|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x y \left (x^{2}+1\right ) y^{\prime }-1-y^{2} = 0
\] |
[_separable] |
✓ |
4.222 |
|
|
\[
{}1+y^{2}-\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{{3}/{2}} y^{\prime } = 0
\] |
[_separable] |
✓ |
2.910 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
3.146 |
|
|
\[
{}\sec \left (x \right )^{2} \tan \left (y\right )+\sec \left (y\right )^{2} \tan \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
38.622 |
|
|
\[
{}\left (y-x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.208 |
|
|
\[
{}\left (2 \sqrt {x y}-x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
24.925 |
|
|
\[
{}x y^{\prime }-y-\sqrt {x^{2}+y^{2}} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
8.448 |
|
|
\[
{}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.747 |
|
|
\[
{}8 y+10 x +\left (5 y+7 x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.429 |
|
|
\[
{}2 x -y+1+\left (2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.723 |
|
|
\[
{}\left (3-3 x +7 y\right ) y^{\prime }+7-7 x +3 y = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.665 |
|
|
\[
{}y^{\prime }+\frac {x y}{x^{2}+1} = \frac {1}{2 x \left (x^{2}+1\right )}
\] |
[_linear] |
✓ |
1.458 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime }+\left (2 x^{2}-1\right ) y = a \,x^{3}
\] |
[_linear] |
✓ |
1.200 |
|
|
\[
{}y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{{3}/{2}}} = \frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}}
\] |
[_linear] |
✓ |
4.172 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2}
\] |
[_linear] |
✓ |
2.520 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+y = \arctan \left (x \right )
\] |
[_linear] |
✓ |
2.137 |
|
|
\[
{}\left (-x^{2}+1\right ) z^{\prime }-x z = a x z^{2}
\] |
[_separable] |
✓ |
2.433 |
|
|
\[
{}3 z^{2} z^{\prime }-a z^{3} = x +1
\] |
[_rational, _Bernoulli] |
✓ |
1.781 |
|
|
\[
{}z^{\prime }+2 x z = 2 a \,x^{3} z^{3}
\] |
[_Bernoulli] |
✓ |
1.413 |
|
|
\[
{}z^{\prime }+z \cos \left (x \right ) = z^{n} \sin \left (2 x \right )
\] |
[_Bernoulli] |
✓ |
6.079 |
|
|
\[
{}x y^{\prime }+y = y^{2} \ln \left (x \right )
\] |
[_Bernoulli] |
✓ |
2.421 |
|
|
\[
{}x^{3}+3 x y^{2}+\left (y^{3}+3 x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
1.036 |
|
|
\[
{}1+\frac {y^{2}}{x^{2}}-\frac {2 y y^{\prime }}{x} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
0.530 |
|
|
\[
{}\frac {3 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.559 |
|
|
\[
{}x +y y^{\prime }+\frac {x y^{\prime }}{x^{2}+y^{2}}-\frac {y}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
0.494 |
|
|
\[
{}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
0.492 |
|
|
\[
{}{\mathrm e}^{x} \left (x^{2}+y^{2}+2 x \right )+2 y \,{\mathrm e}^{x} y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _exact, _rational, _Bernoulli] |
✓ |
0.579 |
|
|
\[
{}n \cos \left (n x +m y\right )-m \sin \left (m x +n y\right )+\left (m \cos \left (n x +m y\right )-n \sin \left (m x +n y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.334 |
|
|
\[
{}\frac {x}{\sqrt {1+x^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y}{x^{2}+y^{2}}-\frac {x y^{\prime }}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact] |
✓ |
0.640 |
|
|
\[
{}\frac {x^{n} y^{\prime }}{b y^{2}-c \,x^{2 a}}-\frac {a y x^{a -1}}{b y^{2}-c \,x^{2 a}}+x^{a -1} = 0
\] |
[_Riccati] |
✗ |
13.390 |
|
|
\[
{}2 x y+\left (-2 x^{2}+y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.523 |
|
|
\[
{}\frac {1}{x}+\frac {y^{\prime }}{y}+\frac {2}{y}-\frac {2 y^{\prime }}{x} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.502 |
