|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime } \sqrt {-x^{2}+1}+\sqrt {1-y^{2}} = 0
\] |
[_separable] |
✓ |
41.809 |
|
|
\[
{}y^{\prime } = \frac {\cos \left (x \right )}{\sin \left (y\right )}
\] |
[_separable] |
✓ |
2.572 |
|
|
\[
{}y^{\prime } = a y-b y^{2}
\] |
[_quadrature] |
✓ |
45.073 |
|
|
\[
{}y^{\prime }+y = \frac {2 x \,{\mathrm e}^{-x}}{1+y \,{\mathrm e}^{x}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.085 |
|
|
\[
{}x y^{\prime }-2 y = \frac {x^{6}}{y+x^{2}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
5.049 |
|
|
\[
{}y^{\prime }-y = \frac {\left (x +1\right ) {\mathrm e}^{4 x}}{\left (y+{\mathrm e}^{x}\right )^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
2.559 |
|
|
\[
{}y^{\prime }-2 y = \frac {x \,{\mathrm e}^{2 x}}{1-y \,{\mathrm e}^{-2 x}}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.145 |
|
|
\[
{}y^{\prime } = \frac {x^{2}+y^{2}}{\sin \left (x \right )}
\] |
[_Riccati] |
✗ |
9.571 |
|
|
\[
{}y^{\prime } = \frac {y+{\mathrm e}^{x}}{x^{2}+y^{2}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.841 |
|
|
\[
{}y^{\prime } = \tan \left (x y\right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.506 |
|
|
\[
{}y^{\prime } = \frac {x^{2}+y^{2}}{\ln \left (x y\right )}
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.793 |
|
|
\[
{}y^{\prime } = \left (x^{2}+y^{2}\right ) y^{{1}/{3}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.065 |
|
|
\[
{}y^{\prime } = 2 x y
\] |
[_separable] |
✓ |
1.627 |
|
|
\[
{}y^{\prime } = \ln \left (1+x^{2}+y^{2}\right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.086 |
|
|
\[
{}y^{\prime } = \frac {2 x +3 y}{x -4 y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.153 |
|
|
\[
{}y^{\prime } = \sqrt {x^{2}+y^{2}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.135 |
|
|
\[
{}y^{\prime } = x \left (y^{2}-1\right )^{{2}/{3}}
\] |
[_separable] |
✓ |
1.494 |
|
|
\[
{}y^{\prime } = \left (x^{2}+y^{2}\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✗ |
0.836 |
|
|
\[
{}y^{\prime } = \sqrt {x +y}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
2.661 |
|
|
\[
{}y^{\prime } = \frac {\tan \left (y\right )}{-1+x}
\] |
[_separable] |
✓ |
2.333 |
|
|
\[
{}y^{\prime } = y^{{2}/{5}}
\] |
[_quadrature] |
✓ |
2.609 |
|
|
\[
{}y^{\prime } = 3 x \left (y-1\right )^{{1}/{3}}
\] |
[_separable] |
✓ |
8.464 |
|
|
\[
{}y^{\prime } = 3 x \left (y-1\right )^{{1}/{3}}
\] |
[_separable] |
✓ |
83.155 |
|
|
\[
{}y^{\prime } = 3 x \left (y-1\right )^{{1}/{3}}
\] |
[_separable] |
✓ |
25.518 |
|
|
\[
{}y^{\prime }-y = x y^{2}
\] |
[_Bernoulli] |
✓ |
1.814 |
|
|
\[
{}y^{\prime } = \frac {y+x \,{\mathrm e}^{-\frac {y}{x}}}{x}
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
15.654 |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y-x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
0.266 |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y-x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
3.612 |
|
|
\[
{}y^{\prime }+y = y^{2}
\] |
[_quadrature] |
✓ |
0.270 |
|
|
\[
{}7 x y^{\prime }-2 y = -\frac {x^{2}}{y^{6}}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
0.327 |
|
|
\[
{}x^{2} y^{\prime }+2 y = 2 \,{\mathrm e}^{\frac {1}{x}} \sqrt {y}
\] |
[_Bernoulli] |
✓ |
0.555 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+2 x y = \frac {1}{\left (x^{2}+1\right ) y}
\] |
[_rational, _Bernoulli] |
✓ |
0.686 |
|
|
\[
{}y^{\prime }-x y = x^{3} y^{3}
\] |
[_Bernoulli] |
✓ |
0.725 |
|
|
\[
{}y^{\prime }-\frac {\left (x +1\right ) y}{3 x} = y^{4}
\] |
[_rational, _Bernoulli] |
✓ |
0.844 |
|
|
\[
{}y^{\prime }-2 y = x y^{3}
\] |
[_Bernoulli] |
✓ |
0.736 |
|
|
\[
{}y^{\prime }-x y = x y^{{3}/{2}}
\] |
[_separable] |
✓ |
24.540 |
|
|
\[
{}x y^{\prime }+y = x^{4} y^{4}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
0.802 |
|
|
\[
{}y^{\prime }-2 y = 2 \sqrt {y}
\] |
[_quadrature] |
✓ |
3.386 |
|
|
\[
{}y^{\prime }-4 y = \frac {48 x}{y^{2}}
\] |
[_rational, _Bernoulli] |
