Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime }-x y^{3} = 0 \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.509 |
|
|
\[ {}\frac {y^{\prime }}{\tan \left (x \right )}-\frac {y}{x^{2}+1} = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.223 |
|
|
\[ {}x^{2} y^{\prime }+x y^{2} = 4 y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.536 |
|
|
\[ {}y \left (2 x^{2} y^{2}+1\right ) y^{\prime }+x \left (y^{4}+1\right ) = 0 \] |
1 |
1 |
4 |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.227 |
|
|
\[ {}2 x y^{\prime }+3 x +y = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.684 |
|
|
\[ {}\left (\cos \left (x \right )^{2}+y \sin \left (2 x \right )\right ) y^{\prime }+y^{2} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.886 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+4 x y = \left (-x^{2}+1\right )^{\frac {3}{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.754 |
|
|
\[ {}y^{\prime }-y \cot \left (x \right )+\frac {1}{\sin \left (x \right )} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.731 |
|
|
\[ {}\left (x +y^{3}\right ) y^{\prime } = y \] |
1 |
1 |
3 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.571 |
|
|
\[ {}y^{\prime } = -\frac {2 x^{2}+y^{2}+x}{x y} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.653 |
|
|
\[ {}\left (y-x \right ) y^{\prime }+2 x +3 y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.97 |
|
|
\[ {}y^{\prime } = \frac {1}{x +2 y+1} \] |
1 |
1 |
1 |
homogeneousTypeC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.549 |
|
|
\[ {}y^{\prime } = -\frac {x +y}{3 x +3 y-4} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime } = \tan \left (x \right ) \cos \left (y\right ) \left (\cos \left (y\right )+\sin \left (y\right )\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.569 |
|
|
\[ {}x \left (1-2 x^{2} y\right ) y^{\prime }+y = 3 x^{2} y^{2} \] |
1 |
1 |
2 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.697 |
|
|
\[ {}y^{\prime }+\frac {x y}{a^{2}+x^{2}} = x \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime } = \frac {4 y^{2}}{x^{2}}-y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.487 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = 1 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.706 |
|
|
\[ {}y^{\prime }-y \tan \left (x \right ) = 1 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.865 |
|
|
\[ {}y^{\prime }-\frac {y^{2}}{x^{2}} = {\frac {1}{4}} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.181 |
|
|
\[ {}y^{\prime }-\frac {y^{2}}{x^{2}} = {\frac {1}{4}} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.727 |
|
|
\[ {}y^{\prime } \sin \left (x \right )+2 y \cos \left (x \right ) = 1 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.245 |
|
|
\[ {}\left (5 x +y-7\right ) y^{\prime } = 3+3 x +3 y \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.904 |
|
|
\[ {}x y^{\prime }+y-\frac {y^{2}}{x^{\frac {3}{2}}} = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.586 |
|
|
\[ {}\left (2 \sin \left (y\right )-x \right ) y^{\prime } = \tan \left (y\right ) \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
6.582 |
|
|
\[ {}\left (2 \sin \left (y\right )-x \right ) y^{\prime } = \tan \left (y\right ) \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
6.447 |
|
|
\[ {}y^{\prime \prime }+{y^{\prime }}^{2}+y^{\prime } = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y, second_order_nonlinear_solved_by_mainardi_lioville_method |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.794 |
|
|
\[ {}x^{\prime \prime }+\omega _{0}^{2} x = a \cos \left (\omega t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.99 |
|
|
\[ {}f^{\prime \prime }+2 f^{\prime }+5 f = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.526 |
|
|
\[ {}f^{\prime \prime }+2 f^{\prime }+5 f = {\mathrm e}^{-t} \cos \left (3 t \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.778 |
|
|
\[ {}f^{\prime \prime }+6 f^{\prime }+9 f = {\mathrm e}^{-t} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.615 |
|
|
\[ {}f^{\prime \prime }+8 f^{\prime }+12 f = 12 \,{\mathrm e}^{-4 t} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.654 |
|
|
\[ {}f^{\prime \prime }+8 f^{\prime }+12 f = 12 \,{\mathrm e}^{-4 t} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.59 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 4 \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.407 |
|
|
\[ {}y^{\prime \prime \prime }-12 y^{\prime }+16 y = 32 x -8 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.554 |
|
|
\[ {}-\frac {{y^{\prime }}^{2}}{y^{2}}+\frac {y^{\prime \prime }}{y}+\frac {2 a \coth \left (2 x a \right ) y^{\prime }}{y} = 2 a^{2} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.431 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = x \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.502 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+3 \left (1+x \right ) y^{\prime }+y = x^{2} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.731 |
|
|
\[ {}\left (-2+x \right ) y^{\prime \prime }+3 y^{\prime }+\frac {4 y}{x^{2}} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.487 |
|
|
\[ {}y^{\prime \prime }-y = x^{n} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.621 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.429 |
|
|
\[ {}2 y y^{\prime \prime \prime }+2 \left (y+3 y^{\prime }\right ) y^{\prime \prime }+2 {y^{\prime }}^{2} = \sin \left (x \right ) \] |
1 |
0 |
2 |
unknown |
[[_3rd_order, _exact, _nonlinear]] |
✗ |
N/A |
0.0 |
|
|
\[ {}x y^{\prime \prime \prime }+2 y^{\prime \prime } = A x \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+6\right ) y = {\mathrm e}^{-x^{2}} \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.322 |
|
|
\[ {}\left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+\lambda y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.898 |
|
|
\[ {}4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.984 |
|
|
\[ {}z y^{\prime \prime }-2 y^{\prime }+9 z^{5} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.749 |
|
|
\[ {}f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.702 |
|
|
\[ {}z^{2} y^{\prime \prime }-\frac {3 z y^{\prime }}{2}+\left (z +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.967 |
|
|
\[ {}z y^{\prime \prime }-2 y^{\prime }+y z = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.766 |
|
|
\[ {}y^{\prime \prime }-2 z y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.505 |
|
|
\[ {}z \left (1-z \right ) y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Jacobi] |
✓ |
✓ |
1.181 |
|
|
\[ {}z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.787 |
|
|
\[ {}\left (z^{2}+5 z +6\right ) y^{\prime \prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.756 |
|
|
\[ {}\left (z^{2}+5 z +7\right ) y^{\prime \prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.793 |
|
|
\[ {}y^{\prime \prime }+\frac {y}{z^{3}} = 0 \] |
1 |
0 |
0 |
second order series method. Irregular singular point |
[[_Emden, _Fowler]] |
❇ |
N/A |
0.141 |
|
|
\[ {}z y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Laguerre] |
✓ |
✓ |
1.276 |
|
|
\[ {}\left (-z^{2}+1\right ) y^{\prime \prime }-z y^{\prime }+m^{2} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.903 |
|
|
|
|||||||||
|
|
|||||||||