|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{2} \left (x^{2} y^{\prime \prime }-x y^{\prime }+y\right ) = x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.140 |
|
|
\[
{}x^{2} y^{2} y^{\prime \prime }-3 x y^{2} y^{\prime }+4 y^{3}+x^{6} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.132 |
|
|
\[
{}y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2} = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
8.194 |
|
|
\[
{}x \left (x^{2} y^{\prime }+2 x y\right ) y^{\prime \prime }+4 x {y^{\prime }}^{2}+8 x y y^{\prime }+4 y^{2}-1 = 0
\] |
[NONE] |
✗ |
0.174 |
|
|
\[
{}x \left (1+x y\right ) y^{\prime \prime }+x^{2} {y^{\prime }}^{2}+\left (4 x y+2\right ) y^{\prime }+y^{2}+1 = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.030 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}-{y^{\prime }}^{4} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
35.493 |
|
|
\[
{}a^{2} y^{\prime \prime } = 2 x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
1.145 |
|
|
\[
{}x^{2} y y^{\prime \prime }+x^{2} {y^{\prime }}^{2}-5 x y y^{\prime } = 4 y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.129 |
|
|
\[
{}y \left (1-\ln \left (y\right )\right ) y^{\prime \prime }+\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.549 |
|
|
\[
{}5 {y^{\prime \prime \prime }}^{2}-3 y^{\prime \prime } y^{\prime \prime \prime \prime } = 0
\] |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries], [_high_order, _reducible, _mu_poly_yn]] |
✓ |
0.707 |
|
|
\[
{}40 {y^{\prime \prime \prime }}^{3}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+9 {y^{\prime \prime }}^{2} y^{\left (5\right )} = 0
\] |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] |
✗ |
0.078 |
|
|
\[
{}{y^{\prime \prime }}^{2}+2 x y^{\prime \prime }-y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.802 |
|
|
\[
{}{y^{\prime \prime }}^{2}-2 x y^{\prime \prime }-y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.776 |
|
|
\[
{}2 x^{3} y^{\prime \prime \prime }-6 x^{2} y^{\prime \prime }+12 x y^{\prime }-12 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.117 |
|
|
\[
{}y^{\prime \prime \prime }-\frac {3 y^{\prime \prime }}{x}+\frac {6 y^{\prime }}{x^{2}}-\frac {6 y}{x^{3}} = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
0.113 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.944 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+y = 0
\] |
[_Lienard] |
✓ |
1.456 |
|
|
\[
{}y^{\prime \prime } \sin \left (x \right )^{2} = 2 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.858 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 x y^{\prime }-6 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.114 |
|
|
\[
{}x y^{\prime \prime \prime }-y^{\prime \prime }+x y^{\prime }-y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.035 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime \prime }-x y^{\prime \prime }+y^{\prime } = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.335 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 2 x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.584 |
|
|
\[
{}y^{\prime \prime }+\frac {x y^{\prime }}{1-x}-\frac {y}{1-x} = -1+x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.848 |
|
|
\[
{}\left (x^{2}+2\right ) y^{\prime \prime \prime }-2 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }-2 x y = x^{4}+12
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✗ |
0.043 |
|
|
\[
{}y^{\prime \prime \prime }+y^{\prime } = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.051 |
|
|
\[
{}y^{\prime \prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.125 |
|
|
\[
{}y^{\prime \prime }+\frac {y}{x^{2} \ln \left (x \right )} = {\mathrm e}^{x} \left (\frac {2}{x}+\ln \left (x \right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
0.323 |
|
|
\[
{}y^{\prime \prime }+p_{1} y^{\prime }+p_{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.531 |
|
|
\[
{}\left (2 x +1\right ) y^{\prime \prime }+\left (4 x -2\right ) y^{\prime }-8 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.087 |
|
|
\[
{}y^{\prime \prime } \sin \left (x \right )^{2}+\sin \left (x \right ) \cos \left (x \right ) y^{\prime } = y
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.099 |
