&=\left (9 x -y\right )^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.319 |
|
| \begin{align*}
y^{\prime }&=\left (4 x +y+2\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.557 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
5.174 |
|
| \begin{align*}
y^{\prime }+\frac {2 y}{x}-y^{2}&=-\frac {2}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.036 |
|
| \begin{align*}
y^{\prime }+\frac {7 y}{x}-3 y^{2}&=\frac {3}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.572 |
|
| \begin{align*}
x^{2} y^{\prime }&=\left (-1+y\right ) x +\left (-1+y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.809 |
|
| \begin{align*}
y^{\prime }&=\frac {x^{2}+y^{2}}{2 x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.316 |
|
| \begin{align*}
2 x y^{\prime }&=1-y^{2} \\
y \left (1\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
6.644 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{x} \left (1+y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.006 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.839 |
|
| \begin{align*}
x y^{\prime }&=x^{5}+x^{3} y^{2}+y \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.530 |
|
| \begin{align*}
x y^{\prime }&=y+x^{2}+9 y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.055 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
25.869 |
|
| \begin{align*}
x^{2}+y+y^{2}-x y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.385 |
|
| \begin{align*}
y^{\prime }+1-x&=y \left (x +y\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.121 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.226 |
|
| \begin{align*}
y^{\prime }&=\left (x -y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.558 |
|
| \begin{align*}
y^{\prime }&=3 y-3 x +3+\left (x -y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.749 |
|
| \begin{align*}
y^{\prime }&=2 x -\left (x^{2}+1\right ) y+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.599 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
3.168 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
3.560 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right )+x f \left (x \right ) y+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.352 |
|
| \begin{align*}
y^{\prime }&=\left (3+x -4 y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
14.872 |
|
| \begin{align*}
y^{\prime }&=\left (1+4 x +9 y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
23.953 |
|
| \begin{align*}
y^{\prime }&=1+a \left (x -y\right ) y \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.340 |
|
| \begin{align*}
y^{\prime }&=1-x -x^{3}+\left (2 x^{2}+1\right ) y-x y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
5.754 |
|
| \begin{align*}
y^{\prime }&=x \left (2+x^{2} y-y^{2}\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
4.739 |
|
| \begin{align*}
y^{\prime }&=x +\left (1-2 x \right ) y-\left (1-x \right ) y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
5.757 |
|
| \begin{align*}
y^{\prime }&=x^{n} \left (a +b y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.178 |
|
| \begin{align*}
y^{\prime }+\tan \left (x \right ) \left (1-y^{2}\right )&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
8.872 |
|
| \begin{align*}
y^{\prime }&=\left (a +b y+c y^{2}\right ) f \left (x \right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
9.682 |
|
| \begin{align*}
x y^{\prime }&=x^{2}+y \left (y+1\right ) \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.934 |
|
| \begin{align*}
x y^{\prime }&=a +b y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.079 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{2}+y+b y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.787 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{2 n}+\left (n +b y\right ) y \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.352 |
|
| \begin{align*}
x y^{\prime }&=x^{3}+\left (2 x^{2}+1\right ) y+x y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.405 |
|
| \begin{align*}
x y^{\prime }+b x +\left (2+a x y\right ) y&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.398 |
|
| \begin{align*}
x y^{\prime }+x^{m}+\frac {\left (n -m \right ) y}{2}+x^{n} y^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.603 |
|
| \begin{align*}
x y^{\prime }&=2 x -y+a \,x^{n} \left (x -y\right )^{2} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
10.126 |
|
| \begin{align*}
x y^{\prime }&=y+\left (x^{2}-y^{2}\right ) f \left (x \right ) \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.819 |
|
| \begin{align*}
3 x y^{\prime }&=3 x^{{2}/{3}}+\left (1-3 y\right ) y \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.582 |
|
| \begin{align*}
x^{2} y^{\prime }+x^{2}+y x +y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
7.309 |
|
| \begin{align*}
x^{2} y^{\prime }&=\left (1+2 x -y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
11.255 |
|
| \begin{align*}
x^{2} y^{\prime }&=a +b y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.750 |
|
| \begin{align*}
x^{2} y^{\prime }+a \,x^{2}+b x y+c y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
28.460 |
|
| \begin{align*}
x^{2} y^{\prime }+2+x y \left (4+y x \right )&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.341 |
|
| \begin{align*}
x^{2} y^{\prime }+2+a x \left (-y x +1\right )-x^{2} y^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.015 |
|
| \begin{align*}
x^{2} y^{\prime }&=a +b \,x^{2} y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
7.020 |
|
| \begin{align*}
x^{2} y^{\prime }&=a +b x y+c \,x^{2} y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.078 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.055 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }&=-1-y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.959 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime }&=1-y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
6.471 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime }&=y^{2}-1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
6.384 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
5.144 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
27.814 |
|
| \begin{align*}
\left (x -a \right ) \left (x -b \right ) y^{\prime }+k \left (y-a \right ) \left (y-b \right )&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
15.760 |
|
| \begin{align*}
\left (x -a \right ) \left (x -b \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] |
✓ |
✓ |
✓ |
✓ |
13.782 |
|
| \begin{align*}
2 x^{2} y^{\prime }+1+2 y x -x^{2} y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
7.562 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
91.787 |
|
| \begin{align*}
x \left (1-2 x \right ) y^{\prime }&=4 x -\left (1+4 x \right ) y+y^{2} \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.914 |
