|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}16 y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
226.235 |
|
|
\[
{}x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-x^{2} y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
182.710 |
|
|
\[
{}y^{3} {y^{\prime }}^{3}-\left (1-3 x \right ) y^{2} {y^{\prime }}^{2}+3 x^{2} y y^{\prime }+x^{3}-y^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
246.753 |
|
|
\[
{}y^{4} {y^{\prime }}^{3}-6 y^{\prime } x +2 y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
272.186 |
|
|
\[
{}{y^{\prime }}^{4} = \left (y-a \right )^{3} \left (y-b \right )^{2}
\] |
[_quadrature] |
✓ |
1.303 |
|
|
\[
{}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{2} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
3.240 |
|
|
\[
{}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
9.604 |
|
|
\[
{}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} \left (y-c \right )^{2} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
5.122 |
|
|
\[
{}{y^{\prime }}^{4}+y^{\prime } x -3 y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
4.003 |
|
|
\[
{}{y^{\prime }}^{4}-4 x^{2} y {y^{\prime }}^{2}+16 x y^{2} y^{\prime }-16 y^{3} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.771 |
|
|
\[
{}{y^{\prime }}^{4}+4 y {y^{\prime }}^{3}+6 y^{2} {y^{\prime }}^{2}-\left (1-4 y^{3}\right ) y^{\prime }-\left (3-y^{3}\right ) y = 0
\] |
[_quadrature] |
✓ |
10.135 |
|
|
\[
{}2 {y^{\prime }}^{4}-y y^{\prime }-2 = 0
\] |
[_quadrature] |
✓ |
1.604 |
|
|
\[
{}x {y^{\prime }}^{4}-2 y {y^{\prime }}^{3}+12 x^{3} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
4.153 |
|
|
\[
{}3 {y^{\prime }}^{5}-y y^{\prime }+1 = 0
\] |
[_quadrature] |
✓ |
0.663 |
|
|
\[
{}{y^{\prime }}^{6} = \left (y-a \right )^{4} \left (y-b \right )^{3}
\] |
[_quadrature] |
✓ |
2.369 |
|
|
\[
{}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{4} \left (y-b \right )^{3} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
13.669 |
|
|
\[
{}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{3} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
13.793 |
|
|
\[
{}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{4} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
14.531 |
|
|
\[
{}x^{2} \left ({y^{\prime }}^{6}+3 y^{4}+3 y^{2}+1\right ) = a^{2}
\] |
[_rational] |
✓ |
10.259 |
|
|
\[
{}2 \sqrt {a y^{\prime }}+y^{\prime } x -y = 0
\] |
[[_homogeneous, ‘class G‘], _Clairaut] |
✓ |
9.809 |
|
|
\[
{}\left (x -y\right ) \sqrt {y^{\prime }} = a \left (1+y^{\prime }\right )
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
7.194 |
|
|
\[
{}2 \left (1+y\right )^{{3}/{2}}+3 y^{\prime } x -3 y = 0
\] |
[_separable] |
✓ |
15.422 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } = x
\] |
[_quadrature] |
✓ |
1.413 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } = y
\] |
[_quadrature] |
✓ |
4.508 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}} = y^{\prime } x
\] |
[_quadrature] |
✓ |
1.397 |
|
|
\[
{}\sqrt {a^{2}+b^{2} {y^{\prime }}^{2}}+y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
37.617 |
|
|
\[
{}a \sqrt {1+{y^{\prime }}^{2}}+y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
33.736 |
|
|
\[
{}a x \sqrt {1+{y^{\prime }}^{2}}+y^{\prime } x -y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✗ |
347.544 |
|
|
\[
{}\sqrt {\left (a \,x^{2}+y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )}-y y^{\prime }-a x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
110.648 |
|
|
\[
{}a \left (1+{y^{\prime }}^{3}\right )^{{1}/{3}}+y^{\prime } x -y = 0
\] |
[_Clairaut] |
✓ |
154.488 |
|
|
\[
{}\cos \left (y^{\prime }\right )+y^{\prime } x = y
\] |
[_Clairaut] |
✓ |
1.273 |
|
|
\[
{}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0
\] |
[_quadrature] |
✓ |
0.538 |
|
|
\[
{}\sin \left (y^{\prime }\right )+y^{\prime } = x
