|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\left (\cos \left (x \right )-\sin \left (x \right )\right ) y^{\prime \prime }-2 \sin \left (x \right ) y^{\prime }+\left (\cos \left (x \right )+\sin \left (x \right )\right ) y = \left (\cos \left (x \right )-\sin \left (x \right )\right )^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
6.046 |
|
|
\[
{}y^{\prime } = x^{2} \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
2.201 |
|
|
\[
{}y^{\prime } = \frac {x^{2}}{1-y^{2}}
\] |
[_separable] |
✓ |
1.226 |
|
|
\[
{}y^{\prime } = \frac {3 x^{2}+4 x +2}{2 y-2}
\] |
[_separable] |
✓ |
2.362 |
|
|
\[
{}y^{\prime } x -2 \sqrt {x y} = y
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
6.435 |
|
|
\[
{}y^{\prime } = \frac {x +y-1}{x -y+3}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.622 |
|
|
\[
{}{\mathrm e}^{x}+y+\left (x -2 \sin \left (y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
2.000 |
|
|
\[
{}3 x +\frac {6}{y}+\left (\frac {x^{2}}{y}+\frac {3 y}{x}\right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.499 |
|
|
\[
{}y^{2}-x y+x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.416 |
|
|
\[
{}x +y-\left (x -y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.676 |
|
|
\[
{}y^{\prime } = \frac {y}{2 x}+\frac {x^{2}}{2 y}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
2.220 |
|
|
\[
{}y^{\prime } = -\frac {2}{t}+\frac {y}{t}+\frac {y^{2}}{t}
\] |
[_separable] |
✓ |
2.811 |
|
|
\[
{}y^{\prime } = -\frac {y}{t}-1-y^{2}
\] |
[_rational, _Riccati] |
✓ |
1.296 |
|
|
\[
{}x +y y^{\prime } = a {y^{\prime }}^{2}
\] |
[_dAlembert] |
✗ |
338.901 |
|
|
\[
{}{y^{\prime }}^{2}-a^{2} y^{2} = 0
\] |
[_quadrature] |
✓ |
1.290 |
|
|
\[
{}{y^{\prime }}^{2} = 4 x^{2}
\] |
[_quadrature] |
✓ |
0.854 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }-3 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.112 |
|
|
\[
{}s^{\prime \prime }+2 s^{\prime }+s = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.530 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }+5 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.020 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }-3 y = 1+3 x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.340 |
|
|
\[
{}y^{\prime \prime }-3 y^{\prime }+2 y = x \,{\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.322 |
|
|
\[
{}y^{\prime \prime }+y = 4 \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.272 |
|
|
\[
{}y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.175 |
|
|
\[
{}p \,x^{2} u^{\prime \prime }+q x u^{\prime }+r u = f \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.448 |
|
|
\[
{}\sin \left (x \right ) u^{\prime \prime }+2 \cos \left (x \right ) u^{\prime }+\sin \left (x \right ) u = 0
\] |
[_Lienard] |
✓ |
3.329 |
|
|
\[
{}3 {y^{\prime \prime }}^{2}-y^{\prime } y^{\prime \prime \prime }-y^{\prime \prime } {y^{\prime }}^{2} = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_exponential_symmetries], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]] |
✓ |
0.747 |
|
|
\[
{}y^{\prime \prime }-\frac {x y^{\prime }}{-x^{2}+1}+\frac {y}{-x^{2}+1} = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.713 |
|
|
\[
{}x^{2} y y^{\prime \prime } = x^{2} {y^{\prime }}^{2}-y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.138 |
|
|
\[
{}y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y = 4 \,{\mathrm e}^{t}
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.138 |
|
|
\[
{}y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y = 3 \sin \left (t \right )-5 \cos \left (t \right )
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
1.391 |
|
|
\[
{}y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y = g \left (t \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.513 |
|
|
\[
{}y^{\left (5\right )}-\frac {y^{\prime \prime \prime \prime }}{t} = 0
\] |
[[_high_order, _missing_y]] |
✓ |
0.243 |
|
|
\[
{}x x^{\prime \prime }-{x^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.260 |
|
|
\[