|
| \begin{align*}
a \,x^{2} y^{\prime }&=x^{2}+a x y+b^{2} y^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
40.164 |
|
| \begin{align*}
\left (b \,x^{2}+a \right ) y^{\prime }&=A +B y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.896 |
|
| \begin{align*}
\left (b \,x^{2}+a \right ) y^{\prime }&=-A -B y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.116 |
|
| \begin{align*}
x^{3} y^{\prime }&=x^{4}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.657 |
|
| \begin{align*}
x^{3} y^{\prime }&=x^{2} \left (-1+y\right )+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
7.974 |
|
| \begin{align*}
x^{3} y^{\prime }+20+x^{2} y \left (1-x^{2} y\right )&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
11.721 |
|
| \begin{align*}
x^{3} y^{\prime }+3+\left (3-2 x \right ) x^{2} y-x^{6} y^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.060 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.774 |
|
| \begin{align*}
\left (-x^{4}+1\right ) y^{\prime }&=2 x \left (1-y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.629 |
|
| \begin{align*}
x \left (-x^{3}+1\right ) y^{\prime }&=x^{2}+\left (1-2 y x \right ) y \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.941 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
5.954 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
8.441 |
|
| \begin{align*}
x^{n} y^{\prime }&=a^{2} x^{2 n -2}+b^{2} y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
13.254 |
|
| \begin{align*}
x^{n} y^{\prime }&=x^{n -1} \left (a \,x^{2 n}+n y-b y^{2}\right ) \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
7.394 |
|
| \begin{align*}
y^{\prime } \sqrt {-x^{2}+1}&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
8.539 |
|
| \begin{align*}
y^{\prime }&=x^{3}+\frac {2 y}{x}-\frac {y^{2}}{x} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
7.971 |
|
| \begin{align*}
y^{\prime }&=\frac {1}{x^{2}}-\frac {y}{x}-y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
32.737 |
|
| \begin{align*}
y^{\prime }&=1+\frac {y}{x}-\frac {y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
46.384 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.685 |
|
| \begin{align*}
x y^{\prime }-y^{2}+1&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.609 |
|
| \begin{align*}
x^{2} y^{\prime }+x^{2}+y x +y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.749 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.376 |
|
| \begin{align*}
y^{\prime }&=x y^{2}-\frac {2 y}{x}-\frac {1}{x^{3}} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.779 |
|
| \begin{align*}
y^{\prime }&=\frac {2 y^{2}}{x}+\frac {y}{x}-2 x \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.376 |
|
| \begin{align*}
y^{\prime }&=y^{2} {\mathrm e}^{-x}+y-{\mathrm e}^{x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.356 |
|
| \begin{align*}
y^{\prime }&=\left (1+y^{2}\right ) \tan \left (x \right ) \\
y \left (0\right ) &= \sqrt {3} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
6.953 |
|
| \begin{align*}
y^{\prime }&=\sqrt {1+\sin \left (x \right )}\, \left (1+y^{2}\right ) \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
25.469 |
|
| \begin{align*}
\left (y-4 x -1\right )^{2}-y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
9.270 |
|
| \begin{align*}
y^{\prime }&=\left (x +y+2\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.559 |
|
| \begin{align*}
y^{\prime }&=\left (x -y+5\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.124 |
|
| \begin{align*}
y^{\prime }&=x^{3} \left (-x +y\right )^{2}+\frac {y}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.400 |
|
| \begin{align*}
y^{\prime }&=\left (2 x +y-1\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.918 |
|
| \begin{align*}
\left (x +1\right )^{2} y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.731 |
|
| \begin{align*}
y^{\prime }+x +x y^{2}&=0 \\
y \left (1\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
4.496 |
|
| \begin{align*}
y^{\prime }&=-2 \left (2 x +3 y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
11.639 |
|
| \begin{align*}
1+y^{2}&=\left (x^{2}+x \right ) y^{\prime } \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.259 |
|
| \begin{align*}
y^{2}-1+x y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.703 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (3+2 y\right )^{2}}{\left (5+4 x \right )^{2}} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
8.510 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}-1}{x^{2}-1} \\
y \left (2\right ) &= 2 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
5.589 |
|
| \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] |
✓ |
✓ |
✗ |
✓ |
5.365 |
|
| \begin{align*}
y^{\prime }&=\left (1+y^{2}\right ) \sqrt {1+\cos \left (x^{3}\right )} \\
y \left (1\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
16.156 |
|
| \begin{align*}
y^{\prime }&=\left (x +y+1\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.703 |
|
| \begin{align*}
x^{2}+y x +y^{2}&=x^{2} y^{\prime } \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.126 |
|
| \begin{align*}
y^{\prime }&=x^{2} \left (1+y^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.161 |
|
| \begin{align*}
y^{\prime }&=-\frac {2}{t}+\frac {y}{t}+\frac {y^{2}}{t} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
14.332 |
|
| \begin{align*}
x y^{\prime }-y&=x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.546 |
|
| \begin{align*}
y^{\prime }&=x^{2} y^{2}-4 x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✗ |
8.346 |
|
| \begin{align*}
y^{\prime }&=\frac {x^{2}+y x +y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.809 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (x +y-1\right )^{2}}{2 \left (x +2\right )^{2}} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.361 |
|
| \begin{align*}
x y^{\prime }&=y+x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.490 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.616 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }+y^{2}&=-1 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
6.827 |
|
| \begin{align*}
y^{\prime }&=\frac {5 x^{2}-y x +y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.928 |
|
| \begin{align*}
x y^{\prime }-2 y+b y^{2}&=c \,x^{4} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.187 |
|
| \begin{align*}
y^{\prime }+y^{2}-2 x^{2} y+x^{4}-2 x -1&=0 \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.186 |
|
| \begin{align*}
y^{\prime }+y^{2}+\left (y x -1\right ) f \left (x \right )&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.789 |
|
| \begin{align*}
y^{\prime }-y^{2}-y x -x +1&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.507 |
|
| \begin{align*}
