|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\left [\begin {array}{c} x^{\prime }=2 x+y-z+5 \sin \left (t \right ) \\ y^{\prime }=y+z-10 \cos \left (t \right ) \\ z^{\prime }=x+z+2 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
1.529 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=-3 x+3 y+z+5 \sin \left (2 t \right ) \\ y^{\prime }=x-5 y-3 z+5 \cos \left (2 t \right ) \\ z^{\prime }=-3 x+7 y+3 z+23 \,{\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
2.445 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=-3 x+y-3 z+2 \,{\mathrm e}^{t} \\ y^{\prime }=4 x-y+2 z+4 \,{\mathrm e}^{t} \\ z^{\prime }=4 x-2 y+3 z+4 \,{\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
1.485 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=x+5 y+10 \sinh \left (t \right ) \\ y^{\prime }=19 x-13 y+24 \sinh \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
1.328 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=9 x-3 y-6 t \\ y^{\prime }=-x+11 y+10 t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.506 |
|
|
\[
{}\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.336 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+x y = 0
\] |
[_Lienard] |
✓ |
0.391 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }+y = x^{{3}/{2}} {\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.137 |
|
|
\[
{}y^{\prime \prime }+4 y = 2 \sec \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.328 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{4 x^{2}}\right ) y = x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
9.547 |
|
|
\[
{}y^{\prime \prime }+y = f \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.905 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (-\frac {1}{2}+x \right ) y^{\prime }+\frac {y}{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.026 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.846 |
|
|
\[
{}x \left (1-x \right ) y^{\prime \prime }+\left (1-5 x \right ) y^{\prime }-4 y = 0
\] |
[_Jacobi] |
✓ |
0.768 |
|
|
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime }+\left (x +1\right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.669 |
|
|
\[
{}x y^{\prime \prime }+4 y^{\prime }-x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.754 |
|
|
\[
{}2 x y^{\prime \prime }+\left (x +1\right ) y^{\prime }-k y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.993 |
|
|
\[
{}x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✗ |
0.118 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime }-2 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✗ |
0.171 |
|
|
\[
{}2 x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.919 |
|
|
\[
{}x \left (x -1\right ) y^{\prime \prime }+3 x y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.229 |
|
|
\[
{}y^{\prime \prime }-x^{2} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.918 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.724 |
|
|
\[
{}x y^{\prime \prime }+x^{2} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.665 |
|
|
\[
{}y^{\prime \prime }+\alpha ^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.626 |
|
|
\[
{}y^{\prime \prime }-\alpha ^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.317 |
|
|
\[
{}y^{\prime \prime }+\beta y^{\prime }+\gamma y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.067 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.945 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-\nu ^{2}+x^{2}\right ) y = \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
13.335 |
|
|
\[
{}y^{\prime }+\cos \left (x \right ) y = \frac {\sin \left (2 x \right )}{2}
\] |