|
|
\[
{}y^{\prime \prime \prime \prime }-2 y^{\prime \prime } = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.054 |
|
|
\[
{}y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.053 |
|
|
\[
{}y^{\prime \prime \prime \prime }+4 y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.067 |
|
|
\[
{}y^{\prime \prime \prime \prime }-y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.058 |
|
|
\[
{}2 y^{\prime \prime }+y^{\prime }-y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.880 |
|
|
\[
{}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+3 y^{\prime \prime }+2 y^{\prime }+y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.087 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime }+4 y = x^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.341 |
|
|
\[
{}y^{\prime \prime }-6 y^{\prime }+8 y = {\mathrm e}^{x}+{\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.207 |
|
|
\[
{}y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y = x \,{\mathrm e}^{x}
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.119 |
|
|
\[
{}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+y = \left (x +1\right ) {\mathrm e}^{x}
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
0.149 |
|
|
\[
{}y^{\prime \prime }+4 y = x \sin \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
5.114 |
|
|
\[
{}y^{\prime \prime }+y^{\prime }+y = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
36.145 |
|
|
\[
{}y^{\prime \prime }-y = \frac {{\mathrm e}^{x}-{\mathrm e}^{-x}}{{\mathrm e}^{x}+{\mathrm e}^{-x}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.502 |
|
|
\[
{}y^{\prime \prime }-2 y = 4 x^{2} {\mathrm e}^{x^{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.295 |
|
|
\[
{}y^{\prime \prime }+y = \sin \left (x \right ) \sin \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
4.265 |
|
|
\[
{}y^{\prime \prime }+9 y = \ln \left (2 \sin \left (\frac {x}{2}\right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
119.984 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-\frac {n \left (n +1\right ) y}{x^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.010 |
|
|
\[
{}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.386 |
|
|
\[
{}x^{2} y^{\prime \prime }-x y^{\prime }+2 y = \ln \left (x \right ) x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
129.907 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 y = x^{2}+\frac {1}{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.029 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = x^{3}+3 x
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.326 |
|
|
\[
{}\left (x +1\right )^{2} y^{\prime \prime }+\left (x +1\right ) y^{\prime }+y = 4 \cos \left (\ln \left (x +1\right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
5.875 |
|
|
\[
{}y^{\prime \prime }-\frac {y^{\prime }}{x}+\left (1-\frac {m^{2}}{x^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.870 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+y = 0
\] |
[_Lienard] |
✓ |
1.414 |
|
|
\[
{}y^{\prime \prime }+\frac {2 p y^{\prime }}{x}+y = 0
\] |
[_Lienard] |
✓ |
0.908 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }-x^{3} y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.210 |
|
|
\[
{}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-1\right ) y = -3 \,{\mathrm e}^{x^{2}} \sin \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
4.162 |
|
|
\[
{}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {y \left (-8+\sqrt {x}+x \right )}{4 x^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.099 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=z \\ z^{\prime }=x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.892 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }=y+z \\ z^{\prime }=y+z+x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.413 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }=\frac {y^{2}}{z} \\ z^{\prime }=\frac {y}{2} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.030 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }=1-\frac {1}{z} \\ z^{\prime }=\frac {1}{y-x} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.029 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }=-z \\ z^{\prime }=y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.537 |