y^{\prime }-\left (x +y\right )^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.546 |
|
| \begin{align*}
y^{\prime }-y^{2}+\left (x^{2}+1\right ) y-2 x&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.485 |
|
| \begin{align*}
y^{\prime }+a y \left (-x +y\right )-1&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.850 |
|
| \begin{align*}
y^{\prime }+x y^{2}-x^{3} y-2 x&=0 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
2.666 |
|
| \begin{align*}
y^{\prime }-a \,x^{n} \left (1+y^{2}\right )&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.225 |
|
| \begin{align*}
y^{\prime }+f \left (x \right ) \left (y^{2}+2 a y+b \right )&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.733 |
|
| \begin{align*}
x y^{\prime }-y^{2}+1&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.477 |
|
| \begin{align*}
x y^{\prime }+a y^{2}-y+b \,x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.364 |
|
| \begin{align*}
x y^{\prime }+a y^{2}-b y+c \,x^{2 b}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.563 |
|
| \begin{align*}
x y^{\prime }+x y^{2}-y-a \,x^{3}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.166 |
|
| \begin{align*}
x y^{\prime }+x y^{2}-\left (2 x^{2}+1\right ) y-x^{3}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.428 |
|
| \begin{align*}
x y^{\prime }+a x y^{2}+2 y+b x&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.447 |
|
| \begin{align*}
x y^{\prime }+x^{a} y^{2}+\frac {\left (a -b \right ) y}{2}+x^{b}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.381 |
|
| \begin{align*}
x y^{\prime }+f \left (x \right ) \left (y^{2}-x^{2}\right )-y&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.102 |
|
| \begin{align*}
x^{2} y^{\prime }+x^{2}+y x +y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.857 |
|
| \begin{align*}
x^{2} y^{\prime }-y^{2}-y x -x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.197 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+y^{2}\right )+4 y x +2&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.134 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+y^{2}\right )+a x y+b&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.416 |
|
| \begin{align*}
x^{2} \left (y^{\prime }-y^{2}\right )-a \,x^{2} y+a x +2&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.583 |
|
| \begin{align*}
x^{2} \left (y^{\prime }+a y^{2}\right )-b&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
4.205 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
4.174 |
|
| \begin{align*}
\left (x -a \right ) \left (x -b \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] |
✓ |
✓ |
✓ |
✓ |
10.246 |
|
| \begin{align*}
2 x^{2} y^{\prime }-2 y^{2}-y x +2 a^{2} x&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.300 |
|
| \begin{align*}
x \left (2 x -1\right ) y^{\prime }+y^{2}-\left (1+4 x \right ) y+4 x&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.742 |
|
| \begin{align*}
3 x^{2} y^{\prime }-7 y^{2}-3 y x -x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.877 |
|
| \begin{align*}
x^{3} y^{\prime }-y^{2}-x^{4}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.681 |
|
| \begin{align*}
x^{3} y^{\prime }-y^{2} x^{4}+x^{2} y+20&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
7.240 |
|
| \begin{align*}
x^{3} y^{\prime }-x^{6} y^{2}-\left (2 x -3\right ) x^{2} y+3&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.361 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
3.869 |
|
| \begin{align*}
\left (a \,x^{2}+b x +c \right ) \left (x y^{\prime }-y\right )-y^{2}+x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.138 |
|
| \begin{align*}
x \left (x^{3}-1\right ) y^{\prime }-2 x y^{2}+y+x^{2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.310 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.481 |
|
| \begin{align*}
x^{n} y^{\prime }-a y^{2}-b \,x^{2 n -2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
8.977 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.390 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (2 y \ln \left (x \right )-1\right )^{2}}{x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
34.607 |
|
| \begin{align*}
y^{\prime }&=\frac {y+x^{3} a \,{\mathrm e}^{x}+a \,x^{4}+a \,x^{3}-x y^{2} {\mathrm e}^{x}-x^{2} y^{2}-x y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.402 |
|
| \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 )-x^{2} y^{2}-x y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.430 |
|
| \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 x^{2} y^{2}+7 x y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.947 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
6.653 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.593 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.402 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.190 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.074 |
|
| \begin{align*}
y^{\prime }&=\frac {y \ln \left (x -1\right )+x^{4}+x^{3}+x^{2} y^{2}+x y^{2}}{\ln \left (x -1\right ) x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.840 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.108 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
20.198 |
|
| \begin{align*}
y^{\prime }&=\frac {-y \,{\mathrm e}^{x}+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] |
✓ |
✓ |
✓ |
✗ |
5.749 |
|
| \begin{align*}
y^{\prime }&=\frac {x y \ln \left (x \right )-y+2 x^{5} b +2 x^{3} a y^{2}}{\left (x \ln \left (x \right )-1\right ) x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.523 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
7.207 |
|
| \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}+x^{2} \ln \left (2 x \right )}{\ln \left (x \right )} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
10.558 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.599 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
6.934 |
|
| \begin{align*}
y^{\prime }&=\frac {-\sinh \left (x \right )+x^{2} \ln \left (x \right )+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] |
✓ |
✓ |
✓ |
✗ |
26.126 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
56.125 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.211 |
|
| \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] |
✓ |
✗ |
✓ |
✗ |
79.925 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.013 |
|
| \begin{align*}
y^{\prime }&=\frac {x +y+y^{2}-2 x y \ln \left (x \right )+x^{2} \ln \left (x \right )^{2}}{x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.209 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
12.619 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
6.378 |
|
| \begin{align*}