{}y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+3 y^{\prime \prime }-4 y^{\prime }-4 y = f \left (x \right )
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
0.609 |
|
|
\[
{}u^{\prime \prime }-\left (2 x +1\right ) u^{\prime }+\left (x^{2}+x -1\right ) u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.940 |
|
|
\[
{}y^{\prime \prime }+6 y^{\prime }+9 y = 50 \,{\mathrm e}^{2 x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.398 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime }+4 y = 50 \,{\mathrm e}^{2 x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.422 |
|
|
\[
{}y^{\prime \prime }+3 y^{\prime }+2 y = \cos \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.692 |
|
|
\[
{}y^{\prime \prime \prime }+6 y^{\prime \prime }+11 y^{\prime }+6 y = 2 \sin \left (3 x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.161 |
|
|
\[
{}y^{\prime \prime }+4 y = x^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.580 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime }+3 y = x^{3}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.411 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }+\left (1+\frac {2}{\left (1+3 x \right )^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.923 |
|
|
\[
{}y+\sqrt {x^{2}+y^{2}}-y^{\prime } x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.674 |
|
|
\[
{}{y^{\prime }}^{2} = a^{2}-y^{2}
\] |
[_quadrature] |
✓ |
0.678 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 y^{\prime } x +\left (x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.528 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-\frac {2 y}{\left (x +1\right )^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.887 |
|
|
\[
{}y \left (x^{2} y^{2}+1\right )+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.117 |
|
|
\[
{}2 x^{3} y^{2}-y+\left (2 x^{2} y^{3}-x \right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.620 |
|
|
\[
{}\frac {1}{y}+\sec \left (\frac {y}{x}\right )-\frac {x y^{\prime }}{y^{2}} = 0
\] |
[[_homogeneous, ‘class D‘]] |
✓ |
17.407 |
|
|
\[
{}\phi ^{\prime }-\frac {\phi ^{2}}{2}-\phi \cot \left (\theta \right ) = 0
\] |
[_Bernoulli] |
✓ |
2.919 |
|
|
\[
{}u^{\prime \prime }-\cot \left (\theta \right ) u^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.792 |
|
|
\[
{}\left (\phi ^{\prime }-\frac {\phi ^{2}}{2}\right ) \sin \left (\theta \right )^{2}-\phi \sin \left (\theta \right ) \cos \left (\theta \right ) = \frac {\cos \left (2 \theta \right )}{2}+1
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
4.875 |
|
|
\[
{}a y^{\prime \prime } y^{\prime \prime \prime } = \sqrt {1+{y^{\prime \prime }}^{2}}
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
0.865 |
|
|
\[
{}a^{2} y^{\prime \prime \prime \prime } = y^{\prime \prime }
\] |
[[_high_order, _missing_x]] |
✓ |
0.088 |
|
|
\[
{}y \,{\mathrm e}^{x y}+x \,{\mathrm e}^{x y} y^{\prime } = 0
\] |
[_separable] |
✓ |
2.184 |
|
|
\[
{}x -2 x y+{\mathrm e}^{y}+\left (y-x^{2}+x \,{\mathrm e}^{y}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
1.851 |
|
|
\[
{}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.307 |
|
|
\[
{}\left (-x^{2}+1\right ) z^{\prime \prime }+\left (1-3 x \right ) z^{\prime }+k z = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.782 |
|
|
\[
{}\left (-x^{2}+1\right ) \eta ^{\prime \prime }-\left (x +1\right ) \eta ^{\prime }+\left (k +1\right ) \eta = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.749 |
|
|
\[
{}x^{2}+y^{2}-2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
8.165 |
|
|
\[
{}x^{2}-y^{2}+2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
7.243 |
|
|
\[
{}-y+y^{\prime } x = x^{2}+y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
1.710 |
|
|
\[
{}-y+y^{\prime } x = x \sqrt {x^{2}-y^{2}}\, y^{\prime }
\] |
[‘y=_G(x,y’)‘] |
✗ |
4.273 |
|
|
\[
