|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}2 x y^{\prime \prime }-\left (x^{3}+1\right ) y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.106 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+2 y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.703 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+2 x y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.569 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 \left (1+x \right ) y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.678 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (1+x \right ) y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.668 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2 x +3\right ) y^{\prime }+4 y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.737 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }+9 y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.888 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}-1\right ) y^{\prime }+\left (-x^{2}+1\right ) y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.715 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2 x -1\right ) y^{\prime }+x \left (-1+x \right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.902 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.886 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-\left (3 x^{2}+2\right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.892 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-9 y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.945 |
|
|
\[ {}\left (-x^{2}+x \right ) y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.819 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -7\right ) y^{\prime }+\left (x +12\right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.456 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (x -4\right ) y^{\prime }+4 y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.497 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-x^{2}+3\right ) y^{\prime }-3 y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.385 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }-y = x \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.286 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }-y = x \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.216 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }-2 x y = x^{2} \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.608 |
|
|
\[ {}x y^{\prime \prime }-x y^{\prime }+y = x^{3} \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.333 |
|
|
\[ {}\left (1-2 x \right ) y^{\prime \prime }+4 x y^{\prime }-4 y = x^{2}-x \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.522 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x +12\right ) y = x^{2}+x \] |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.238 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (x^{2}+3\right ) y^{\prime }+y = -2 x^{2}+x \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.102 |
|
|
\[ {}3 x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+\left (2 x^{2}-1\right ) y = -x^{3}+x \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.207 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+\left (3 x +2\right ) y = x^{4}+x^{2} \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.793 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+10 x y^{\prime }+y = -1+x \] |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.684 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+\left (-x^{2}+x \right ) y^{\prime }-y = x^{3}+1 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.829 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 6 \left (-x^{2}+1\right )^{2} \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.392 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+2 y = x^{2} \left (2+x \right )^{2} \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.933 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (1+x \right ) y = x \left (x^{2}+x +1\right ) \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.869 |
|
|
\[ {}\left (x^{3}+2 x^{2}\right ) y^{\prime \prime }-x y^{\prime }+\left (1-x \right ) y = x^{2} \left (1+x \right )^{2} \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.01 |
|
|
\[ {}y^{\prime } = 2 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.062 |
|
|
\[ {}y^{\prime } = 2 \,{\mathrm e}^{3 x} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.082 |
|
|
\[ {}y^{\prime } = \frac {2}{\sqrt {-x^{2}+1}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.306 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x^{2}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.076 |
|
|
\[ {}y^{\prime } = x \,{\mathrm e}^{x^{2}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.091 |
|
|
\[ {}y^{\prime } = \arcsin \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.09 |
|
|
\[ {}y^{\prime } = x y \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.436 |
|
|
\[ {}y^{\prime } = x^{2} y^{2} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.388 |
|
|
\[ {}y^{\prime } = -x \,{\mathrm e}^{y} \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.436 |
|
|
\[ {}y^{\prime } \sin \left (y\right ) = x^{2} \] |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.825 |
|
|
\[ {}x y^{\prime } = \sqrt {1-y^{2}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.892 |
|
|
\[ {}{y^{\prime }}^{2}-y^{2} = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.22 |
|
|
\[ {}{y^{\prime }}^{2}-3 y^{\prime }+2 = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.122 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.096 |
|
|
\[ {}y^{\prime } \sin \left (x \right ) = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.128 |
|
|
\[ {}y^{\prime } = t^{2}+3 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.073 |
|
|
