|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x y \left (x^{2}+1\right ) y^{\prime }-1-y^{2} = 0
\] |
[_separable] |
✓ |
3.272 |
|
|
\[
{}1+y^{2}-\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{{3}/{2}} y^{\prime } = 0
\] |
[_separable] |
✓ |
2.972 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.847 |
|
|
\[
{}\sec \left (x \right )^{2} \tan \left (y\right )+\sec \left (y\right )^{2} \tan \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
37.282 |
|
|
\[
{}\left (y-x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.106 |
|
|
\[
{}\left (2 \sqrt {x y}-x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
95.435 |
|
|
\[
{}y^{\prime } x -y-\sqrt {x^{2}+y^{2}} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.686 |
|
|
\[
{}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.994 |
|
|
\[
{}8 y+10 x +\left (7 x +5 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.174 |
|
|
\[
{}2 x -y+1+\left (2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.312 |
|
|
\[
{}\left (3-3 x +7 y\right ) y^{\prime }+7-7 x +3 y = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.308 |
|
|
\[
{}y^{\prime }+\frac {x y}{x^{2}+1} = \frac {1}{2 x \left (x^{2}+1\right )}
\] |
[_linear] |
✓ |
1.366 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime }+\left (2 x^{2}-1\right ) y = a \,x^{3}
\] |
[_linear] |
✓ |
1.151 |
|
|
\[
{}y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{{3}/{2}}} = \frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}}
\] |
[_linear] |
✓ |
3.510 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2}
\] |
[_linear] |
✓ |
2.309 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+y = \arctan \left (x \right )
\] |
[_linear] |
✓ |
1.978 |
|
|
\[
{}\left (-x^{2}+1\right ) z^{\prime }-x z = a x z^{2}
\] |
[_separable] |
✓ |
2.183 |
|
|
\[
{}3 z^{2} z^{\prime }-a z^{3} = x +1
\] |
[_rational, _Bernoulli] |
✓ |
1.640 |
|
|
\[
{}z^{\prime }+2 x z = 2 a \,x^{3} z^{3}
\] |
[_Bernoulli] |
✓ |
1.209 |
|
|
\[
{}z^{\prime }+z \cos \left (x \right ) = z^{n} \sin \left (2 x \right )
\] |
[_Bernoulli] |
✓ |
4.960 |
|
|
\[
{}y^{\prime } x +y = y^{2} \ln \left (x \right )
\] |
[_Bernoulli] |
✓ |
2.121 |
|
|
\[
{}x^{3}+3 x y^{2}+\left (y^{3}+3 x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
0.561 |
|
|
\[
{}1+\frac {y^{2}}{x^{2}}-\frac {2 y y^{\prime }}{x} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
0.375 |
|
|
\[
{}\frac {3 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.483 |
|
|
\[
{}x +y y^{\prime }+\frac {x y^{\prime }}{x^{2}+y^{2}}-\frac {y}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
0.449 |
|
|
\[
{}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
0.432 |
|
|
\[
{}{\mathrm e}^{x} \left (x^{2}+y^{2}+2 x \right )+2 y \,{\mathrm e}^{x} y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _exact, _rational, _Bernoulli] |
✓ |
0.448 |
|
|
\[
{}n \cos \left (n x +m y\right )-m \sin \left (m x +n y\right )+\left (m \cos \left (n x +m y\right )-n \sin \left (m x +n y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.287 |
|
|
\[
{}\frac {x}{\sqrt {1+x^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y}{x^{2}+y^{2}}-\frac {x y^{\prime }}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact] |
✓ |
0.722 |
|
|
\[
{}\frac {x^{n} y^{\prime }}{b y^{2}-c \,x^{2 a}}-\frac {a y x^{a -1}}{b y^{2}-c \,x^{2 a}}+x^{a -1} = 0
\] |
[_Riccati] |
✗ |
7.141 |
|
|
\[
{}2 x y+\left (y^{2}-2 x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.500 |
|
|
\[
{}\frac {1}{x}+\frac {y^{\prime }}{y}+\frac {2}{y}-\frac {2 y^{\prime }}{x} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.451 |