y^{\prime }&=-F \left (x \right ) \left (-a \,x^{2}+y^{2}\right )+\frac {y}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.872 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
6.132 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.587 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.571 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
4.585 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.796 |
|
| \begin{align*}
y^{\prime }&=\left (y-{\mathrm e}^{x}\right )^{2}+{\mathrm e}^{x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
10.676 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
7.222 |
|
| \begin{align*}
y^{\prime }&=y^{2}-a^{2} x^{2}+3 a \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.226 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
50.579 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{-n -2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.412 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
56.734 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
5.169 |
|
| \begin{align*}
x^{2} y^{\prime }&=x^{2} y^{2}-a^{2} x^{4}+a \left (1-2 b \right ) x^{2}-b \left (b +1\right ) \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.235 |
|
| \begin{align*}
\left (a \,x^{n}+b \right ) y^{\prime }&=b y^{2}+a \,x^{-2+n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
9.942 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,x^{n} y+a \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.699 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,x^{n} y-a b \,x^{n}-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.165 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} y+b c \,x^{m}-a \,c^{2} x^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.713 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
10.526 |
|
| \begin{align*}
x y^{\prime }&=a y^{2}+b y+c \,x^{2 b} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.380 |
|
| \begin{align*}
x y^{\prime }&=a y^{2}+\left (n +b \,x^{n}\right ) y+c \,x^{2 n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.365 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{n} y^{2}+b y+c \,x^{-n} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.975 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{n} y^{2}+m y-a \,b^{2} x^{n +2 m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.649 |
|
| \begin{align*}
x y^{\prime }&=x^{2 n} y^{2}+\left (m -n \right ) y+x^{2 m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.739 |
|
| \begin{align*}
x y^{\prime }&=x^{2 n} a y^{2}+\left (b \,x^{n}-n \right ) y+c \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.856 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{2 n +m} y^{2}+\left (b \,x^{n +m}-n \right ) y+c \,x^{m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
30.513 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
9.631 |
|
| \begin{align*}
2 x^{2} y^{\prime }&=2 y^{2}+y x -2 a^{2} x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.052 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.107 |
|
| \begin{align*}
\left (a \,x^{2}+b \right ) y^{\prime }+y^{2}-2 y x +\left (1-a \right ) x^{2}-b&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.503 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.358 |
|
| \begin{align*}
\left (x -a \right ) \left (x -b \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] |
✓ |
✓ |
✓ |
✓ |
11.657 |
|
| \begin{align*}
\left (a \,x^{2}+b x +e \right ) \left (x y^{\prime }-y\right )-y^{2}+x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.822 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
39.050 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
1.030 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
95.883 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
3.641 |
|
| \begin{align*}
\left (a \,x^{n}+b \,x^{m}+c \right ) \left (x y^{\prime }-y\right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right )&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
39.348 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.538 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
4.944 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,{\mathrm e}^{\lambda x} y-a \,{\mathrm e}^{\lambda x} b -b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.099 |
|
| \begin{align*}
y^{\prime }&=b \,{\mathrm e}^{\mu x} y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✓ |
✗ |
42.379 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
35.550 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
26.634 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \,{\mathrm e}^{\lambda x} y+a \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.996 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b \,{\mathrm e}^{-\lambda x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.902 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b n \,x^{n -1}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
6.365 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
55.463 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b n \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
10.898 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
45.667 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.968 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} {\mathrm e}^{\lambda x} y-a \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.382 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
9.815 |
|
| \begin{align*}
x y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+k y+a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.922 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y+c \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
13.052 |
|
| \begin{align*}
y^{\prime }&=y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
68.014 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+\lambda x y+a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.736 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \sinh \left (\beta x \right ) y+a b \sinh \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.877 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \sinh \left (b x \right )^{m} y+a \sinh \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
13.343 |
|
| \begin{align*}
\left (\sinh \left (\lambda x \right ) a +b \right ) y^{\prime }&=y^{2}+c \sinh \left (\mu x \right ) y-d^{2}+c d \sinh \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
93.860 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
9.965 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \cosh \left (b x \right )^{m} y+a \cosh \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
10.261 |
|
| \begin{align*}
\left (a \cosh \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \cosh \left (\mu x \right ) y-d^{2}+c d \cosh \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