✓ |
1.035 |
|
|
\[
{}x^{2} y^{\prime }+2 x y = y^{3}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
1.060 |
|
|
\[
{}y^{\prime }-y = x \sqrt {y}
\] |
[_Bernoulli] |
✓ |
0.858 |
|
|
\[
{}y^{\prime } = \frac {x +y}{x}
\] |
[_linear] |
✓ |
1.513 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+2 x y}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.932 |
|
|
\[
{}x y^{3} y^{\prime } = y^{4}+x^{4}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
6.181 |
|
|
\[
{}y^{\prime } = \frac {y}{x}+\sec \left (\frac {y}{x}\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.260 |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y+x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
2.493 |
|
|
\[
{}x y y^{\prime } = x^{2}+2 y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
86.752 |
|
|
\[
{}y^{\prime } = \frac {2 y^{2}+x^{2} {\mathrm e}^{-\frac {y^{2}}{x^{2}}}}{2 x y}
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
5.283 |
|
|
\[
{}y^{\prime } = \frac {x y+y^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.722 |
|
|
\[
{}y^{\prime } = \frac {x^{3}+y^{3}}{x y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
290.692 |
|
|
\[
{}x y y^{\prime }+x^{2}+y^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
714.499 |
|
|
\[
{}y^{\prime } = \frac {y^{2}-3 x y-5 x^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
4.796 |
|
|
\[
{}x^{2} y^{\prime } = 2 x^{2}+y^{2}+4 x y
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
3.226 |
|
|
\[
{}x y y^{\prime } = 3 x^{2}+4 y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
88.963 |
|
|
\[
{}y^{\prime } = \frac {x +y}{x -y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
39.413 |
|
|
\[
{}\left (x y^{\prime }-y\right ) \left (\ln \left (y\right )-\ln \left (x \right )\right ) = x
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
12.823 |
|
|
\[
{}y^{\prime } = \frac {y^{3}+2 x y^{2}+x^{2} y+x^{3}}{x \left (x +y\right )^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
799.400 |
|
|
\[
{}y^{\prime } = \frac {x +2 y}{2 x +y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
6.735 |
|
|
\[
{}y^{\prime } = \frac {y}{y-2 x}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.651 |
|
|
\[
{}y^{\prime } = \frac {x y^{2}+2 y^{3}}{x^{3}+x^{2} y+x y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1268.050 |
|
|
\[
{}y^{\prime } = \frac {x^{3}+x^{2} y+3 y^{3}}{x^{3}+3 x y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
87.271 |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y-4 x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
4.674 |
|
|
\[
{}x y y^{\prime } = x^{2}-x y+y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
5.860 |
|
|
\[
{}y^{\prime } = \frac {2 y^{2}-x y+2 x^{2}}{x y+2 x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
103.635 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+x y+x^{2}}{x y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
37.773 |
|
|
\[
{}y^{\prime } = \frac {-6 x +y-3}{2 x -y-1}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.759 |
|
|
\[
{}y^{\prime } = \frac {2 x +y+1}{x +2 y-4}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
9.960 |
|
|
\[
{}y^{\prime } = \frac {-x +3 y-14}{x +y-2}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.785 |
|
|
\[
{}3 x y^{2} y^{\prime } = y^{3}+x
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
2.340 |
|
|
\[
{}x y y^{\prime } = 3 x^{6}+6 y^{2}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
2.570 |
|
|
\[
{}x^{3} y^{\prime } = 2 y^{2}+2 x^{2} y-2 x^{4}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
2.162 |
|
|
\[
{}y^{\prime } = y^{2} {\mathrm e}^{-x}+4 y+2 \,{\mathrm e}^{x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
1.659 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+y \tan \left (x \right )+\tan \left (x \right )^{2}}{\sin \left (x \right )^{2}}
\] |
[_Riccati] |
✓ |
45.931 |
|
|
\[
