|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime } = \frac {1}{y^{2}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
71.387 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.250 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2} = 1
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.798 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }-1-{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
0.543 |
|
|
\[
{}x y^{\prime \prime }+3 y^{\prime } = 3 x
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.144 |
|
|
\[
{}x = y^{\prime \prime }+y^{\prime }
\] |
[[_2nd_order, _missing_y]] |
✓ |
2.093 |
|
|
\[
{}x = y+{y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
0.456 |
|
|
\[
{}y = y^{\prime } x -{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.316 |
|
|
\[
{}V^{\prime \prime }+\frac {2 V^{\prime }}{r} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.665 |
|
|
\[
{}V^{\prime \prime }+\frac {V^{\prime }}{r} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.765 |
|
|
\[
{}\left [\begin {array}{c} z^{\prime }+7 y-3 z=0 \\ 7 y^{\prime }+63 y-36 z=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.405 |
|
|
\[
{}\left [\begin {array}{c} z^{\prime }+2 y^{\prime }+3 y=0 \\ y^{\prime }+3 y-2 z=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.417 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }+3 y+z=0 \\ z^{\prime }+3 y+5 z=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.421 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }+3 y+2 z=0 \\ z^{\prime }+2 y-4 z=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.594 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }-3 y-2 z=0 \\ z^{\prime }+y-2 z=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.706 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }+z^{\prime }+6 y=0 \\ z^{\prime }+5 y+z=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.595 |
|
|
\[
{}\left [\begin {array}{c} z^{\prime }+y^{\prime }+5 y-3 z=x +{\mathrm e}^{x} \\ y^{\prime }+2 y-z={\mathrm e}^{x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.490 |
|
|
\[
{}\left [\begin {array}{c} z^{\prime }+y+3 z={\mathrm e}^{x} \\ y^{\prime }+3 y+4 z={\mathrm e}^{2 x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.531 |
|
|
\[
{}\left [\begin {array}{c} z^{\prime }-3 y+2 z={\mathrm e}^{x} \\ y^{\prime }+2 y-z={\mathrm e}^{3 x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.711 |
|
|
\[
{}\left [\begin {array}{c} z^{\prime }+5 y-2 z=x \\ y^{\prime }+4 y+z=x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.822 |
|
|
\[
{}\left [\begin {array}{c} z^{\prime }+7 y-9 z={\mathrm e}^{x} \\ y^{\prime }-y-3 z={\mathrm e}^{2 x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
1.201 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }-2 y-2 z={\mathrm e}^{3 x} \\ z^{\prime }+5 y-2 z={\mathrm e}^{4 x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.966 |
|
|
\[
{}{y^{\prime }}^{2}+y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.324 |
|
|
\[
{}y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\frac {2 y}{x^{2}} = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.382 |
|
|
\[
{}v^{\prime \prime }+\frac {2 v^{\prime }}{r} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.668 |
|
|
\[
{}y^{\prime \prime }-k^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.058 |
|
|
\[
{}\left (1-x \right ) y^{\prime }-1-y = 0
\] |
[_separable] |
✓ |
1.915 |
|
|
\[
{}y^{\prime }+\sqrt {\frac {1-y^{2}}{-x^{2}+1}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
25.759 |
|
|
\[
{}y-y^{\prime } x = a \left (y^{2}+y^{\prime }\right )
\] |
[_separable] |
✓ |
1.158 |
|
|
\[
{}3 \,{\mathrm e}^{x} \tan \left (y\right )+\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime } = 0
\] |
[_separable] |
✓ |
2.990 |
|
|
\[