|
|
\[
{}-y+y^{\prime } x = \sqrt {x^{2}+y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
7.216 |
|
|
\[
{}8 y+10 x +\left (7 x +5 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.307 |
|
|
\[
{}x^{2}+2 x y-y^{2}+\left (y^{2}+2 x y-x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.388 |
|
|
\[
{}y^{2}+\left (x y+x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
6.510 |
|
|
\[
{}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y+\left (x \cos \left (\frac {y}{x}\right )-y \sin \left (\frac {y}{x}\right )\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
0.525 |
|
|
\[
{}\left (x^{2} y^{2}+x y\right ) y+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.686 |
|
|
\[
{}\left (x^{3} y^{3}+x^{2} y^{2}+x y+1\right ) y+\left (x^{3} y^{3}-x^{2} y^{2}-x y+1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.132 |
|
|
\[
{}2 y y^{\prime }+2 x +x^{2}+y^{2} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
0.513 |
|
|
\[
{}x^{2}+y^{2}-2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
0.444 |
|
|
\[
{}2 x y+\left (y^{2}-3 x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.424 |
|
|
\[
{}y+\left (2 y-x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.469 |
|
|
\[
{}y^{\prime } x -a y+y^{2} = x^{-2 a}
\] |
[_rational, _Riccati] |
✓ |
0.687 |
|
|
\[
{}y^{\prime } x -a y+y^{2} = x^{-\frac {2 a}{3}}
\] |
[_rational, _Riccati] |
✓ |
2.572 |
|
|
\[
{}u^{\prime }+u^{2} = \frac {c}{x^{{4}/{3}}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.344 |
|
|
\[
{}u^{\prime }+b u^{2} = \frac {c}{x^{4}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.289 |
|
|
\[
{}u^{\prime }-u^{2} = \frac {2}{x^{{8}/{3}}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.414 |
|
|
\[
{}\frac {\sqrt {f \,x^{4}+c \,x^{3}+c \,x^{2}+b x +a}\, y^{\prime }}{\sqrt {a +b y+c y^{2}+c y^{3}+f y^{4}}} = -1
\] |
[_separable] |
✓ |
10.859 |
|
|
\[
{}{y^{\prime }}^{2}-5 y^{\prime }+6 = 0
\] |
[_quadrature] |
✓ |
1.433 |
|
|
\[
{}{y^{\prime }}^{2}-\frac {a^{2}}{x^{2}} = 0
\] |
[_quadrature] |
✓ |
0.411 |
|
|
\[
{}{y^{\prime }}^{2} = \frac {1-x}{x}
\] |
[_quadrature] |
✓ |
0.306 |
|
|
\[
{}{y^{\prime }}^{2}+\frac {2 x y^{\prime }}{y}-1 = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
2.013 |
|
|
\[
{}y = a y^{\prime }+b {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
0.625 |
|
|
\[
{}x = a y^{\prime }+b {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
0.234 |
|
|
\[
{}y = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }
\] |
[_quadrature] |
✓ |
1.776 |
|
|
\[
{}x = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }
\] |
[_quadrature] |
✓ |
1.101 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {1+{y^{\prime }}^{2}}}{x} = 0
\] |
[_quadrature] |
✓ |
1.137 |
|
|
\[
{}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0
\] |
[_quadrature] |
✓ |
2.302 |
|
|
\[
{}1+{y^{\prime }}^{2} = \frac {\left (x +a \right )^{2}}{2 a x +x^{2}}
\] |
[_quadrature] |
✓ |
0.647 |
|
|
\[
{}y = y^{\prime } x +y^{\prime }-{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.401 |
|
|
\[
{}y = y^{\prime } x +\sqrt {b^{2}-a^{2} {y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
3.598 |
|
|
\[
{}y = y^{\prime } x +x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
33.130 |
|
|
\[
{}y = y^{\prime } x +a x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✗ |
315.136 |
|
|
\[
{}x -y y^{\prime } = a {y^{\prime }}^{2}
\] |
[_dAlembert] |
✓ |
509.929 |
|
|
\[
{}x +y y^{\prime } = a \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✗ |
459.137 |
|
|
\[