{}x \ln \left (x \right )^{2} y^{\prime } = -4 \ln \left (x \right )^{2}+y \ln \left (x \right )+y^{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Riccati] |
✓ |
1.954 |
|
|
\[
{}2 x \left (y+2 \sqrt {x}\right ) y^{\prime } = \left (y+\sqrt {x}\right )^{2}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.723 |
|
|
\[
{}\left (y+{\mathrm e}^{x^{2}}\right ) y^{\prime } = 2 x \left (y^{2}+y \,{\mathrm e}^{x^{2}}+{\mathrm e}^{2 x^{2}}\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.861 |
|
|
\[
{}y^{\prime }+\frac {2 y}{x} = \frac {3 y^{2} x^{2}+6 x y+2}{x^{2} \left (2 x y+3\right )}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
8.116 |
|
|
\[
{}y^{\prime }+\frac {3 y}{x} = \frac {3 x^{4} y^{2}+10 x^{2} y+6}{x^{3} \left (2 x^{2} y+5\right )}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
6.071 |
|
|
\[
{}y^{\prime } = 1+x -\left (2 x +1\right ) y+x y^{2}
\] |
[_Riccati] |
✓ |
2.265 |
|
|
\[
{}6 y^{2} x^{2}+4 x^{3} y y^{\prime } = 0
\] |
[_separable] |
✓ |
2.276 |
|
|
\[
{}3 \cos \left (x \right ) y+4 x \,{\mathrm e}^{x}+2 x^{3} y+\left (3 \sin \left (x \right )+3\right ) y^{\prime } = 0
\] |
[_linear] |
✓ |
127.794 |
|
|
\[
{}14 x^{2} y^{3}+21 x^{2} y^{2} y^{\prime } = 0
\] |
[_quadrature] |
✓ |
1.660 |
|
|
\[
{}2 x -2 y^{2}+\left (12 y^{2}-4 x y\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.245 |
|
|
\[
{}\left (x +y\right )^{2}+\left (x +y\right )^{2} y^{\prime } = 0
\] |
[_quadrature] |
✓ |
0.969 |
|
|
\[
{}4 x +7 y+\left (3 x +4 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
33.942 |
|
|
\[
{}-2 \sin \left (x \right ) y^{2}+3 y^{3}-2 x +\left (4 \cos \left (x \right ) y+9 x y^{2}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
34.225 |
|
|
\[
{}2 x +y+\left (2 y+2 x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.488 |
|
|
\[
{}3 x^{2}+2 x y+4 y^{2}+\left (x^{2}+8 x y+18 y\right ) y^{\prime } = 0
\] |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.608 |
|
|
\[
{}2 x^{2}+8 x y+y^{2}+\left (2 x^{2}+\frac {x y^{3}}{3}\right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
1.658 |
|
|
\[
{}\frac {1}{x}+2 x +\left (\frac {1}{y}+2 y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.033 |
|
|
\[
{}y \sin \left (x y\right )+x y^{2} \cos \left (x y\right )+\left (x \sin \left (x y\right )+x y^{2} \cos \left (x y\right )\right ) y^{\prime } = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
78.587 |
|
|
\[
{}\frac {x}{\left (x^{2}+y^{2}\right )^{{3}/{2}}}+\frac {y y^{\prime }}{\left (x^{2}+y^{2}\right )^{{3}/{2}}} = 0
\] |
[_separable] |
✓ |
4.055 |
|
|
\[
{}{\mathrm e}^{x} \left (y^{2} x^{2}+2 x y^{2}\right )+6 x +\left (2 x^{2} y \,{\mathrm e}^{x}+2\right ) y^{\prime } = 0
\] |
[_exact, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.167 |
|
|
\[
{}x^{2} {\mathrm e}^{y+x^{2}} \left (2 x^{2}+3\right )+4 x +\left (x^{3} {\mathrm e}^{y+x^{2}}-12 y^{2}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
3.836 |
|
|
\[
{}{\mathrm e}^{x y} \left (x^{4} y+4 x^{3}\right )+3 y+\left (x^{5} {\mathrm e}^{x y}+3 x \right ) y^{\prime } = 0
\] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
7.665 |
|
|
\[
{}3 x^{2} \cos \left (x \right ) y-x^{3} y \sin \left (x \right )+4 x +\left (8 y-x^{4} \sin \left (x \right ) y\right ) y^{\prime } = 0
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
56.742 |
|
|
\[
{}4 x^{3} y^{2}-6 x^{2} y-2 x -3+\left (2 x^{4} y-2 x^{3}\right ) y^{\prime } = 0
\] |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.283 |
|
|
\[
{}-4 \cos \left (x \right ) y+4 \sin \left (x \right ) \cos \left (x \right )+\sec \left (x \right )^{2}+\left (4 y-4 \sin \left (x \right )\right ) y^{\prime } = 0
\] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
7.455 |
|
|
\[
{}\left (y^{3}-1\right ) {\mathrm e}^{x}+3 y^{2} \left ({\mathrm e}^{x}+1\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
3.276 |
|
|
\[
{}\sin \left (x \right )-\sin \left (x \right ) y-2 \cos \left (x \right )+\cos \left (x \right ) y^{\prime } = 0
\] |
[_linear] |
✓ |
2.571 |
|