{}x^{2}+y^{2}-2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
7.732 |
|
|
\[
{}y^{2}+\left (x y+x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
6.111 |
|
|
\[
{}x^{2} y-\left (x^{3}+y^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
9.111 |
|
|
\[
{}\left (4 y+3 x \right ) y^{\prime }+y-2 x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
11.043 |
|
|
\[
{}3 y-7 x +7+\left (7 y-3 x +3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.085 |
|
|
\[
{}\left (y-3 x +3\right ) y^{\prime } = 2 y-x -4
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
6.623 |
|
|
\[
{}x^{2}-4 x y-2 y^{2}+\left (y^{2}-4 x y-2 x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
4.733 |
|
|
\[
{}x +y y^{\prime }+\frac {-y+y^{\prime } x}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
1.736 |
|
|
\[
{}a^{2}-2 x y-y^{2}-\left (x +y\right )^{2} y^{\prime } = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
1.260 |
|
|
\[
{}2 a x +b y+g +\left (2 c y+b x +e \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.677 |
|
|
\[
{}\left (2 x^{2} y+4 x^{3}-12 x y^{2}+3 y^{2}-x \,{\mathrm e}^{y}+{\mathrm e}^{2 x}\right ) y^{\prime }+12 x^{2} y+2 x y^{2}+4 x^{3}-4 y^{3}+2 y \,{\mathrm e}^{2 x}-{\mathrm e}^{y} = 0
\] |
[_exact] |
✓ |
2.586 |
|
|
\[
{}y-y^{\prime } x +\ln \left (x \right ) = 0
\] |
[_linear] |
✓ |
1.150 |
|
|
\[
{}\left (1+x y\right ) y-\left (1-x y\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.748 |
|
|
\[
{}a \left (y^{\prime } x +2 y\right ) = x y y^{\prime }
\] |
[_separable] |
✓ |
1.371 |
|
|
\[
{}x^{4} {\mathrm e}^{x}-2 m x y^{2}+2 m \,x^{2} y y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _Bernoulli] |
✓ |
1.702 |
|
|
\[
{}y \left (2 x y+{\mathrm e}^{x}\right )-{\mathrm e}^{x} y^{\prime } = 0
\] |
[_Bernoulli] |
✓ |
2.222 |
|
|
\[
{}x^{2} y-2 x y^{2}-\left (x^{3}-3 x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
4.023 |
|
|
\[
{}y \left (x y+2 x^{2} y^{2}\right )+x \left (x y-x^{2} y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.798 |
|
|
\[
{}x^{2}+y^{2}+2 x +2 y y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.159 |
|
|
\[
{}x^{2}+y^{2}-x^{2} y y^{\prime } = 0
\] |
[_rational, _Bernoulli] |
✓ |
1.579 |
|
|
\[
{}3 x^{2} y^{4}+2 x y+\left (2 x^{3} y^{3}-x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
3.515 |
|
|
\[
{}y^{4}+2 y+\left (x y^{3}+2 y^{4}-4 x \right ) y^{\prime } = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
3.036 |
|
|
\[
{}y^{3}-2 x^{2} y+\left (2 x y^{2}-x^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
28.099 |
|
|
\[
{}2 x^{2} y-3 y^{4}+\left (3 x^{3}+2 x y^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
6.800 |
|
|
\[
{}y^{2}+2 x^{2} y+\left (2 x^{3}-x y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.654 |
|
|
\[
{}y^{\prime } x -a y = x +1
\] |
[_linear] |
✓ |
1.254 |
|
|
\[
{}y^{\prime }+y = {\mathrm e}^{-x}
\] |
[[_linear, ‘class A‘]] |
✓ |
1.182 |
|
|
\[
{}\cos \left (x \right )^{2} y^{\prime }+y = \tan \left (x \right )
\] |
[_linear] |
✓ |
3.766 |
|
|
\[
{}\left (x +1\right ) y^{\prime }-n y = {\mathrm e}^{x} \left (x +1\right )^{n +1}
\] |
[_linear] |
✓ |
1.344 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+2 x y = 4 x^{2}
\] |
[_linear] |
✓ |
1.233 |
|
|
\[
{}y^{\prime }+\frac {y}{x} = x^{2} y^{6}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
2.454 |
|
|
\[
{}1+y^{2} = \left (\arctan \left (y\right )-x \right ) y^{\prime }
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
5.340 |
|
|
\[
{}y^{\prime }+\frac {2 y}{x} = 3 x^{2} y^{{1}/{3}}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
4.369 |
|
|
\[
{}y^{\prime }+\frac {x y}{-x^{2}+1} = x \sqrt {y}
\] |
[_rational, _Bernoulli] |
✓ |
2.790 |
|
|
\[
{}3 x \left (-x^{2}+1\right ) y^{2} y^{\prime }+\left (2 x^{2}-1\right ) y^{3} = a \,x^{3}
