|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime } \sqrt {-x^{2}+1}+\sqrt {1-y^{2}} = 0
\] |
[_separable] |
✓ |
18.164 |
|
|
\[
{}y^{\prime } = \frac {\cos \left (x \right )}{\sin \left (y\right )}
\] |
[_separable] |
✓ |
2.437 |
|
|
\[
{}y^{\prime } = a y-b y^{2}
\] |
[_quadrature] |
✓ |
1.940 |
|
|
\[
{}y^{\prime }+y = \frac {2 x \,{\mathrm e}^{-x}}{1+y \,{\mathrm e}^{x}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.970 |
|
|
\[
{}x y^{\prime }-2 y = \frac {x^{6}}{y+x^{2}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
4.363 |
|
|
\[
{}y^{\prime }-y = \frac {\left (x +1\right ) {\mathrm e}^{4 x}}{\left (y+{\mathrm e}^{x}\right )^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
2.623 |
|
|
\[
{}y^{\prime }-2 y = \frac {x \,{\mathrm e}^{2 x}}{1-y \,{\mathrm e}^{-2 x}}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.963 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+x^{2}}{\sin \left (x \right )}
\] |
[_Riccati] |
✗ |
8.820 |
|
|
\[
{}y^{\prime } = \frac {y+{\mathrm e}^{x}}{y^{2}+x^{2}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.860 |
|
|
\[
{}y^{\prime } = \tan \left (x y\right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.372 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+x^{2}}{\ln \left (x y\right )}
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.748 |
|
|
\[
{}y^{\prime } = \left (y^{2}+x^{2}\right ) y^{{1}/{3}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.173 |
|
|
\[
{}y^{\prime } = 2 x y
\] |
[_separable] |
✓ |
1.170 |
|
|
\[
{}y^{\prime } = \ln \left (1+x^{2}+y^{2}\right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.041 |
|
|
\[
{}y^{\prime } = \frac {2 x +3 y}{x -4 y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.017 |
|
|
\[
{}y^{\prime } = \sqrt {y^{2}+x^{2}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.139 |
|
|
\[
{}y^{\prime } = x \left (y^{2}-1\right )^{{2}/{3}}
\] |
[_separable] |
✓ |
1.503 |
|
|
\[
{}y^{\prime } = \left (y^{2}+x^{2}\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✗ |
0.898 |
|
|
\[
{}y^{\prime } = \sqrt {x +y}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
2.419 |
|
|
\[
{}y^{\prime } = \frac {\tan \left (y\right )}{x -1}
\] |
[_separable] |
✓ |
1.936 |
|
|
\[
{}y^{\prime } = y^{{2}/{5}}
\] |
[_quadrature] |
✓ |
2.315 |
|
|
\[
{}y^{\prime } = 3 x \left (y-1\right )^{{1}/{3}}
\] |
[_separable] |
✓ |
6.898 |
|
|
\[
{}y^{\prime } = 3 x \left (y-1\right )^{{1}/{3}}
\] |
[_separable] |
✓ |
22.707 |
|
|
\[
{}y^{\prime } = 3 x \left (y-1\right )^{{1}/{3}}
\] |
[_separable] |
✓ |
7.513 |
|
|
\[
{}y^{\prime }-y = x y^{2}
\] |
[_Bernoulli] |
✓ |
1.500 |
|
|
\[
{}y^{\prime } = \frac {y+x \,{\mathrm e}^{-\frac {y}{x}}}{x}
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
10.658 |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y-x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
0.201 |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y-x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
3.193 |
|
|
\[
{}y^{\prime }+y = y^{2}
\] |
[_quadrature] |
✓ |
0.206 |
|
|
\[
{}7 x y^{\prime }-2 y = -\frac {x^{2}}{y^{6}}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
0.303 |
|
|
\[
{}x^{2} y^{\prime }+2 y = 2 \,{\mathrm e}^{\frac {1}{x}} \sqrt {y}
\] |
[_Bernoulli] |
✓ |
0.488 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+2 x y = \frac {1}{\left (x^{2}+1\right ) y}
\] |
[_rational, _Bernoulli] |
✓ |
0.465 |
|
|
\[
{}y^{\prime }-x y = x^{3} y^{3}
\] |
[_Bernoulli] |
✓ |
0.443 |
|
|
\[
{}y^{\prime }-\frac {\left (x +1\right ) y}{3 x} = y^{4}
\] |
[_rational, _Bernoulli] |
✓ |
0.679 |
|
|
\[
{}y^{\prime }-2 y = x y^{3}
\] |
[_Bernoulli] |
✓ |
0.654 |
|
|
\[
{}y^{\prime }-x y = x y^{{3}/{2}}
\] |
[_separable] |
✓ |
8.280 |
|
|
\[
{}x y^{\prime }+y = x^{4} y^{4}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
0.734 |
|
|
\[
{}y^{\prime }-2 y = 2 \sqrt {y}
\] |
[_quadrature] |
✓ |
2.732 |
|
|
\[
{}y^{\prime }-4 y = \frac {48 x}{y^{2}}
