|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x y \left (x^{2}+1\right ) y^{\prime }-1-y^{2} = 0
\] |
[_separable] |
✓ |
2.957 |
|
|
\[
{}1+y^{2}-\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{{3}/{2}} y^{\prime } = 0
\] |
[_separable] |
✓ |
2.781 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.521 |
|
|
\[
{}\sec \left (x \right )^{2} \tan \left (y\right )+\sec \left (y\right )^{2} \tan \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
37.183 |
|
|
\[
{}\left (y-x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.423 |
|
|
\[
{}\left (2 \sqrt {x y}-x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
90.099 |
|
|
\[
{}x y^{\prime }-y-\sqrt {y^{2}+x^{2}} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.477 |
|
|
\[
{}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.707 |
|
|
\[
{}8 y+10 x +\left (7 x +5 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.090 |
|
|
\[
{}2 x -y+1+\left (2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.082 |
|
|
\[
{}3 y-7 x +7+\left (7 y-3 x +3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.409 |
|
|
\[
{}y^{\prime }+\frac {x y}{x^{2}+1} = \frac {1}{2 x \left (x^{2}+1\right )}
\] |
[_linear] |
✓ |
1.178 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime }+\left (2 x^{2}-1\right ) y = a \,x^{3}
\] |
[_linear] |
✓ |
1.138 |
|
|
\[
{}y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{{3}/{2}}} = \frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}}
\] |
[_linear] |
✓ |
3.297 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2}
\] |
[_linear] |
✓ |
2.103 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+y = \arctan \left (x \right )
\] |
[_linear] |
✓ |
1.713 |
|
|
\[
{}\left (-x^{2}+1\right ) z^{\prime }-x z = a x z^{2}
\] |
[_separable] |
✓ |
2.164 |
|
|
\[
{}3 z^{2} z^{\prime }-a z^{3} = x +1
\] |
[_rational, _Bernoulli] |
✓ |
1.636 |
|
|
\[
{}z^{\prime }+2 x z = 2 a \,x^{3} z^{3}
\] |
[_Bernoulli] |
✓ |
1.204 |
|
|
\[
{}z^{\prime }+z \cos \left (x \right ) = z^{n} \sin \left (2 x \right )
\] |
[_Bernoulli] |
✓ |
5.260 |
|
|
\[
{}x y^{\prime }+y = y^{2} \ln \left (x \right )
\] |
[_Bernoulli] |
✓ |
1.962 |
|
|
\[
{}x^{3}+3 x y^{2}+\left (y^{3}+3 x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
0.548 |
|
|
\[
{}1+\frac {y^{2}}{x^{2}}-\frac {2 y y^{\prime }}{x} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
0.311 |
|
|
\[
{}\frac {3 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.382 |
|
|
\[
{}x +y y^{\prime }+\frac {x y^{\prime }}{y^{2}+x^{2}}-\frac {y}{y^{2}+x^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
0.402 |
|
|
\[
{}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
0.336 |
|
|
\[
{}{\mathrm e}^{x} \left (x^{2}+y^{2}+2 x \right )+2 y \,{\mathrm e}^{x} y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _exact, _rational, _Bernoulli] |
✓ |
0.371 |
|
|
\[
{}n \cos \left (n x +m y\right )-m \sin \left (m x +n y\right )+\left (m \cos \left (n x +m y\right )-n \sin \left (m x +n y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.276 |
|
|
\[
{}\frac {x}{\sqrt {1+x^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y}{y^{2}+x^{2}}-\frac {x y^{\prime }}{y^{2}+x^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact] |
✓ |
0.635 |
|
|
\[
{}\frac {x^{n} y^{\prime }}{b y^{2}-c \,x^{2 a}}-\frac {a y x^{a -1}}{b y^{2}-c \,x^{2 a}}+x^{a -1} = 0
\] |
[_Riccati] |
✗ |
6.634 |
|
|
\[
{}2 x y+\left (y^{2}-2 x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.365 |
|
|
\[
{}\frac {1}{x}+\frac {y^{\prime }}{y}+\frac {2}{y}-\frac {2 y^{\prime }}{x} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.332 |
