|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x y^{\prime } = k y
\] |
[_separable] |
✓ |
1.349 |
|
|
\[
{}i^{\prime } = p \left (t \right ) i
\] |
[_separable] |
✓ |
1.086 |
|
|
\[
{}x^{\prime } = \lambda x
\] |
[_quadrature] |
✓ |
0.728 |
|
|
\[
{}m v^{\prime } = -m g +k v^{2}
\] |
[_quadrature] |
✓ |
0.819 |
|
|
\[
{}x^{\prime } = k x-x^{2}
\] |
[_quadrature] |
✓ |
1.750 |
|
|
\[
{}x^{\prime } = -x \left (k^{2}+x^{2}\right )
\] |
[_quadrature] |
✓ |
11.521 |
|
|
\[
{}y^{\prime }+\frac {y}{x} = x^{2}
\] |
[_linear] |
✓ |
1.284 |
|
|
\[
{}x^{\prime }+x t = 4 t
\] |
[_separable] |
✓ |
1.841 |
|
|
\[
{}z^{\prime } = z \tan \left (y \right )+\sin \left (y \right )
\] |
[_linear] |
✓ |
1.633 |
|
|
\[
{}y^{\prime }+{\mathrm e}^{-x} y = 1
\] |
[_linear] |
✓ |
1.292 |
|
|
\[
{}x^{\prime }+x \tanh \left (t \right ) = 3
\] |
[_linear] |
✓ |
1.170 |
|
|
\[
{}y^{\prime }+2 y \cot \left (x \right ) = 5
\] |
[_linear] |
✓ |
1.566 |
|
|
\[
{}x^{\prime }+5 x = t
\] |
[[_linear, ‘class A‘]] |
✓ |
0.993 |
|
|
\[
{}x^{\prime }+\left (a +\frac {1}{t}\right ) x = b
\] |
[_linear] |
✓ |
1.156 |
|
|
\[
{}T^{\prime } = -k \left (T-\mu -a \cos \left (\omega \left (t -\phi \right )\right )\right )
\] |
[[_linear, ‘class A‘]] |
✓ |
1.672 |
|
|
\[
{}2 x y-\sec \left (x \right )^{2}+\left (x^{2}+2 y\right ) y^{\prime } = 0
\] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.230 |
|
|
\[
{}1+y \,{\mathrm e}^{x}+x \,{\mathrm e}^{x} y+\left (x \,{\mathrm e}^{x}+2\right ) y^{\prime } = 0
\] |
[_linear] |
✓ |
1.654 |
|
|
\[
{}\left (x \cos \left (y\right )+\cos \left (x \right )\right ) y^{\prime }+\sin \left (y\right )-y \sin \left (x \right ) = 0
\] |
[_exact] |
✓ |
28.484 |
|
|
\[
{}{\mathrm e}^{x} \sin \left (y\right )+y+\left ({\mathrm e}^{x} \cos \left (y\right )+x +{\mathrm e}^{y}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
2.482 |
|
|
\[
{}{\mathrm e}^{-y} \sec \left (x \right )+2 \cos \left (x \right )-{\mathrm e}^{-y} y^{\prime } = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
2.984 |
|
|
\[
{}V^{\prime }\left (x \right )+2 y^{\prime } y = 0
\] |
[_separable] |
✓ |
0.622 |
|
|
\[
{}\left (\frac {1}{y}-a \right ) y^{\prime }+\frac {2}{x}-b = 0
\] |
[_separable] |
✓ |
1.481 |
|
|
\[
{}x y+y^{2}+x^{2}-x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
2.121 |
|
|
\[
{}x^{\prime } = \frac {x^{2}+t \sqrt {t^{2}+x^{2}}}{x t}
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
39.153 |
|
|
\[
{}x^{\prime } = k x-x^{2}
\] |
[_quadrature] |
✓ |
1.412 |
|
|
\[
{}x^{\prime \prime }-3 x^{\prime }+2 x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.079 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime }+4 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.155 |
|
|
\[
{}z^{\prime \prime }-4 z^{\prime }+13 z = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.700 |
|
|
\[
{}y^{\prime \prime }+y^{\prime }-6 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.420 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime } = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.381 |
|
|
\[
{}\theta ^{\prime \prime }+4 \theta = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.660 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }+10 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.622 |
|
|
\[
{}2 z^{\prime \prime }+7 z^{\prime }-4 z = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.415 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.155 |
|
|
\[
{}x^{\prime \prime }+6 x^{\prime }+10 x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.117 |
|
|
\[
{}4 x^{\prime \prime }-20 x^{\prime }+21 x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.414 |
|
|
\[
{}y^{\prime \prime }+y^{\prime }-2 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.417 |
|
|
\[