{}y y^{\prime } = x +y^{2}-y^{2} {y^{\prime }}^{2}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
7.529 |
|
|
\[
{}y-\frac {1}{\sqrt {1+{y^{\prime }}^{2}}} = x +\frac {y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.731 |
|
|
\[
{}y-2 y^{\prime } x = x {y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.271 |
|
|
\[
{}\frac {y-y^{\prime } x}{y^{2}+y^{\prime }} = \frac {y-y^{\prime } x}{1+x^{2} y^{\prime }}
\] |
[_separable] |
✓ |
0.555 |
|
|
\[
{}2 x y+\left (x^{2}+y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
4.198 |
|
|
\[
{}\left (x +\sqrt {y^{2}-x y}\right ) y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.023 |
|
|
\[
{}x +y-\left (x -y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.686 |
|
|
\[
{}y^{\prime } x -y-x \sin \left (\frac {y}{x}\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
3.885 |
|
|
\[
{}2 x^{2} y+y^{3}+\left (x y^{2}-2 x^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
24.315 |
|
|
\[
{}y^{2}+\left (x \sqrt {y^{2}-x^{2}}-x y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
10.418 |
|
|
\[
{}\frac {y \cos \left (\frac {y}{x}\right )}{x}-\left (\frac {x \sin \left (\frac {y}{x}\right )}{y}+\cos \left (\frac {y}{x}\right )\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.609 |
|
|
\[
{}y+x \ln \left (\frac {y}{x}\right ) y^{\prime }-2 y^{\prime } x = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
6.105 |
|
|
\[
{}2 y \,{\mathrm e}^{\frac {x}{y}}+\left (y-2 x \,{\mathrm e}^{\frac {x}{y}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.119 |
|
|
\[
{}x \,{\mathrm e}^{\frac {y}{x}}-y \sin \left (\frac {y}{x}\right )+x \sin \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.844 |
|
|
\[
{}x^{2}+y^{2} = 2 x y y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
9.949 |
|
|
\[
{}x \,{\mathrm e}^{\frac {y}{x}}+y = y^{\prime } x
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.591 |
|
|
\[
{}y^{\prime }-\frac {y}{x}+\csc \left (\frac {y}{x}\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.199 |
|
|
\[
{}x y-y^{2}-x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
3.422 |
|
|
\[
{}x +2 y-4-\left (2 x -4 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.629 |
|
|
\[
{}3 x +2 y+1-\left (3 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.957 |
|
|
\[
{}x +y+1+\left (2 x +2 y+2\right ) y^{\prime } = 0
\] |
[_quadrature] |
✓ |
0.867 |
|
|
\[
{}x +y-1+\left (2 x +2 y-3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.834 |
|
|
\[
{}x +y-1-\left (x -y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.499 |
|
|
\[
{}x +y+\left (2 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.796 |
|
|
\[
{}7 y-3+\left (2 x +1\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.128 |
|
|
\[
{}x +2 y+\left (3 x +6 y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.898 |
|
|
\[
{}x +2 y+\left (-1+y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.005 |
|
|
\[
{}3 x -2 y+4-\left (2 x +7 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
50.262 |
|
|
\[
{}x +y+\left (3 x +3 y-4\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.935 |
|
|
\[
{}3 x +2 y+3-\left (x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.215 |
|
|
\[
{}y+7+\left (2 x +y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
23.307 |
|
|
\[
{}x +y+2-\left (x -y-4\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.541 |
|
|
\[
{}3 x^{2} y+8 x y^{2}+\left (x^{3}+8 x^{2} y+12 y^{2}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
0.379 |
|
|
\[
{}\frac {1+2 x y}{y}+\frac {\left (y-x \right ) y^{\prime }}{y^{2}} = 0
\] |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.433 |
|