|
|
\[
{}-y+x y^{\prime } = \sqrt {y^{2}+x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.605 |
|
|
\[
{}8 y+10 x +\left (7 x +5 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.116 |
|
|
\[
{}x^{2}+2 x y-y^{2}+\left (y^{2}+2 x y-x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.681 |
|
|
\[
{}y^{2}+\left (x y+x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
6.019 |
|
|
\[
{}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y+\left (x \cos \left (\frac {y}{x}\right )-y \sin \left (\frac {y}{x}\right )\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
0.531 |
|
|
\[
{}\left (y^{2} x^{2}+x y\right ) y+\left (y^{2} x^{2}-1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.414 |
|
|
\[
{}\left (x^{3} y^{3}+y^{2} x^{2}+x y+1\right ) y+\left (x^{3} y^{3}-y^{2} x^{2}-x y+1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.860 |
|
|
\[
{}x^{2}+y^{2}+2 x +2 y y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
0.388 |
|
|
\[
{}x^{2}+y^{2}-2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
0.338 |
|
|
\[
{}2 x y+\left (y^{2}-3 x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.310 |
|
|
\[
{}y+\left (2 y-x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.351 |
|
|
\[
{}x y^{\prime }-a y+y^{2} = x^{-2 a}
\] |
[_rational, _Riccati] |
✓ |
0.637 |
|
|
\[
{}x y^{\prime }-a y+y^{2} = x^{-\frac {2 a}{3}}
\] |
[_rational, _Riccati] |
✓ |
2.435 |
|
|
\[
{}u^{\prime }+u^{2} = \frac {c}{x^{{4}/{3}}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.313 |
|
|
\[
{}u^{\prime }+b u^{2} = \frac {c}{x^{4}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.278 |
|
|
\[
{}u^{\prime }-u^{2} = \frac {2}{x^{{8}/{3}}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.350 |
|
|
\[
{}\frac {\sqrt {f \,x^{4}+c \,x^{3}+c \,x^{2}+b x +a}\, y^{\prime }}{\sqrt {a +b y+c y^{2}+c y^{3}+f y^{4}}} = -1
\] |
[_separable] |
✓ |
8.758 |
|
|
\[
{}{y^{\prime }}^{2}-5 y^{\prime }+6 = 0
\] |
[_quadrature] |
✓ |
0.839 |
|
|
\[
{}{y^{\prime }}^{2}-\frac {a^{2}}{x^{2}} = 0
\] |
[_quadrature] |
✓ |
0.383 |
|
|
\[
{}{y^{\prime }}^{2} = \frac {1-x}{x}
\] |
[_quadrature] |
✓ |
0.290 |
|
|
\[
{}{y^{\prime }}^{2}+\frac {2 x y^{\prime }}{y}-1 = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.931 |
|
|
\[
{}y = a y^{\prime }+b {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
0.601 |
|
|
\[
{}x = a y^{\prime }+b {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
0.227 |
|
|
\[
{}y = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }
\] |
[_quadrature] |
✓ |
1.756 |
|
|
\[
{}x = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }
\] |
[_quadrature] |
✓ |
1.094 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {1+{y^{\prime }}^{2}}}{x} = 0
\] |
[_quadrature] |
✓ |
1.209 |
|
|
\[
{}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0
\] |
[_quadrature] |
✓ |
2.121 |
|
|
\[
{}1+{y^{\prime }}^{2} = \frac {\left (x +a \right )^{2}}{2 a x +x^{2}}
\] |
[_quadrature] |
✓ |
0.608 |
|
|
\[
{}y = x y^{\prime }+y^{\prime }-{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.380 |
|
|
\[
{}y = x y^{\prime }+\sqrt {b^{2}-a^{2} {y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
3.269 |
|
|
\[
{}y = x y^{\prime }+x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
32.394 |
|
|
\[
{}y = x y^{\prime }+a x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✗ |
337.064 |
|
|
\[
{}x -y y^{\prime } = a {y^{\prime }}^{2}
\] |
[_dAlembert] |
✗ |
641.628 |
|
|
\[
{}x +y y^{\prime } = a \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✗ |
488.152 |
|
|
\[