|
|
\[
{}-y+x y^{\prime } = \sqrt {x^{2}+y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
8.414 |
|
|
\[
{}8 y+10 x +\left (5 y+7 x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.311 |
|
|
\[
{}x^{2}+2 x y-y^{2}+\left (y^{2}+2 x y-x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
34.085 |
|
|
\[
{}y^{2}+\left (x y+x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
159.136 |
|
|
\[
{}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y+\left (x \cos \left (\frac {y}{x}\right )-y \sin \left (\frac {y}{x}\right )\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
0.594 |
|
|
\[
{}\left (y^{2} x^{2}+x y\right ) y+\left (y^{2} x^{2}-1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.725 |
|
|
\[
{}\left (x^{3} y^{3}+y^{2} x^{2}+x y+1\right ) y+\left (x^{3} y^{3}-y^{2} x^{2}-x y+1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.180 |
|
|
\[
{}x^{2}+y^{2}+2 x +2 y y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
0.563 |
|
|
\[
{}x^{2}+y^{2}-2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
0.538 |
|
|
\[
{}2 x y+\left (y^{2}-3 x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.369 |
|
|
\[
{}y+\left (2 y-x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.493 |
|
|
\[
{}x y^{\prime }-a y+y^{2} = x^{-2 a}
\] |
[_rational, _Riccati] |
✓ |
0.770 |
|
|
\[
{}x y^{\prime }-a y+y^{2} = x^{-\frac {2 a}{3}}
\] |
[_rational, _Riccati] |
✓ |
2.750 |
|
|
\[
{}u^{\prime }+u^{2} = \frac {c}{x^{{4}/{3}}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.370 |
|
|
\[
{}u^{\prime }+b u^{2} = \frac {c}{x^{4}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.342 |
|
|
\[
{}u^{\prime }-u^{2} = \frac {2}{x^{{8}/{3}}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.492 |
|
|
\[
{}\frac {\sqrt {f \,x^{4}+c \,x^{3}+c \,x^{2}+b x +a}\, y^{\prime }}{\sqrt {a +b y+c y^{2}+c y^{3}+f y^{4}}} = -1
\] |
[_separable] |
✓ |
10.788 |
|
|
\[
{}{y^{\prime }}^{2}-5 y^{\prime }+6 = 0
\] |
[_quadrature] |
✓ |
0.346 |
|
|
\[
{}{y^{\prime }}^{2}-\frac {a^{2}}{x^{2}} = 0
\] |
[_quadrature] |
✓ |
0.694 |
|
|
\[
{}{y^{\prime }}^{2} = \frac {1-x}{x}
\] |
[_quadrature] |
✓ |
0.378 |
|
|
\[
{}{y^{\prime }}^{2}+\frac {2 x y^{\prime }}{y}-1 = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.838 |
|
|
\[
{}y = a y^{\prime }+b {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
0.516 |
|
|
\[
{}x = a y^{\prime }+b {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
0.284 |
|
|
\[
{}y = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }
\] |
[_quadrature] |
✓ |
2.144 |
|
|
\[
{}x = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }
\] |
[_quadrature] |
✓ |
1.282 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {1+{y^{\prime }}^{2}}}{x} = 0
\] |
[_quadrature] |
✓ |
1.285 |
|
|
\[
{}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0
\] |
[_quadrature] |
✓ |
4.279 |
|
|
\[
{}1+{y^{\prime }}^{2} = \frac {\left (x +a \right )^{2}}{2 a x +x^{2}}
\] |
[_quadrature] |
✓ |
0.756 |
|
|
\[
{}y = x y^{\prime }+y^{\prime }-{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.441 |
|
|
\[
{}y = x y^{\prime }+\sqrt {b^{2}-a^{2} {y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
3.414 |
|
|
\[
{}y = x y^{\prime }+x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
8.924 |
|
|
\[
{}y = x y^{\prime }+a x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
32.298 |
|
|
\[
{}x -y y^{\prime } = a {y^{\prime }}^{2}
\] |
[_dAlembert] |
✓ |
75.804 |
|
|
\[
{}y y^{\prime }+x = a \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
34.554 |
|
|
\[