\] |
[_rational, _Bernoulli] |
✓ |
0.990 |
|
|
\[
{}x^{2} y^{\prime }+2 x y = y^{3}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
0.918 |
|
|
\[
{}y^{\prime }-y = x \sqrt {y}
\] |
[_Bernoulli] |
✓ |
0.761 |
|
|
\[
{}y^{\prime } = \frac {x +y}{x}
\] |
[_linear] |
✓ |
1.148 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+2 x y}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.240 |
|
|
\[
{}x y^{3} y^{\prime } = y^{4}+x^{4}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
5.530 |
|
|
\[
{}y^{\prime } = \frac {y}{x}+\sec \left (\frac {y}{x}\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.757 |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y+x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
2.129 |
|
|
\[
{}x y y^{\prime } = x^{2}+2 y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
4.322 |
|
|
\[
{}y^{\prime } = \frac {2 y^{2}+x^{2} {\mathrm e}^{-\frac {y^{2}}{x^{2}}}}{2 x y}
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
2.888 |
|
|
\[
{}y^{\prime } = \frac {x y+y^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.191 |
|
|
\[
{}y^{\prime } = \frac {x^{3}+y^{3}}{x y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
95.176 |
|
|
\[
{}x y y^{\prime }+x^{2}+y^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
6.204 |
|
|
\[
{}y^{\prime } = \frac {y^{2}-3 x y-5 x^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
3.440 |
|
|
\[
{}x^{2} y^{\prime } = 2 x^{2}+y^{2}+4 x y
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
3.023 |
|
|
\[
{}x y y^{\prime } = 3 x^{2}+4 y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
7.737 |
|
|
\[
{}y^{\prime } = \frac {x +y}{x -y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.727 |
|
|
\[
{}\left (-y+x y^{\prime }\right ) \left (\ln \left (y\right )-\ln \left (x \right )\right ) = x
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
10.687 |
|
|
\[
{}y^{\prime } = \frac {y^{3}+2 x y^{2}+x^{2} y+x^{3}}{x \left (x +y\right )^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
45.013 |
|
|
\[
{}y^{\prime } = \frac {x +2 y}{2 x +y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.496 |
|
|
\[
{}y^{\prime } = \frac {y}{y-2 x}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.666 |
|
|
\[
{}y^{\prime } = \frac {x y^{2}+2 y^{3}}{x^{3}+x^{2} y+x y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
416.310 |
|
|
\[
{}y^{\prime } = \frac {x^{3}+x^{2} y+3 y^{3}}{x^{3}+3 x y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
38.042 |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+x y-4 x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
3.286 |
|
|
\[
{}x y y^{\prime } = x^{2}-x y+y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
4.563 |
|
|
\[
{}y^{\prime } = \frac {2 y^{2}-x y+2 x^{2}}{x y+2 x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
49.509 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+x y+x^{2}}{x y}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
37.189 |
|
|
\[
{}y^{\prime } = \frac {-6 x +y-3}{2 x -y-1}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.397 |
|
|
\[
{}y^{\prime } = \frac {2 x +y+1}{x +2 y-4}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
12.144 |
|
|
\[
{}y^{\prime } = \frac {-x +3 y-14}{x +y-2}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.993 |
|
|
\[
{}3 x y^{2} y^{\prime } = y^{3}+x
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
2.148 |
|
|
\[
{}x y y^{\prime } = 3 x^{6}+6 y^{2}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
2.169 |
|
|
\[
{}x^{3} y^{\prime } = 2 y^{2}+2 x^{2} y-2 x^{4}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
1.840 |
|
|
\[
{}y^{\prime } = y^{2} {\mathrm e}^{-x}+4 y+2 \,{\mathrm e}^{x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
1.404 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+y \tan \left (x \right )+\tan \left (x \right )^{2}}{\sin \left (x \right )^{2}}
\] |
[_Riccati] |
✓ |
44.389 |
|
|
\[