\[ {}y^{\prime } = t \,{\mathrm e}^{2 t} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.097 |
|
|
\[ {}y^{\prime } = \sin \left (3 t \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.154 |
|
|
\[ {}y^{\prime } = \sin \left (t \right )^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.213 |
|
|
\[ {}y^{\prime } = \frac {t}{t^{2}+4} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.089 |
|
|
\[ {}y^{\prime } = \ln \left (t \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.171 |
|
|
\[ {}y^{\prime } = \frac {t}{\sqrt {t}+1} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.242 |
|
|
\[ {}y^{\prime } = 2 y-4 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.346 |
|
|
\[ {}y^{\prime } = -y^{3} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.267 |
|
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{t}}{y} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.207 |
|
|
\[ {}y^{\prime } = t \,{\mathrm e}^{2 t} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.204 |
|
|
\[ {}y^{\prime } = \sin \left (t \right )^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.236 |
|
|
\[ {}y^{\prime } = 8 \,{\mathrm e}^{4 t}+t \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.22 |
|
|
\[ {}y^{\prime } = \frac {y}{t} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.388 |
|
|
\[ {}y^{\prime } = -\frac {t}{y} \] |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.953 |
|
|
\[ {}y^{\prime } = y^{2}-y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.25 |
|
|
\[ {}y^{\prime } = y-1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.119 |
|
|
\[ {}y^{\prime } = 1-y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.121 |
|
|
\[ {}y^{\prime } = y^{3}-y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.237 |
|
|
\[ {}y^{\prime } = 1-y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.12 |
|
|
\[ {}y^{\prime } = \left (t^{2}+1\right ) y \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.48 |
|
|
\[ {}y^{\prime } = -y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.089 |
|
|
\[ {}y^{\prime } = 2 y+{\mathrm e}^{-3 t} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.663 |
|
|
\[ {}y^{\prime } = 2 y+{\mathrm e}^{2 t} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.45 |
|
|
\[ {}y^{\prime } = t -y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.454 |
|
|
\[ {}t y^{\prime }+2 y = \sin \left (t \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.585 |
|
|
\[ {}y^{\prime } = \tan \left (t \right ) y+\sec \left (t \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.578 |
|
|
\[ {}y^{\prime } = \frac {2 t y}{t^{2}+1}+t +1 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.637 |
|
|
\[ {}y^{\prime } = \tan \left (t \right ) y+\sec \left (t \right )^{3} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.651 |
|
|
\[ {}y^{\prime } = y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.191 |
|
|
\[ {}y^{\prime } = 2 y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.226 |
|
|
\[ {}t y^{\prime } = y+t^{3} \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.724 |
|
|
\[ {}y^{\prime } = -\tan \left (t \right ) y+\sec \left (t \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.819 |
|
|
\[ {}y^{\prime } = \frac {2 y}{t +1} \] |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.977 |
|
|
\[ {}t y^{\prime } = -y+t^{3} \] |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.793 |
|
|
\[ {}y^{\prime }+4 \tan \left (2 t \right ) y = \tan \left (2 t \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.954 |
|
|
\[ {}t \ln \left (t \right ) y^{\prime } = \ln \left (t \right ) t -y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.948 |
|
|
\[ {}y^{\prime } = \frac {2 y}{-t^{2}+1}+3 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.002 |
|
|
\[ {}y^{\prime } = -\cot \left (t \right ) y+6 \cos \left (t \right )^{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.224 |
|
|
\[ {}y^{\prime }-x y^{3} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.509 |
|
|
\[ {}\frac {y^{\prime }}{\tan \left (x \right )}-\frac {y}{x^{2}+1} = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.223 |
|
|
\[ {}x^{2} y^{\prime }+x y^{2} = 4 y^{2} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.536 |
|
|
\[ {}y \left (2 x^{2} y^{2}+1\right ) y^{\prime }+x \left (y^{4}+1\right ) = 0 \] |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.227 |
|
|
\[ {}2 x y^{\prime }+3 x +y = 0 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.684 |
|
|
\[ {}\left (\cos \left (x \right )^{2}+y \sin \left (2 x \right )\right ) y^{\prime }+y^{2} = 0 \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.886 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+4 x y = \left (-x^{2}+1\right )^{\frac {3}{2}} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.754 |
|
|
\[ {}y^{\prime }-y \cot \left (x \right )+\frac {1}{\sin \left (x \right )} = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.731 |
|
|
\[ {}\left (x +y^{3}\right ) y^{\prime } = y \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.571 |
|
|
\[ {}y^{\prime } = -\frac {2 x^{2}+y^{2}+x}{x y} \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.653 |
|
|
\[ {}\left (y-x \right ) y^{\prime }+2 x +3 y = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.97 |
|
|
\[ {}y^{\prime } = \frac {1}{x +2 y+1} \] |
homogeneousTypeC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.549 |
|
|
\[ {}y^{\prime } = -\frac {x +y}{3 x +3 y-4} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime } = \tan \left (x \right ) \cos \left (y\right ) \left (\cos \left (y\right )+\sin \left (y\right )\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.569 |
|
|
\[ {}x \left (1-2 x^{2} y\right ) y^{\prime }+y = 3 x^{2} y^{2} \] |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.697 |
|
|
|
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