|
|
\[
{}y^{\prime \prime } = x +y^{2}
\] |
[NONE] |
✗ |
0.122 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }+y^{2} = 0
\] |
[[_2nd_order, _missing_x], [_Emden, _modified]] |
✗ |
1.527 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }=\frac {z^{2}}{y} \\ z^{\prime }=\frac {y^{2}}{z} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.036 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }=\frac {y^{2}}{z} \\ z^{\prime }=\frac {z^{2}}{y} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.029 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=y+z-x \\ y^{\prime }=x-y+z \\ z^{\prime }=x+y-z \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.389 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+x+y=t^{2} \\ y^{\prime }+y+z=2 t \\ z^{\prime }+z=t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.522 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+5 x+y=7 \,{\mathrm e}^{t}-27 \\ -2 x+y^{\prime }+3 y=-3 \,{\mathrm e}^{t}+12 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.885 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime \prime }+z^{\prime }-2 z={\mathrm e}^{2 x} \\ z^{\prime }+2 y^{\prime }-3 y=0 \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.030 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=x+{\mathrm e}^{t}+{\mathrm e}^{-t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.470 |
|
|
\[
{}\left [\begin {array}{c} y^{\prime }+\frac {2 z}{x^{2}}=1 \\ z^{\prime }+y=x \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.030 |
|
|
\[
{}\left [\begin {array}{c} t x^{\prime }-x-3 y=t \\ t y^{\prime }-x+y=0 \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.033 |
|
|
\[
{}\left [\begin {array}{c} t x^{\prime }+6 x-y-3 z=0 \\ t y^{\prime }+23 x-6 y-9 z=0 \\ t z^{\prime }+x+y-2 z=0 \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.040 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+5 x+y={\mathrm e}^{t} \\ y^{\prime }-x+3 y={\mathrm e}^{2 t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.499 |
|
|
\[
{}y^{\prime } = 2 x
\] |
[_quadrature] |
✓ |
0.439 |
|
|
\[
{}x y^{\prime } = 2 y
\] |
[_separable] |
✓ |
2.239 |
|
|
\[
{}y y^{\prime } = {\mathrm e}^{2 x}
\] |
[_separable] |
✓ |
2.203 |
|
|
\[
{}y^{\prime } = k y
\] |
[_quadrature] |
✓ |
0.839 |
|
|
\[
{}y^{\prime \prime }+4 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.219 |
|
|
\[
{}y^{\prime \prime }-4 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.028 |
|
|
\[
{}x y^{\prime }+y = y^{\prime } \sqrt {1-y^{2} x^{2}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
4.434 |
|
|
\[
{}x y^{\prime } = y+x^{2}+y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
1.734 |
|
|
\[
{}y^{\prime } = \frac {x y}{x^{2}+y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
15.999 |
|
|
\[
{}2 x y y^{\prime } = x^{2}+y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
7.282 |
|
|
\[
{}x y^{\prime }+y = x^{4} {y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.843 |
|
|
\[
{}y^{\prime } = \frac {y^{2}}{x y-x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
7.458 |
|
|
\[
{}\left (y \cos \left (y\right )-\sin \left (y\right )+x \right ) y^{\prime } = y
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.092 |
|
|
\[
{}1+y^{2}+y^{2} y^{\prime } = 0
\] |
[_quadrature] |
✓ |
1.362 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{3 x}-x
\] |
[_quadrature] |
✓ |
0.474 |
|
|
\[
{}x y^{\prime } = 1
\] |
[_quadrature] |
✓ |
0.441 |
|
|
\[
{}y^{\prime } = x \,{\mathrm e}^{x^{2}}
\] |
[_quadrature] |
✓ |
0.536 |
|
|
\[
{}y^{\prime } = \arcsin \left (x \right )
\] |
[_quadrature] |
✓ |
0.422 |
|
|
\[
{}\left (x +1\right ) y^{\prime } = x
\] |
[_quadrature] |
✓ |
0.571 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime } = x
\] |
[_quadrature] |
✓ |
0.573 |
|
|
\[
{}\left (x^{3}+1\right ) y^{\prime } = x
\] |
[_quadrature] |
✓ |
0.721 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime } = \arctan \left (x \right )
\] |
[_quadrature] |
✓ |
0.651 |
|
|
\[
{}x y y^{\prime } = y-1
\] |
[_separable] |
✓ |
1.710 |
|
|
\[
{}x^{5} y^{\prime }+y^{5} = 0
\] |
[_separable] |
✓ |
8.397 |
|