{}x \ln \left (x \right )^{2} y^{\prime } = -4 \ln \left (x \right )^{2}+y \ln \left (x \right )+y^{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Riccati] |
✓ |
2.082 |
|
|
\[
{}2 x \left (y+2 \sqrt {x}\right ) y^{\prime } = \left (y+\sqrt {x}\right )^{2}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.264 |
|
|
\[
{}\left (y+{\mathrm e}^{x^{2}}\right ) y^{\prime } = 2 x \left (y^{2}+y \,{\mathrm e}^{x^{2}}+{\mathrm e}^{2 x^{2}}\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.506 |
|
|
\[
{}y^{\prime }+\frac {2 y}{x} = \frac {3 y^{2} x^{2}+6 x y+2}{x^{2} \left (2 x y+3\right )}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
7.950 |
|
|
\[
{}y^{\prime }+\frac {3 y}{x} = \frac {3 x^{4} y^{2}+10 x^{2} y+6}{x^{3} \left (2 x^{2} y+5\right )}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
6.789 |
|
|
\[
{}y^{\prime } = 1+x -\left (2 x +1\right ) y+x y^{2}
\] |
[_Riccati] |
✓ |
1.969 |
|
|
\[
{}6 y^{2} x^{2}+4 x^{3} y y^{\prime } = 0
\] |
[_separable] |
✓ |
1.768 |
|
|
\[
{}3 \cos \left (x \right ) y+4 x \,{\mathrm e}^{x}+2 x^{3} y+\left (3 \sin \left (x \right )+3\right ) y^{\prime } = 0
\] |
[_linear] |
✓ |
102.764 |
|
|
\[
{}14 x^{2} y^{3}+21 x^{2} y^{2} y^{\prime } = 0
\] |
[_quadrature] |
✓ |
1.069 |
|
|
\[
{}2 x -2 y^{2}+\left (12 y^{2}-4 x y\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.183 |
|
|
\[
{}\left (x +y\right )^{2}+\left (x +y\right )^{2} y^{\prime } = 0
\] |
[_quadrature] |
✓ |
0.532 |
|
|
\[
{}4 x +7 y+\left (3 x +4 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.591 |
|
|
\[
{}-2 y^{2} \sin \left (x \right )+3 y^{3}-2 x +\left (4 \cos \left (x \right ) y+9 x y^{2}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
33.710 |
|
|
\[
{}2 x +y+\left (2 x +2 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.250 |
|
|
\[
{}3 x^{2}+2 x y+4 y^{2}+\left (x^{2}+8 x y+18 y\right ) y^{\prime } = 0
\] |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.446 |
|
|
\[
{}2 x^{2}+8 x y+y^{2}+\left (2 x^{2}+\frac {x y^{3}}{3}\right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
1.640 |
|
|
\[
{}\frac {1}{x}+2 x +\left (\frac {1}{y}+2 y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
1.875 |
|
|
\[
{}y \sin \left (x y\right )+x y^{2} \cos \left (x y\right )+\left (x \sin \left (x y\right )+x y^{2} \cos \left (x y\right )\right ) y^{\prime } = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
87.903 |
|
|
\[
{}\frac {x}{\left (y^{2}+x^{2}\right )^{{3}/{2}}}+\frac {y y^{\prime }}{\left (y^{2}+x^{2}\right )^{{3}/{2}}} = 0
\] |
[_separable] |
✓ |
4.681 |
|
|
\[
{}{\mathrm e}^{x} \left (y^{2} x^{2}+2 x y^{2}\right )+6 x +\left (2 x^{2} y \,{\mathrm e}^{x}+2\right ) y^{\prime } = 0
\] |
[_exact, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.966 |
|
|
\[
{}x^{2} {\mathrm e}^{y+x^{2}} \left (2 x^{2}+3\right )+4 x +\left (x^{3} {\mathrm e}^{y+x^{2}}-12 y^{2}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
2.761 |
|
|
\[
{}{\mathrm e}^{x y} \left (x^{4} y+4 x^{3}\right )+3 y+\left (x^{5} {\mathrm e}^{x y}+3 x \right ) y^{\prime } = 0
\] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
6.742 |
|
|
\[
{}3 x^{2} \cos \left (x \right ) y-x^{3} y \sin \left (x \right )+4 x +\left (8 y-x^{4} \sin \left (x \right ) y\right ) y^{\prime } = 0
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
53.563 |
|
|
\[
{}4 x^{3} y^{2}-6 x^{2} y-2 x -3+\left (2 x^{4} y-2 x^{3}\right ) y^{\prime } = 0
\] |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.221 |
|
|
\[
{}-4 \cos \left (x \right ) y+4 \sin \left (x \right ) \cos \left (x \right )+\sec \left (x \right )^{2}+\left (4 y-4 \sin \left (x \right )\right ) y^{\prime } = 0
\] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
6.694 |
|
|
\[
{}\left (y^{3}-1\right ) {\mathrm e}^{x}+3 y^{2} \left (1+{\mathrm e}^{x}\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.977 |
|
|
\[
{}\sin \left (x \right )-y \sin \left (x \right )-2 \cos \left (x \right )+\cos \left (x \right ) y^{\prime } = 0
\] |
[_linear] |
✓ |
2.395 |
|