{}y^{\prime \prime }-4 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.583 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime }+4 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.152 |
|
|
\[
{}y^{\prime \prime }+\omega ^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.864 |
|
|
\[
{}x^{\prime \prime }-4 x = t^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.143 |
|
|
\[
{}x^{\prime \prime }-4 x^{\prime } = t^{2}
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.628 |
|
|
\[
{}x^{\prime \prime }+x^{\prime }-2 x = 3 \,{\mathrm e}^{-t}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.126 |
|
|
\[
{}x^{\prime \prime }+x^{\prime }-2 x = {\mathrm e}^{t}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.104 |
|
|
\[
{}x^{\prime \prime }+2 x^{\prime }+x = {\mathrm e}^{-t}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.010 |
|
|
\[
{}x^{\prime \prime }+\omega ^{2} x = \sin \left (\alpha t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.408 |
|
|
\[
{}x^{\prime \prime }+\omega ^{2} x = \sin \left (\omega t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.799 |
|
|
\[
{}x^{\prime \prime }+2 x^{\prime }+10 x = {\mathrm e}^{-t}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.024 |
|
|
\[
{}x^{\prime \prime }+2 x^{\prime }+10 x = {\mathrm e}^{-t} \cos \left (3 t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
18.664 |
|
|
\[
{}x^{\prime \prime }+6 x^{\prime }+10 x = {\mathrm e}^{-2 t} \cos \left (t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
7.251 |
|
|
\[
{}x^{\prime \prime }+4 x^{\prime }+4 x = {\mathrm e}^{2 t}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.060 |
|
|
\[
{}x^{\prime \prime }+x^{\prime }-2 x = 12 \,{\mathrm e}^{-t}-6 \,{\mathrm e}^{t}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.264 |
|
|
\[
{}x^{\prime \prime }+4 x = 289 t \,{\mathrm e}^{t} \sin \left (2 t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.762 |
|
|
\[
{}x^{\prime \prime }+\omega ^{2} x = \cos \left (\alpha t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.395 |
|
|
\[
{}x^{\prime \prime }+\omega ^{2} x = \cos \left (\omega t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.943 |
|
|
\[
{}x^{\prime \prime \prime }-6 x^{\prime \prime }+11 x^{\prime }-6 x = {\mathrm e}^{-t}
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.115 |
|
|
\[
{}y^{\prime \prime \prime }-3 y^{\prime \prime }+2 y = \sin \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.139 |
|
|
\[
{}x^{\prime \prime \prime \prime }-4 x^{\prime \prime \prime }+8 x^{\prime \prime }-8 x^{\prime }+4 x = \sin \left (t \right )
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
0.163 |
|
|
\[
{}x^{\prime \prime \prime \prime }-5 x^{\prime \prime }+4 x = {\mathrm e}^{t}
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
0.121 |
|
|
\[
{}t^{2} y^{\prime \prime }-\left (t^{2}+2 t \right ) y^{\prime }+\left (t +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.335 |
|
|
\[
{}\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.336 |
|
|
\[
{}\left (t \cos \left (t \right )-\sin \left (t \right )\right ) x^{\prime \prime }-x^{\prime } t \sin \left (t \right )-x \sin \left (t \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.530 |
|
|
\[
{}\left (-t^{2}+t \right ) x^{\prime \prime }+\left (-t^{2}+2\right ) x^{\prime }+\left (-t +2\right ) x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.348 |
|
|
\[
{}y^{\prime \prime }-x y^{\prime }+y = 0
\] |
[_Hermite] |
✓ |
0.343 |
|
|
\[
{}\tan \left (t \right ) x^{\prime \prime }-3 x^{\prime }+\left (\tan \left (t \right )+3 \cot \left (t \right )\right ) x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.325 |
|
|
\[
{}y^{\prime \prime }-y^{\prime }-6 y = {\mathrm e}^{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.090 |
|
|
\[