{}y y^{\prime } = x +y^{2}-y^{2} {y^{\prime }}^{2}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
7.134 |
|
|
\[
{}y-\frac {1}{\sqrt {1+{y^{\prime }}^{2}}} = x +\frac {y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.657 |
|
|
\[
{}y-2 x y^{\prime } = x {y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.224 |
|
|
\[
{}\frac {y-x y^{\prime }}{y^{2}+y^{\prime }} = \frac {y-x y^{\prime }}{1+x^{2} y^{\prime }}
\] |
[_separable] |
✓ |
0.527 |
|
|
\[
{}2 x y+\left (y^{2}+x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
3.475 |
|
|
\[
{}\left (x +\sqrt {y^{2}-x y}\right ) y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.411 |
|
|
\[
{}x +y-\left (x -y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.649 |
|
|
\[
{}x y^{\prime }-y-x \sin \left (\frac {y}{x}\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
3.191 |
|
|
\[
{}2 x^{2} y+y^{3}+\left (x y^{2}-2 x^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
25.534 |
|
|
\[
{}y^{2}+\left (x \sqrt {y^{2}-x^{2}}-x y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
10.810 |
|
|
\[
{}\frac {y \cos \left (\frac {y}{x}\right )}{x}-\left (\frac {x \sin \left (\frac {y}{x}\right )}{y}+\cos \left (\frac {y}{x}\right )\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.965 |
|
|
\[
{}y+x \ln \left (\frac {y}{x}\right ) y^{\prime }-2 x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.711 |
|
|
\[
{}2 y \,{\mathrm e}^{\frac {x}{y}}+\left (y-2 x \,{\mathrm e}^{\frac {x}{y}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
3.463 |
|
|
\[
{}x \,{\mathrm e}^{\frac {y}{x}}-y \sin \left (\frac {y}{x}\right )+x \sin \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.197 |
|
|
\[
{}y^{2}+x^{2} = 2 x y y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
9.121 |
|
|
\[
{}x \,{\mathrm e}^{\frac {y}{x}}+y = x y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.743 |
|
|
\[
{}y^{\prime }-\frac {y}{x}+\csc \left (\frac {y}{x}\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.519 |
|
|
\[
{}x y-y^{2}-x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.744 |
|
|
\[
{}x +2 y-4-\left (2 x -4 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.850 |
|
|
\[
{}3 x +2 y+1-\left (3 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.393 |
|
|
\[
{}x +y+1+\left (2 x +2 y+2\right ) y^{\prime } = 0
\] |
[_quadrature] |
✓ |
0.519 |
|
|
\[
{}x +y-1+\left (2 x +2 y-3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.264 |
|
|
\[
{}x +y-1-\left (x -y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.777 |
|
|
\[
{}x +y+\left (2 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.233 |
|
|
\[
{}7 y-3+\left (2 x +1\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
1.620 |
|
|
\[
{}x +2 y+\left (3 x +6 y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.283 |
|
|
\[
{}x +2 y+\left (y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.687 |
|
|
\[
{}3 x -2 y+4-\left (2 x +7 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
45.924 |
|
|
\[
{}x +y+\left (3 x +3 y-4\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.662 |
|
|
\[
{}3 x +2 y+3-\left (x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.992 |
|
|
\[
{}y+7+\left (2 x +y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
21.966 |
|
|
\[
{}x +y+2-\left (x -y-4\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.837 |
|
|
\[
{}3 x^{2} y+8 x y^{2}+\left (x^{3}+8 x^{2} y+12 y^{2}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
0.330 |
|
|
\[
{}\frac {2 x y+1}{y}+\frac {\left (y-x \right ) y^{\prime }}{y^{2}} = 0
\] |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.345 |
|