\] |
[_quadrature] |
✓ |
0.553 |
|
|
\[
{}y^{\prime } \sin \left (y^{\prime }\right )+\cos \left (y^{\prime }\right ) = y
\] |
[_quadrature] |
✓ |
146.988 |
|
|
\[
{}{y^{\prime }}^{2} \left (x +\sin \left (y^{\prime }\right )\right ) = y
\] |
[_dAlembert] |
✓ |
2.205 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right ) \sin \left (-y+y^{\prime } x \right )^{2} = 1
\] |
[_Clairaut] |
✓ |
36.615 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+a x \right )+y^{\prime } = 0
\] |
[_quadrature] |
✓ |
1.559 |
|
|
\[
{}{\mathrm e}^{y^{\prime }-y}-{y^{\prime }}^{2}+1 = 0
\] |
[_quadrature] |
✓ |
0.464 |
|
|
\[
{}\ln \left (y^{\prime }\right )+y^{\prime } x +a = 0
\] |
[_quadrature] |
✓ |
7.513 |
|
|
\[
{}\ln \left (y^{\prime }\right )+y^{\prime } x +a = y
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
3.872 |
|
|
\[
{}\ln \left (y^{\prime }\right )+y^{\prime } x +a +b y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
18.619 |
|
|
\[
{}\ln \left (y^{\prime }\right )+4 y^{\prime } x -2 y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
11.400 |
|
|
\[
{}\ln \left (y^{\prime }\right )+a \left (-y+y^{\prime } x \right ) = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
3.994 |
|
|
\[
{}a \left (\ln \left (y^{\prime }\right )-y^{\prime }\right )-x +y = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
14.013 |
|
|
\[
{}y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-y x = 0
\] |
[_separable] |
✓ |
19.402 |
|
|
\[
{}y^{\prime } \ln \left (y^{\prime }\right )-\left (x +1\right ) y^{\prime }+y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
9.639 |
|
|
\[
{}y^{\prime } \ln \left (y^{\prime }+\sqrt {a +{y^{\prime }}^{2}}\right )-\sqrt {1+{y^{\prime }}^{2}}-y^{\prime } x +y = 0
\] |
[_Clairaut] |
✓ |
38.781 |
|
|
\[
{}\ln \left (\cos \left (y^{\prime }\right )\right )+y^{\prime } \tan \left (y^{\prime }\right ) = y
\] |
[_dAlembert] |
✓ |
0.362 |
|
|
\[
{}y^{\prime } = \frac {x y}{x^{2}-y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
22.459 |
|
|
\[
{}y^{\prime } = \frac {x +y-3}{x -y-1}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
16.319 |
|
|
\[
{}y^{\prime } = \frac {2 x +y-1}{4 x +2 y+5}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.877 |
|
|
\[
{}y^{\prime }-\frac {2 y}{x +1} = \left (x +1\right )^{2}
\] |
[_linear] |
✓ |
3.132 |
|
|
\[
{}y^{\prime }+y x = x^{3} y^{3}
\] |
[_Bernoulli] |
✓ |
13.860 |
|
|
\[
{}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
31.360 |
|
|
\[
{}y+x y^{2}-y^{\prime } x = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
18.710 |
|
|
\[
{}y^{2} \left (1+{y^{\prime }}^{2}\right ) = R^{2}
\] |
[_quadrature] |
✓ |
21.341 |
|
|
\[
{}y = y^{\prime } x +\frac {a y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}}
\] |
[_Clairaut] |
✓ |
343.872 |
|
|
\[
{}y = x {y^{\prime }}^{2}+{y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
110.058 |
|
|
\[
{}\left (x +1\right ) y+\left (1-y\right ) x y^{\prime } = 0
\] |
[_separable] |
✓ |
49.588 |
|
|
\[
{}y^{2}+x y^{2}+\left (x^{2}-x^{2} y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
53.114 |
|
|
\[
{}x y \left (x^{2}+1\right ) y^{\prime }-1-y^{2} = 0
\] |
[_separable] |
✓ |
98.201 |
|
|
\[
{}1+y^{2}-\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{{3}/{2}} y^{\prime } = 0
\] |
[_separable] |
✓ |
293.333 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
101.372 |
|
|
\[
{}\sec \left (x \right )^{2} \tan \left (y\right )+\sec \left (y\right )^{2} \tan \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
256.403 |
|
|
\[
{}\left (-x +y\right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
11.572 |
|
|
\[
{}\left (2 \sqrt {y x}-x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
467.590 |
|
|
\[
{}y^{\prime } x -y-\sqrt {x^{2}+y^{2}} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