89.907 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \tanh \left (b x \right )^{m} y+a \tanh \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
9.541 |
|
| \begin{align*}
\left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \tanh \left (\mu x \right ) y-d^{2}+c d \tanh \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
95.595 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \coth \left (b x \right )^{m} y+a \coth \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
9.583 |
|
| \begin{align*}
\left (a \coth \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
106.514 |
|
| \begin{align*}
y^{\prime }&=a \ln \left (x \right )^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
7.734 |
|
| \begin{align*}
x y^{\prime }&=x y^{2}-a^{2} x \ln \left (\beta x \right )^{2}+a \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
96.599 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{n} y^{2}+b -a \,b^{2} x^{n} \ln \left (x \right )^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
32.735 |
|
| \begin{align*}
x^{2} \ln \left (a x \right ) \left (y^{\prime }-y^{2}\right )&=1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
16.290 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \ln \left (\beta x \right ) y-a b \ln \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
5.038 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \ln \left (b x \right )^{m} y+a \ln \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.550 |
|
| \begin{align*}
y^{\prime }&=-\left (n +1\right ) x^{n} y^{2}+a \,x^{n +1} \ln \left (x \right )^{m} y-a \ln \left (x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.226 |
|
| \begin{align*}
y^{\prime }&=a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y+b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.586 |
|
| \begin{align*}
x y^{\prime }&=\left (a y+b \ln \left (x \right )\right )^{2} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
31.234 |
|
| \begin{align*}
x y^{\prime }&=a \ln \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \ln \left (\lambda x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
35.612 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{n} \left (y+b \ln \left (x \right )\right )^{2}-b \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
38.474 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{2 n} \ln \left (x \right ) y^{2}+\left (b \,x^{n} \ln \left (x \right )-n \right ) y+c \ln \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.761 |
|
| \begin{align*}
\left (a \ln \left (x \right )+b \right ) y^{\prime }&=y^{2}+c \ln \left (x \right )^{n} y-\lambda ^{2}+\lambda c \ln \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
10.179 |
|
| \begin{align*}
\left (a \ln \left (x \right )+b \right ) y^{\prime }&=\ln \left (x \right )^{n} y^{2}+c y-\lambda ^{2} \ln \left (x \right )^{n}+\lambda c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
17.178 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \sin \left (\beta x \right ) y+a b \sin \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.940 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} \sin \left (x \right )^{m} y-a \sin \left (x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
47.472 |
|
| \begin{align*}
x y^{\prime }&=a \sin \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \sin \left (\lambda x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
67.467 |
|
| \begin{align*}
\left (\sin \left (\lambda x \right ) a +b \right ) y^{\prime }&=y^{2}+c \sin \left (\mu x \right ) y-d^{2}+c d \sin \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.709 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \cos \left (\beta x \right ) y+a b \cos \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
8.061 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} \cos \left (x \right )^{m} y-a \cos \left (x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
32.185 |
|
| \begin{align*}
x y^{\prime }&=a \cos \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cos \left (\lambda x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
68.044 |
|
| \begin{align*}
\left (\cos \left (\lambda x \right ) a +b \right ) y^{\prime }&=y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.710 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \tan \left (\beta x \right ) y+a b \tan \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
41.803 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \tan \left (b x \right )^{m} y+a \tan \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.190 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} \tan \left (x \right )^{m} y-a \tan \left (x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
23.705 |
|
| \begin{align*}
x y^{\prime }&=a \tan \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \tan \left (\lambda x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
90.255 |
|
| \begin{align*}
\left (a \tan \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+k \tan \left (\mu x \right ) y-d^{2}+k d \tan \left (\mu x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
79.035 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \cot \left (\beta x \right ) y+a b \cot \left (\beta x \right )-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
12.240 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a x \cot \left (b x \right )^{m} y+a \cot \left (b x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
14.111 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+a \,x^{1+k} \cot \left (x \right )^{m} y-a \cot \left (x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
30.016 |
|
| \begin{align*}
x y^{\prime }&=a \cot \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cot \left (\lambda x \right )^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
92.992 |
|
| \begin{align*}
\left (a \cot \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right ) \\
\end{align*} |
unknown |
✓ |
✓ |
✓ |
✗ |
132.317 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda \arcsin \left (x \right )^{n} y-a^{2}+a \lambda \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
11.642 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arcsin \left (x \right )^{n} y+\arcsin \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
25.560 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+\lambda \arcsin \left (x \right )^{n} \left (x^{1+k} y-1\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
58.553 |
|
| \begin{align*}
y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
17.052 |
|
| \begin{align*}
y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arcsin \left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