{}x +y y^{\prime }+y-y^{\prime } x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.647 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}-y^{2} y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.481 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=3 x_{1}-18 x_{2} \\ x_{2}^{\prime }=2 x_{1}-9 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.535 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=x_{1}+3 x_{2} \\ x_{2}^{\prime }=5 x_{1}+3 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.440 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=-x_{1}+3 x_{2} \\ x_{2}^{\prime }=-3 x_{1}+5 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.520 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=4 x_{1}-x_{2} \\ x_{2}^{\prime }=5 x_{1}+2 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.590 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=-2 x_{1}+x_{2} \\ x_{2}^{\prime }=x_{1}-2 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.428 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=-2 x_{1}+x_{2}+2 \,{\mathrm e}^{-t} \\ x_{2}^{\prime }=x_{1}-2 x_{2}+3 t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.524 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=3 x_{1}-x_{2} \\ x_{2}^{\prime }=16 x_{1}-5 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.555 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=x_{1}-2 x_{2} \\ x_{2}^{\prime }=3 x_{1}-4 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.587 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=3 x_{1}-18 x_{2} \\ x_{2}^{\prime }=2 x_{1}-9 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.535 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=-x_{1}+3 x_{2} \\ x_{2}^{\prime }=-3 x_{1}+5 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.515 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=3 x_{1}-18 x_{2} \\ x_{2}^{\prime }=2 x_{1}-9 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.534 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=3 x_{1}-x_{2} \\ x_{2}^{\prime }=4 x_{1}-2 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.589 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=x_{1}+x_{2}-8 \\ x_{2}^{\prime }=x_{1}+x_{2}+3 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.529 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=x_{1}+x_{2}-8 \\ x_{2}^{\prime }=x_{1}+x_{2}+3 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.701 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{3 x}+\sin \left (x \right )
\] |
[_quadrature] |
✓ |
0.556 |
|
|
\[
{}y^{\prime \prime } = x +2
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.942 |
|
|
\[
{}y^{\prime \prime \prime } = x^{2}
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.108 |
|
|
\[
{}y^{\prime }+\cos \left (x \right ) y = 0
\] |
[_separable] |
✓ |
1.828 |
|
|
\[
{}y^{\prime }+\cos \left (x \right ) y = \sin \left (x \right ) \cos \left (x \right )
\] |
[_linear] |
✓ |
1.848 |
|
|
\[
{}y^{\prime \prime }-y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.276 |
|
|
\[
{}y^{\prime \prime }+4 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.368 |
|
|
\[
{}y^{\prime \prime }+k^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.756 |
|
|
\[
{}y^{\prime }+5 y = 2
\] |
[_quadrature] |
✓ |
1.316 |
|
|
\[
{}y^{\prime \prime } = 1+3 x
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.958 |
|
|
\[
{}y^{\prime } = k y
\] |
[_quadrature] |
✓ |
0.697 |
|
|
\[
{}y^{\prime }-2 y = 1
\] |
[_quadrature] |
✓ |
1.181 |
|
|
\[
{}y^{\prime }+y = {\mathrm e}^{x}
\] |
[[_linear, ‘class A‘]] |
✓ |
1.253 |
|
|
\[
{}y^{\prime }-2 y = x^{2}+x
\] |
[[_linear, ‘class A‘]] |
✓ |
1.264 |
|
|
\[
{}3 y^{\prime }+y = 2 \,{\mathrm e}^{-x}
\] |
[[_linear, ‘class A‘]] |
✓ |
1.360 |
|
|
\[
{}y^{\prime }+3 y = {\mathrm e}^{i x}
\] |
[[_linear, ‘class A‘]] |
✓ |
0.852 |
|
|
\[
{}y^{\prime }+i y = x
\] |
[[_linear, ‘class A‘]] |
✓ |
0.938 |
|
|
\[
{}L y^{\prime }+R y = E
\] |
[_quadrature] |
✓ |
0.759 |
|
|
\[
{}L y^{\prime }+R y = E \sin \left (\omega x \right )
\] |
[[_linear, ‘class A‘]] |
✓ |
1.530 |
|
|
\[
{}L y^{\prime }+R y = E \,{\mathrm e}^{i \omega x}
\] |
[[_linear, ‘class A‘]] |
✓ |
1.167 |
|
|
\[
{}y^{\prime }+a y = b \left (x \right )
\] |
[[_linear, ‘class A‘]] |
✓ |
1.209 |
|