[_linear] |
✓ |
2.164 |
|
|
\[
{}{y^{\prime }}^{2}-y^{\prime }-x y^{\prime }+y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.402 |
|
|
\[
{}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
2.005 |
|
|
\[
{}x y \left (1-{y^{\prime }}^{2}\right ) = \left (x^{2}-y^{2}-a^{2}\right ) y^{\prime }
\] |
[_rational] |
✓ |
117.937 |
|
|
\[
{}y^{\prime \prime \prime }+\frac {3 y^{\prime \prime }}{x} = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.174 |
|
|
\[
{}y^{\prime \prime }-2 k y^{\prime }+k^{2} y = {\mathrm e}^{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.873 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }-a^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.073 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.687 |
|
|
\[
{}y-x y^{\prime } = 0
\] |
[_separable] |
✓ |
1.270 |
|
|
\[
{}\left (1+u \right ) v+\left (1-v\right ) u v^{\prime } = 0
\] |
[_separable] |
✓ |
1.377 |
|
|
\[
{}1+y-\left (1-x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
1.496 |
|
|
\[
{}\left (t^{2}+x t^{2}\right ) x^{\prime }+x^{2}+t x^{2} = 0
\] |
[_separable] |
✓ |
1.631 |
|
|
\[
{}y-a +x^{2} y^{\prime } = 0
\] |
[_separable] |
✓ |
0.915 |
|
|
\[
{}z-\left (-a^{2}+t^{2}\right ) z^{\prime } = 0
\] |
[_separable] |
✓ |
1.546 |
|
|
\[
{}y^{\prime } = \frac {1+y^{2}}{x^{2}+1}
\] |
[_separable] |
✓ |
1.825 |
|
|
\[
{}1+s^{2}-\sqrt {t}\, s^{\prime } = 0
\] |
[_separable] |
✓ |
2.017 |
|
|
\[
{}r^{\prime }+r \tan \left (t \right ) = 0
\] |
[_separable] |
✓ |
1.338 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }-\sqrt {1-y^{2}} = 0
\] |
[_separable] |
✓ |
2.023 |
|
|
\[
{}\sqrt {-x^{2}+1}\, y^{\prime }-\sqrt {1-y^{2}} = 0
\] |
[_separable] |
✓ |
4.750 |
|
|
\[
{}3 \,{\mathrm e}^{x} \tan \left (y\right )+\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime } = 0
\] |
[_separable] |
✓ |
3.141 |
|
|
\[
{}x -x y^{2}+\left (y-x^{2} y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
1.877 |
|
|
\[
{}y-x +\left (x +y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.719 |
|
|
\[
{}x +y+x y^{\prime } = 0
\] |
[_linear] |
✓ |
1.879 |
|
|
\[
{}x +y+\left (y-x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.734 |
|
|
\[
{}-y+x y^{\prime } = \sqrt {y^{2}+x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.910 |
|
|
\[
{}8 y+10 x +\left (5 y+7 x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.266 |
|
|
\[
{}2 \sqrt {s t}-s+t s^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
6.562 |
|
|
\[
{}t -s+t s^{\prime } = 0
\] |
[_linear] |
✓ |
1.253 |
|
|
\[
{}x y^{2} y^{\prime } = x^{3}+y^{3}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
7.945 |
|
|
\[
{}x \cos \left (\frac {y}{x}\right ) \left (x y^{\prime }+y\right ) = y \sin \left (\frac {y}{x}\right ) \left (-y+x y^{\prime }\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
6.721 |
|
|
\[
{}3 y-7 x +7-\left (3 x -7 y-3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.632 |
|
|
\[
{}x +2 y+1-\left (4 y+2 x +3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.406 |
|
|
\[
{}x +2 y+1-\left (2 x -3\right ) y^{\prime } = 0
\] |
[_linear] |
✓ |
1.222 |
|
|
\[
{}\frac {y-x y^{\prime }}{\sqrt {y^{2}+x^{2}}} = m
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
15.253 |
|
|
\[
{}\frac {x +y^{\prime } y}{\sqrt {y^{2}+x^{2}}} = m
\] |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
72.728 |
|
|
\[
{}y+\frac {x}{y^{\prime }} = \sqrt {y^{2}+x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.286 |
|
|
\[
{}y^{\prime } y = -x +\sqrt {y^{2}+x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.615 |
|
|
\[
{}y^{\prime }-\frac {2 y}{x +1} = \left (x +1\right )^{3}