38.853 |
|
| \begin{align*}
y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
44.690 |
|
| \begin{align*}
x y^{\prime }&=\lambda \arcsin \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arcsin \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
39.107 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda \arccos \left (x \right )^{n} y-a^{2}+a \lambda \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
12.383 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arccos \left (x \right )^{n} y+\arccos \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
27.589 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+\lambda \arccos \left (x \right )^{n} \left (x^{1+k} y-1\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
50.768 |
|
| \begin{align*}
y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
24.370 |
|
| \begin{align*}
y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arccos \left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
69.658 |
|
| \begin{align*}
y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
46.966 |
|
| \begin{align*}
x y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arccos \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
63.324 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda \arctan \left (x \right )^{n} y-a^{2}+a \lambda \arctan \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
9.517 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \arctan \left (x \right )^{n} y+\arctan \left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.381 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+\lambda \arctan \left (x \right )^{n} \left (x^{1+k} y-1\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
46.244 |
|
| \begin{align*}
y^{\prime }&=\lambda \arctan \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arctan \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
10.900 |
|
| \begin{align*}
y^{\prime }&=\lambda \arctan \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arctan \left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
44.935 |
|
| \begin{align*}
x y^{\prime }&=\lambda \arctan \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arctan \left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
35.202 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} y-a^{2}+a \lambda \operatorname {arccot}\left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.296 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\lambda x \operatorname {arccot}\left (x \right )^{n} y+\operatorname {arccot}\left (x \right )^{n} \lambda \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.941 |
|
| \begin{align*}
y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} \left (x^{1+k} y-1\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
57.400 |
|
| \begin{align*}
y^{\prime }&=\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \operatorname {arccot}\left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
14.887 |
|
| \begin{align*}
y^{\prime }&=\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}-b \lambda \,x^{m} \operatorname {arccot}\left (x \right )^{n} y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
57.690 |
|
| \begin{align*}
x y^{\prime }&=\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \operatorname {arccot}\left (x \right )^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
40.376 |
|
| \begin{align*}
y^{\prime }&=y^{2}+f \left (x \right ) y-a^{2}-a f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.877 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a y-a b -b^{2} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
7.430 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right )+x f \left (x \right ) y+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.190 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,x^{n} f \left (x \right ) y+a n \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
6.932 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
7.693 |
|
| \begin{align*}
y^{\prime }&=-\left (n +1\right ) x^{n} y^{2}+x^{n +1} f \left (x \right ) y-f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✗ |
✗ |
8.298 |
|
| \begin{align*}
x y^{\prime }&=f \left (x \right ) y^{2}+n y+a \,x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.838 |
|
| \begin{align*}
x y^{\prime }&=x^{2 n} f \left (x \right ) y^{2}+\left (a \,x^{n} f \left (x \right )-n \right ) y+f \left (x \right ) b \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
10.366 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+g \left (x \right ) y-a^{2} f \left (x \right )-a g \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.588 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+g \left (x \right ) y+a n \,x^{n -1}-a \,x^{n} g \left (x \right )-a^{2} x^{2 n} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
18.026 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,x^{n} g \left (x \right ) y+a n \,x^{n -1}+a^{2} x^{2 n} \left (g \left (x \right )-f \left (x \right )\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
10.881 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
9.124 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+\lambda y+a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.479 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{\lambda x} f \left (x \right ) y^{2}+\left (a f \left (x \right )-\lambda \right ) y+b \,{\mathrm e}^{-\lambda x} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
11.276 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{\lambda x} g \left (x \right )-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
10.001 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}-a \,{\mathrm e}^{\lambda x} g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}+a^{2} {\mathrm e}^{2 \lambda x} \left (g \left (x \right )-f \left (x \right )\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
117.582 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} f \left (x \right ) {\mathrm e}^{2 \lambda \,x^{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
16.354 |
|
| \begin{align*}
y^{\prime }&=f^{\prime }\left (x \right ) y^{2}+a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+a \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
15.751 |
|
| \begin{align*}
x y^{\prime }-y&=x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.086 |
|
| \begin{align*}
x y^{\prime }-a y+b y^{2}&=c \,x^{2 a} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
7.320 |
|
| \begin{align*}
y^{\prime }+2 y x&=x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.775 |
|
| \begin{align*}
R^{\prime }&=\left (t +1\right ) \left (1+R^{2}\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.027 |
|
| \begin{align*}