{}y y^{\prime } = x +y^{2}-y^{2} {y^{\prime }}^{2}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
8.602 |
|
|
\[
{}y-\frac {1}{\sqrt {1+{y^{\prime }}^{2}}} = x +\frac {y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.737 |
|
|
\[
{}y-2 x y^{\prime } = x {y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
13.089 |
|
|
\[
{}\frac {y-x y^{\prime }}{y^{\prime }+y^{2}} = \frac {y-x y^{\prime }}{1+x^{2} y^{\prime }}
\] |
[_separable] |
✓ |
0.495 |
|
|
\[
{}2 x y+\left (x^{2}+y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
17.593 |
|
|
\[
{}\left (x +\sqrt {y^{2}-x y}\right ) y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
159.056 |
|
|
\[
{}x +y-\left (x -y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
39.329 |
|
|
\[
{}x y^{\prime }-y-x \sin \left (\frac {y}{x}\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
6.524 |
|
|
\[
{}2 x^{2} y+y^{3}+\left (x y^{2}-2 x^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
45.921 |
|
|
\[
{}y^{2}+\left (x \sqrt {y^{2}-x^{2}}-x y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
13.211 |
|
|
\[
{}\frac {y \cos \left (\frac {y}{x}\right )}{x}-\left (\frac {x \sin \left (\frac {y}{x}\right )}{y}+\cos \left (\frac {y}{x}\right )\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.442 |
|
|
\[
{}y+x \ln \left (\frac {y}{x}\right ) y^{\prime }-2 x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
9.961 |
|
|
\[
{}2 y \,{\mathrm e}^{\frac {x}{y}}+\left (y-2 x \,{\mathrm e}^{\frac {x}{y}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
3.984 |
|
|
\[
{}x \,{\mathrm e}^{\frac {y}{x}}-y \sin \left (\frac {y}{x}\right )+x \sin \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
6.221 |
|
|
\[
{}x^{2}+y^{2} = 2 x y y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
9.250 |
|
|
\[
{}x \,{\mathrm e}^{\frac {y}{x}}+y = x y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.174 |
|
|
\[
{}y^{\prime }-\frac {y}{x}+\csc \left (\frac {y}{x}\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.470 |
|
|
\[
{}x y-y^{2}-x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
3.418 |
|
|
\[
{}x +2 y-4-\left (2 x -4 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
37.100 |
|
|
\[
{}3 x +2 y+1-\left (3 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.986 |
|
|
\[
{}x +y+1+\left (2 x +2 y+2\right ) y^{\prime } = 0
\] |
[_quadrature] |
✓ |
0.990 |
|
|
\[
{}x +y-1+\left (2 x +2 y-3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.904 |
|
|
\[
{}x +y-1-\left (x -y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
36.842 |
|
|
\[
{}x +y+\left (2 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.895 |
|
|
\[
{}7 y-3+\left (2 x +1\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.554 |
|
|
\[
{}x +2 y+\left (3 x +6 y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.875 |
|
|
\[
{}x +2 y+\left (y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.168 |
|
|
\[
{}3 x -2 y+4-\left (2 x +7 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
40.550 |
|
|
\[
{}x +y+\left (3 x +3 y-4\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.477 |
|
|
\[
{}3 x +2 y+3-\left (x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.226 |
|
|
\[
{}y+7+\left (2 x +y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
10.878 |
|
|
\[
{}x +y+2-\left (x -y-4\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
37.286 |
|
|
\[
{}3 x^{2} y+8 x y^{2}+\left (x^{3}+8 x^{2} y+12 y^{2}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
0.342 |
|
|
\[
{}\frac {2 x y+1}{y}+\frac {\left (y-x \right ) y^{\prime }}{y^{2}} = 0
\] |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.502 |
|