374.535 |
|
|
\[
{}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
243.765 |
|
|
\[
{}8 y+10 x +\left (5 y+7 x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
82.801 |
|
|
\[
{}2 x -y+1+\left (2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.090 |
|
|
\[
{}\left (7 y-3 x +3\right ) y^{\prime }+7-7 x +3 y = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.495 |
|
|
\[
{}y^{\prime }+\frac {x y}{x^{2}+1} = \frac {1}{2 x \left (x^{2}+1\right )}
\] |
[_linear] |
✓ |
1.188 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime }+\left (2 x^{2}-1\right ) y = a \,x^{3}
\] |
[_linear] |
✓ |
1.160 |
|
|
\[
{}y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{{3}/{2}}} = \frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}}
\] |
[_linear] |
✓ |
3.988 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2}
\] |
[_linear] |
✓ |
2.545 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+y = \arctan \left (x \right )
\] |
[_linear] |
✓ |
1.728 |
|
|
\[
{}\left (-x^{2}+1\right ) z^{\prime }-x z = a x z^{2}
\] |
[_separable] |
✓ |
2.383 |
|
|
\[
{}3 z^{2} z^{\prime }-a z^{3} = x +1
\] |
[_rational, _Bernoulli] |
✓ |
1.638 |
|
|
\[
{}z^{\prime }+2 x z = 2 a \,x^{3} z^{3}
\] |
[_Bernoulli] |
✓ |
1.296 |
|
|
\[
{}z^{\prime }+z \cos \left (x \right ) = z^{n} \sin \left (2 x \right )
\] |
[_Bernoulli] |
✓ |
4.595 |
|
|
\[
{}y^{\prime } x +y = y^{2} \ln \left (x \right )
\] |
[_Bernoulli] |
✓ |
2.336 |
|
|
\[
{}x^{3}+3 x y^{2}+\left (y^{3}+3 x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
0.546 |
|
|
\[
{}1+\frac {y^{2}}{x^{2}}-\frac {2 y y^{\prime }}{x} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
0.293 |
|
|
\[
{}\frac {3 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.380 |
|
|
\[
{}x +y y^{\prime }+\frac {x y^{\prime }}{x^{2}+y^{2}}-\frac {y}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
0.353 |
|
|
\[
{}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
0.301 |
|
|
\[
{}{\mathrm e}^{x} \left (x^{2}+y^{2}+2 x \right )+2 y \,{\mathrm e}^{x} y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _exact, _rational, _Bernoulli] |
✓ |
0.346 |
|
|
\[
{}n \cos \left (n x +m y\right )-m \sin \left (m x +n y\right )+\left (m \cos \left (n x +m y\right )-n \sin \left (m x +n y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.262 |
|
|
\[
{}\frac {x}{\sqrt {1+x^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y}{x^{2}+y^{2}}-\frac {x y^{\prime }}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact] |
✓ |
0.611 |
|
|
\[
{}\frac {x^{n} y^{\prime }}{b y^{2}-c \,x^{2 a}}-\frac {a y x^{a -1}}{b y^{2}-c \,x^{2 a}}+x^{a -1} = 0
\] |
[_Riccati] |
✗ |
4.788 |
|
|
\[
{}2 y x +\left (y^{2}-2 x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.344 |
|
|
\[
{}\frac {1}{x}+\frac {y^{\prime }}{y}+\frac {2}{y}-\frac {2 y^{\prime }}{x} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.306 |
|
|
\[
{}-y+y^{\prime } x = \sqrt {x^{2}+y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.803 |
|
|
\[
{}8 y+10 x +\left (5 y+7 x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.888 |
|
|
\[
{}x^{2}+2 y x -y^{2}+\left (y^{2}+2 y x -x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.564 |
|
|
\[
{}y^{2}+\left (y x +x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
6.035 |
|
|
\[
{}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y+\left (x \cos \left (\frac {y}{x}\right )-y \sin \left (\frac {y}{x}\right )\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
0.408 |
|
|
\[
{}\left (x^{2} y^{2}+y x \right ) y+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.348 |
|
|
\[
{}\left (x^{3} y^{3}+x^{2} y^{2}+y x +1\right ) y+\left (x^{3} y^{3}-x^{2} y^{2}-y x +1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.783 |
|
|
\[
{}x^{2}+y^{2}+2 x +2 y y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
0.342 |
|