x^{\prime }&=\left (4 t -x\right )^{2} \\
x \left (0\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.445 |
|
| \begin{align*}
T^{\prime }&=2 a t \left (T^{2}-a^{2}\right ) \\
T \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✗ |
9.295 |
|
| \begin{align*}
x^{\prime }&=\left (t +x\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.963 |
|
| \begin{align*}
2 r \left (s^{2}+1\right )+\left (r^{4}+1\right ) s^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.433 |
|
| \begin{align*}
y^{\prime }&=\left (1-x \right ) y^{2}+\left (2 x -1\right ) y-x \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
8.241 |
|
| \begin{align*}
y^{\prime }&=-y^{2}+y x +1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.886 |
|
| \begin{align*}
y^{\prime }&=-8 x y^{2}+4 x \left (1+4 x \right ) y-8 x^{3}-4 x^{2}+1 \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
12.453 |
|
| \begin{align*}
2 x^{2}+y x +y^{2}+2 x^{2} y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
66.481 |
|
| \begin{align*}
y^{\prime }&=\left (1+y^{2}\right ) \tan \left (x \right ) \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
12.720 |
|
| \begin{align*}
y x +y^{2}+x^{2}-x^{2} y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
12.531 |
|
| \begin{align*}
x^{2} y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
6.618 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
7.939 |
|
| \begin{align*}
1+s^{2}-\sqrt {t}\, s^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
9.344 |
|
| \begin{align*}
y^{\prime }&=\left (1+y^{2}\right ) t \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
3.787 |
|
| \begin{align*}
y^{\prime }&=\left (y+\frac {1}{2}\right ) \left (t +y\right ) \\
y \left (0\right ) &= {\frac {1}{2}} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.829 |
|
| \begin{align*}
y^{\prime }&=\left (-2+y\right ) \left (y+1-\cos \left (t \right )\right ) \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
6.006 |
|
| \begin{align*}
x^{2} y^{\prime }+x y^{2}&=x \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.740 |
|
| \begin{align*}
y^{\prime }+\left (8-x \right ) y-y^{2}&=-8 x \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.159 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.756 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.703 |
|
| \begin{align*}
y^{\prime }-3 x^{2} y^{2}&=-3 x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.412 |
|
| \begin{align*}
y^{\prime }-3 x^{2} y^{2}&=3 x^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.756 |
|
| \begin{align*}
y^{\prime }&=1+\left (-x +y\right )^{2} \\
y \left (0\right ) &= {\frac {1}{4}} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.332 |
|
| \begin{align*}
y^{\prime }&=\left (x -y+3\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.896 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
4.122 |
|
| \begin{align*}
y^{\prime }&=x^{2}-2 y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.584 |
|
| \begin{align*}
y^{\prime }&=x y^{2}+3 y^{2}+x +3 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.528 |
|
| \begin{align*}
y^{\prime }&=y^{2} t^{2}+y^{2}-t^{2}-1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✗ |
6.022 |
|
| \begin{align*}
4 \left (x -1\right )^{2} y^{\prime }-3 \left (y+3\right )^{2}&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
9.948 |
|
| \begin{align*}
y^{\prime }&=\left (x +y-4\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.647 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.685 |
|
| \begin{align*}
1+y^{2}&=x y^{\prime } \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
6.719 |
|
| \begin{align*}
x^{2} y^{\prime }&=x^{2}-y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.401 |
|
| \begin{align*}
2 x^{2} y^{\prime }&=x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.297 |
|
| \begin{align*}
x y^{\prime }-y^{2}+\left (2 x +1\right ) y&=x^{2}+2 x \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.030 |
|
| \begin{align*}
x^{2} y^{\prime }&=1+y x +x^{2} y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.086 |
|
| \begin{align*}
y^{\prime }&=\left (x -y\right )^{2}+1 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.815 |
|
| \begin{align*}
y^{\prime }&=\left (1+y^{2}\right ) \tan \left (2 x \right ) \\
y \left (0\right ) &= -\sqrt {3} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
15.498 |
|
| \begin{align*}
y^{\prime }&=2 \left (x +1\right ) \left (1+y^{2}\right ) \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
7.580 |
|
| \begin{align*}
y^{\prime }+3 y t&=4-4 t^{2}+y^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.642 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}}{3}+\frac {2}{3 x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
3.444 |
|
| \begin{align*}
y^{\prime }+y^{2}+\frac {y}{x}-\frac {4}{x^{2}}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.398 |
|
| \begin{align*}
x y^{\prime }-3 y+y^{2}&=4 x^{2}-4 x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.757 |
|
| \begin{align*}
x y^{\prime }&=y+x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.190 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.510 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.143 |
|
| \begin{align*}
x y^{\prime }&=x^{5}+x^{3} y^{2}+y \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.614 |
|
| \begin{align*}
x y^{\prime }&=y+x^{2}+9 y^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.178 |
|
| \begin{align*}
y^{\prime }&=1+\frac {y}{x}-\frac {y^{2}}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
11.951 |
|
| \begin{align*}
x^{2} y^{\prime }&=x^{2}+y x +y^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.813 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.252 |
|
| \begin{align*}
y^{\prime }&=x \left (a y^{2}+b \right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.588 |
|
| \begin{align*}
n^{\prime }&=\left (n^{2}+1\right ) x \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.010 |
|
| \begin{align*}
3 x^{2} y^{\prime }+2 x^{2}-3 y^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.783 |
|
| \begin{align*}
y^{\prime }+2 y x&=x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.556 |
|
| \begin{align*}
y^{\prime }&=\left (4 x +y+1\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.637 |
|
| \begin{align*}
x y^{\prime }-y^{2}+\left (2 x +1\right ) y&=x^{2}+2 x \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
5.092 |
|
| \begin{align*}
y^{\prime }&=\left (x -y+3\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.252 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
4.569 |
|
| \begin{align*}
x^{\prime }&={\mathrm e}^{t} \left (x^{2}+1\right ) \\
x \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.875 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