{}x^{\prime \prime }-x = \frac {1}{t}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.156 |
|
|
\[
{}y^{\prime \prime }+4 y = \cot \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
5.386 |
|
|
\[
{}t^{2} x^{\prime \prime }-2 x = t^{3}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.981 |
|
|
\[
{}x^{\prime \prime }-4 x^{\prime } = \tan \left (t \right )
\] |
[[_2nd_order, _missing_y]] |
✓ |
2.629 |
|
|
\[
{}\left (\tan \left (x \right )^{2}-1\right ) y^{\prime \prime }-4 \tan \left (x \right )^{3} y^{\prime }+2 y \sec \left (x \right )^{4} = \left (\tan \left (x \right )^{2}-1\right ) \left (1-2 \sin \left (x \right )^{2}\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.627 |
|
|
\[
{}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.999 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.977 |
|
|
\[
{}t^{2} x^{\prime \prime }-5 x^{\prime } t +10 x = 0
\] |
[[_Emden, _Fowler]] |
✓ |
4.508 |
|
|
\[
{}t^{2} x^{\prime \prime }+x^{\prime } t -x = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.681 |
|
|
\[
{}x^{2} z^{\prime \prime }+3 x z^{\prime }+4 z = 0
\] |
[[_Emden, _Fowler]] |
✓ |
4.145 |
|
|
\[
{}x^{2} y^{\prime \prime }-x y^{\prime }-3 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.991 |
|
|
\[
{}4 t^{2} x^{\prime \prime }+8 x^{\prime } t +5 x = 0
\] |
[[_Emden, _Fowler]] |
✓ |
3.916 |
|
|
\[
{}x^{2} y^{\prime \prime }-5 x y^{\prime }+5 y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
2.117 |
|
|
\[
{}3 x^{2} z^{\prime \prime }+5 x z^{\prime }-z = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.973 |
|
|
\[
{}t^{2} x^{\prime \prime }+3 x^{\prime } t +13 x = 0
\] |
[[_Emden, _Fowler]] |
✓ |
5.130 |
|
|
\[
{}a y^{\prime \prime }+\left (b -a \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.740 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
0.745 |
|
|
\[
{}y^{\prime \prime }-x y^{\prime }+y = 0
\] |
[_Hermite] |
✓ |
0.478 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.588 |
|
|
\[
{}2 x y^{\prime \prime }+y^{\prime }-2 y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.813 |
|
|
\[
{}y^{\prime \prime }-2 x y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.509 |
|
|
\[
{}y^{\prime \prime }-2 x y^{\prime }+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.497 |
|
|
\[
{}x \left (1-x \right ) y^{\prime \prime }-3 x y^{\prime }-y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.257 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.674 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-1\right ) y = 0
\] |
[_Bessel] |
✓ |
1.202 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-n^{2}+x^{2}\right ) y = 0
\] |
[_Bessel] |
✓ |
0.806 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=4 x-y \\ y^{\prime }=2 x+y+t^{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.625 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=x-4 y+\cos \left (2 t \right ) \\ y^{\prime }=x+y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.835 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=2 x+2 y \\ y^{\prime }=6 x+3 y+{\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.664 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=5 x-4 y+{\mathrm e}^{3 t} \\ y^{\prime }=x+y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.582 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=2 x+5 y \\ y^{\prime }=-2 x+\cos \left (3 t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
1.012 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=x+y+{\mathrm e}^{-t} \\ y^{\prime }=4 x-2 y+{\mathrm e}^{2 t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.661 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=8 x+14 y \\ y^{\prime }=7 x+y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.447 |
|
|
\(\left [\begin {array}{cc} 2 & 2 \\ 0 & -4 \end {array}\right ]\) |
Eigenvectors |
✓ |
0.145 |
|