\] |
[_rational, _Bernoulli] |
✓ |
2.014 |
|
|
\[
{}\left (x +y\right )^{2} y^{\prime } = a^{2}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
3.315 |
|
|
\[
{}-y+y^{\prime } x = \sqrt {x^{2}+y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.302 |
|
|
\[
{}-y+y^{\prime } x = x \sqrt {x^{2}+y^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
4.037 |
|
|
\[
{}\sec \left (x \right )^{2} \tan \left (y\right )+\sec \left (y\right )^{2} \tan \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
37.316 |
|
|
\[
{}\left (x^{2}-x^{2} y\right ) y^{\prime }+y^{2}+x y^{2} = 0
\] |
[_separable] |
✓ |
1.658 |
|
|
\[
{}y^{\prime }+\frac {\left (1-2 x \right ) y}{x^{2}} = 1
\] |
[_linear] |
✓ |
1.625 |
|
|
\[
{}3 y^{\prime }+\frac {2 y}{x +1} = \frac {x^{3}}{y^{2}}
\] |
[_rational, _Bernoulli] |
✓ |
2.323 |
|
|
\[
{}2 x -y+1+\left (2 y-x -1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.856 |
|
|
\[
{}y^{\prime }+\frac {y}{\sqrt {-x^{2}+1}} = \frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}}
\] |
[_linear] |
✓ |
3.091 |
|
|
\[
{}y^{\prime } x +\frac {y^{2}}{x} = y
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.316 |
|
|
\[
{}x \left (x^{2}+y^{2}-a^{2}\right )+y \left (x^{2}-y^{2}-b^{2}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.710 |
|
|
\[
{}y^{\prime }+\frac {4 x y}{x^{2}+1} = \frac {1}{\left (x^{2}+1\right )^{3}}
\] |
[_linear] |
✓ |
2.246 |
|
|
\[
{}x^{2} y-\left (x^{3}+y^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
8.644 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime }+\left (2 x^{2}-1\right ) y = a \,x^{3}
\] |
[_linear] |
✓ |
1.091 |
|
|
\[
{}x^{2}+y^{2}+1-2 x y y^{\prime } = 0
\] |
[_rational, _Bernoulli] |
✓ |
2.041 |
|
|
\[
{}x +y y^{\prime } = m \left (-y+y^{\prime } x \right )
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.771 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = y^{n} \sin \left (2 x \right )
\] |
[_Bernoulli] |
✓ |
4.711 |
|
|
\[
{}\left (x +1\right ) y^{\prime }+1 = 2 \,{\mathrm e}^{y}
\] |
[_separable] |
✓ |
1.907 |
|
|
\[
{}y^{\prime } = x^{3} y^{3}-x y
\] |
[_Bernoulli] |
✓ |
1.262 |
|
|
\[
{}y+\left (a \,x^{2} y^{n}-2 x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.329 |
|
|
\[
{}\left (1+6 y^{2}-3 x^{2} y\right ) y^{\prime } = 3 x y^{2}-x^{2}
\] |
[_exact, _rational] |
✓ |
1.500 |
|
|
\[
{}y \left (x^{2}+y^{2}+a^{2}\right ) y^{\prime }+x \left (x^{2}+y^{2}-a^{2}\right ) = 0
\] |
[_exact, _rational] |
✓ |
1.667 |
|
|
\[
{}\left (x^{2} y^{3}+x y\right ) y^{\prime } = 1
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.457 |
|
|
\[
{}y y^{\prime } = a x
\] |
[_separable] |
✓ |
2.169 |
|
|
\[
{}\sqrt {a^{2}+x^{2}}\, y^{\prime }+y = \sqrt {a^{2}+x^{2}}-x
\] |
[_linear] |
✓ |
1.459 |
|
|
\[
{}\left (x +y\right ) y^{\prime }+x -y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.517 |
|
|
\[
{}y y^{\prime }+b y^{2} = a \cos \left (x \right )
\] |
[_Bernoulli] |
✓ |
2.272 |
|
|
\[
{}2 x y+\left (y^{2}-x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
4.758 |
|
|
\[
{}y-y^{\prime } x = b \left (1+x^{2} y^{\prime }\right )
\] |
[_separable] |
✓ |
0.955 |
|
|
\[
{}3 y+2 x +4-\left (4 x +6 y+5\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.990 |
|
|
\[
{}\left (x^{3} y^{3}+x^{2} y^{2}+x y+1\right ) y+\left (x^{3} y^{3}-x^{2} y^{2}-x y+1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.029 |
|
|
\[
{}2 x^{2} y^{2}+y-\left (x^{3} y-3 x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.239 |
|
|
\[
{}x^{2} y^{\prime }+y^{2} = x y y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
37.239 |
|
|
\[
{}y^{\prime }+\frac {n y}{x} = a \,x^{-n}
\] |
[_linear] |
✓ |
0.924 |
|
|
\[
{}\left (x -y\right )^{2} y^{\prime } = a^{2}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
2.250 |
|