4.786 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.128 |
|
| \begin{align*}
y^{\prime }&=-x^{2}-x -1-\left (2 x +1\right ) y-y^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.164 |
|
| \begin{align*}
y^{\prime }&=\frac {y+x^{2}+y^{2}}{x} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.836 |
|
| \begin{align*}
y^{\prime }+x \left (-x +y\right )+x^{3} \left (-x +y\right )^{2}&=1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
5.305 |
|
| \begin{align*}
1+y+y^{2}+x \left (x^{2}-4\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
8.018 |
|
| \begin{align*}
x^{2} y^{\prime }-y x&=x^{2}-y^{2} \\
y \left (1\right ) &= 0 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
20.820 |
|
| \begin{align*}
x^{2}+y+y^{2}-x y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.358 |
|
| \begin{align*}
y+x^{3}+x y^{2}-x y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
7.211 |
|
| \begin{align*}
x y^{\prime }&=1+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
8.016 |
|
| \begin{align*}
y^{\prime }&=\left (x +y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.071 |
|
| \begin{align*}
y+x^{3}+x y^{2}-x y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
9.264 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (y+3\right )^{2}}{4 x^{2}} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
19.984 |
|
| \begin{align*}
y^{\prime }&=1-\left (x -y\right )^{2} \\
y \left (0\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
11.848 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✗ |
✓ |
11.714 |
|
| \begin{align*}
y^{\prime }&=x y^{2}-2 y+4-4 x \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✓ |
9.914 |
|
| \begin{align*}
y^{\prime }+y^{2}&=x^{2}+1 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.707 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}}{x -1}-\frac {x y}{x -1}+1 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.026 |
|
| \begin{align*}
x y^{\prime }&=x^{2} y^{2}-y+1 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.556 |
|
| \begin{align*}
1+y^{2}&=\left (x^{2}+1\right ) y^{\prime } \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
9.913 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime }&=1-y^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
12.932 |
|
| \begin{align*}
x y^{\prime }+y^{2}&=1 \\
y \left (-2\right ) &= 1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
6.333 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
2.899 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
4.050 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
6.977 |
|
| \begin{align*}
a^{2} \left (y^{\prime }-1\right )&=x^{2} y^{\prime }+y^{2} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.727 |
|
| \begin{align*}
y^{\prime }&=\left (9 x +4 y+1\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
20.426 |
|
| \begin{align*}
y^{\prime }&=2 \left (3 x +y\right )^{2}-1 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
38.848 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}-4 y t +6 t^{2}}{t^{2}} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.174 |
|
| \begin{align*}
y^{\prime }&=t y^{2}-y^{2}+t -1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.442 |
|
| \begin{align*}
y^{\prime }&=\frac {1+y^{2}}{t} \\
y \left (1\right ) &= \sqrt {3} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.843 |
|
| \begin{align*}
t^{2} y^{\prime }&=y^{2}+y t +t^{2} \\
y \left (1\right ) &= 1 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.897 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}-4 y t +6 t^{2}}{t^{2}} \\
y \left (2\right ) &= 4 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
60.368 |
|
| \begin{align*}
y^{\prime }&=\left (t -y\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.748 |
|
| \begin{align*}
y^{\prime }&=1+\left (t -y\right )^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.761 |
|
| \begin{align*}
y^{2}-1+x y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
9.536 |
|
| \begin{align*}
y^{\prime }&=\frac {\left (3+2 y\right )^{2}}{\left (5+4 x \right )^{2}} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
13.648 |
|
| \begin{align*}
1+y^{2}+\left (x^{2}+1\right ) y^{\prime }&=0 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
2.927 |
|
| \begin{align*}
1+y^{2}&=x y^{\prime } \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
3.199 |
|
| \begin{align*}
x^{2} y^{2}+1+2 x^{2} y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
2.877 |
|
| \begin{align*}
y^{\prime }&=\left (x -y\right )^{2}+1 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.404 |
|
| \begin{align*}
2 y^{\prime }+y^{2}+\frac {1}{x^{2}}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
3.220 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}}{x^{2}}-\frac {y}{x}+1 \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.650 |
|
| \begin{align*}
y^{\prime }&=\frac {y^{2}}{2 x}-\frac {y}{x}-\frac {4}{x} \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.398 |
|
| \begin{align*}
x^{2} y^{\prime }&=x^{2}+y^{2} \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.668 |
|
| \begin{align*}
y^{\prime }&=-y^{2} {\mathrm e}^{-x}+y+{\mathrm e}^{x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
2.398 |
|
| \begin{align*}
x^{3} \left (y^{\prime }-x \right )&=y^{2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.253 |
|
| \begin{align*}
y^{\prime }&=y^{2}-\frac {2}{x^{2}} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
5.911 |
|
| \begin{align*}
x^{2} y^{\prime }+y x +x^{2} y^{2}&=4 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
4.691 |
|
| \begin{align*}
3 y^{\prime }+y^{2}+\frac {2}{x^{2}}&=0 \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
3.141 |
|
| \begin{align*}
x y^{\prime }-\left (2 x +1\right ) y+y^{2}&=-x^{2} \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.292 |
|
| \begin{align*}
y^{\prime }-2 y x +y^{2}&=-x^{2}+5 \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.125 |
|
| \begin{align*}
y^{\prime }+2 y \,{\mathrm e}^{x}-y^{2}&={\mathrm e}^{x}+{\mathrm e}^{2 x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
10.431 |
|
| \begin{align*}
x^{2}+y+y^{2}-x y^{\prime }&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
1.342 |
|
| \begin{align*}
2 x y^{\prime }+y^{2}&=1 \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
5.702 |
|
| \begin{align*}
x^{2} \left (y^{\prime }-1\right )&=y \left (x +y\right ) \\
\end{align*} |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
9.744 |
|
| \begin{align*}
y^{\prime }&=\left (4 x +y-3\right )^{2} \\
\end{align*} |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
13.030 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
3.549 |
|