\] |
[_linear] |
✓ |
1.330 |
|
|
\[
{}y^{\prime }-\frac {a y}{x} = \frac {x +1}{x}
\] |
[_linear] |
✓ |
1.306 |
|
|
\[
{}\left (-x^{2}+x \right ) y^{\prime }+\left (2 x^{2}-1\right ) y-a \,x^{3} = 0
\] |
[_linear] |
✓ |
1.457 |
|
|
\[
{}s^{\prime } \cos \left (t \right )+s \sin \left (t \right ) = 1
\] |
[_linear] |
✓ |
1.848 |
|
|
\[
{}s^{\prime }+s \cos \left (t \right ) = \frac {\sin \left (2 t \right )}{2}
\] |
[_linear] |
✓ |
2.144 |
|
|
\[
{}y^{\prime }-\frac {n y}{x} = {\mathrm e}^{x} x^{n}
\] |
[_linear] |
✓ |
1.355 |
|
|
\[
{}y^{\prime }+\frac {n y}{x} = a \,x^{-n}
\] |
[_linear] |
✓ |
1.000 |
|
|
\[
{}y^{\prime }+y = {\mathrm e}^{-x}
\] |
[[_linear, ‘class A‘]] |
✓ |
0.964 |
|
|
\[
{}y^{\prime }+\frac {\left (-2 x +1\right ) y}{x^{2}}-1 = 0
\] |
[_linear] |
✓ |
1.464 |
|
|
\[
{}y^{\prime }+x y = x^{3} y^{3}
\] |
[_Bernoulli] |
✓ |
1.198 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime }-x y+a x y^{2} = 0
\] |
[_separable] |
✓ |
2.261 |
|
|
\[
{}3 y^{2} y^{\prime }-a y^{3}-x -1 = 0
\] |
[_rational, _Bernoulli] |
✓ |
1.683 |
|
|
\[
{}y^{\prime } \left (y^{3} x^{2}+x y\right ) = 1
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.457 |
|
|
\[
{}x y^{\prime } = \left (y \ln \left (x \right )-2\right ) y
\] |
[_Bernoulli] |
✓ |
1.936 |
|
|
\[
{}y-y^{\prime } \cos \left (x \right ) = y^{2} \cos \left (x \right ) \left (-\sin \left (x \right )+1\right )
\] |
[_Bernoulli] |
✓ |
6.039 |
|
|
\[
{}x^{2}+y+\left (x -2 y\right ) y^{\prime } = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.152 |
|
|
\[
{}y-3 x^{2}-\left (4 y-x \right ) y^{\prime } = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.194 |
|
|
\[
{}\left (y^{3}-x \right ) y^{\prime } = y
\] |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
6.652 |
|
|
\[
{}\frac {y^{2}}{\left (x -y\right )^{2}}-\frac {1}{x}+\left (\frac {1}{y}-\frac {x^{2}}{\left (x -y\right )^{2}}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.961 |
|
|
\[
{}6 x y^{2}+4 x^{3}+3 \left (2 x^{2} y+y^{2}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.379 |
|
|
\[
{}\frac {x}{\left (x +y\right )^{2}}+\frac {\left (2 x +y\right ) y^{\prime }}{\left (x +y\right )^{2}} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
2.302 |
|
|
\[
{}\frac {1}{x^{2}}+\frac {3 y^{2}}{x^{4}} = \frac {2 y y^{\prime }}{x^{3}}
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
3.931 |
|
|
\[
{}\frac {x^{2} y^{\prime }}{\left (x -y\right )^{2}}-\frac {y^{2}}{\left (x -y\right )^{2}} = 0
\] |
[_separable] |
✓ |
2.339 |
|
|
\[
{}x +y^{\prime } y = \frac {y}{y^{2}+x^{2}}-\frac {x y^{\prime }}{y^{2}+x^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
1.684 |
|
|
\[
{}y = 2 x y^{\prime }+{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.436 |
|
|
\[
{}y = x {y^{\prime }}^{2}+{y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
0.507 |
|
|
\[
{}y = x \left (1+y^{\prime }\right )+{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.496 |
|
|
\[
{}y = y {y^{\prime }}^{2}+2 x y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.958 |
|
|
\[
{}y = y^{\prime } y+y^{\prime }-{y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
1.353 |
|
|
\[
{}y = x y^{\prime }+\sqrt {1-{y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
1.583 |
|
|
\[
{}y = x y^{\prime }+y^{\prime }
\] |
[_separable] |
✓ |
1.371 |
|
|
\[
{}y = x y^{\prime }+\frac {1}{y^{\prime }}
\] |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
0.394 |
|
|
\[
{}y = x y^{\prime }-\frac {1}{{y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.701 |
|
|
\[
{}y^{\prime } = \frac {2 y}{x}-\sqrt {3}
\] |
[_linear] |
✓ |
2.026 |
|