|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime }+\frac {x y}{a^{2}+x^{2}} = x \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime } = \frac {4 y^{2}}{x^{2}}-y^{2} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.487 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = 1 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.706 |
|
|
\[ {}y^{\prime }-y \tan \left (x \right ) = 1 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.865 |
|
|
\[ {}y^{\prime }-\frac {y^{2}}{x^{2}} = {\frac {1}{4}} \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.181 |
|
|
\[ {}y^{\prime }-\frac {y^{2}}{x^{2}} = {\frac {1}{4}} \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.727 |
|
|
\[ {}y^{\prime } \sin \left (x \right )+2 y \cos \left (x \right ) = 1 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.245 |
|
|
\[ {}\left (5 x +y-7\right ) y^{\prime } = 3+3 x +3 y \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.904 |
|
|
\[ {}x y^{\prime }+y-\frac {y^{2}}{x^{\frac {3}{2}}} = 0 \] |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.586 |
|
|
\[ {}\left (2 \sin \left (y\right )-x \right ) y^{\prime } = \tan \left (y\right ) \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
6.582 |
|
|
\[ {}\left (2 \sin \left (y\right )-x \right ) y^{\prime } = \tan \left (y\right ) \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
6.447 |
|
|
\[ {}y^{\prime \prime }+{y^{\prime }}^{2}+y^{\prime } = 0 \] |
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 |
|
|
\[ {}x^{\prime \prime }+\omega _{0}^{2} x = a \cos \left (\omega t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.99 |
|
|
\[ {}f^{\prime \prime }+2 f^{\prime }+5 f = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.526 |
|
|
\[ {}f^{\prime \prime }+2 f^{\prime }+5 f = {\mathrm e}^{-t} \cos \left (3 t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.778 |
|
|
\[ {}f^{\prime \prime }+6 f^{\prime }+9 f = {\mathrm e}^{-t} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.615 |
|
|
\[ {}f^{\prime \prime }+8 f^{\prime }+12 f = 12 \,{\mathrm e}^{-4 t} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.654 |
|
|
\[ {}f^{\prime \prime }+8 f^{\prime }+12 f = 12 \,{\mathrm e}^{-4 t} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.59 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 4 \,{\mathrm e}^{-x} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.407 |
|
|
\[ {}y^{\prime \prime \prime }-12 y^{\prime }+16 y = 32 x -8 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.554 |
|
|
\[ {}-\frac {{y^{\prime }}^{2}}{y^{2}}+\frac {y^{\prime \prime }}{y}+\frac {2 a \coth \left (2 x a \right ) y^{\prime }}{y} = 2 a^{2} \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.431 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = x \] |
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 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+3 \left (1+x \right ) y^{\prime }+y = x^{2} \] |
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 |
|
|
\[ {}\left (-2+x \right ) y^{\prime \prime }+3 y^{\prime }+\frac {4 y}{x^{2}} = 0 \] |
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 |
|
|
\[ {}y^{\prime \prime }-y = x^{n} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.621 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 x \,{\mathrm e}^{x} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.429 |
|
|
\[ {}2 y y^{\prime \prime \prime }+2 \left (y+3 y^{\prime }\right ) y^{\prime \prime }+2 {y^{\prime }}^{2} = \sin \left (x \right ) \] |
unknown |
[[_3rd_order, _exact, _nonlinear]] |
✗ |
N/A |
0.0 |
|
|
\[ {}x y^{\prime \prime \prime }+2 y^{\prime \prime } = A x \] |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+6\right ) y = {\mathrm e}^{-x^{2}} \sin \left (2 x \right ) \] |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.322 |
|
|
\[ {}\left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+\lambda y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.898 |
|
|
\[ {}4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.984 |
|
|
\[ {}z y^{\prime \prime }-2 y^{\prime }+9 z^{5} y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.749 |
|
|
\[ {}f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.702 |
|
|
\[ {}z^{2} y^{\prime \prime }-\frac {3 z y^{\prime }}{2}+\left (z +1\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.967 |
|
|
\[ {}z y^{\prime \prime }-2 y^{\prime }+y z = 0 \] |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.766 |
|
|
\[ {}y^{\prime \prime }-2 z y^{\prime }-2 y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.505 |
|
|
\[ {}z \left (1-z \right ) y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y = 0 \] |
second order series method. Regular singular point. Repeated root |
[_Jacobi] |
✓ |
✓ |
1.181 |
|
|
\[ {}z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z} = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.787 |
|
|
\[ {}\left (z^{2}+5 z +6\right ) y^{\prime \prime }+2 y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.756 |
|
|
\[ {}\left (z^{2}+5 z +7\right ) y^{\prime \prime }+2 y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.793 |
|
|
\[ {}y^{\prime \prime }+\frac {y}{z^{3}} = 0 \] |
second order series method. Irregular singular point |
[[_Emden, _Fowler]] |
❇ |
N/A |
0.141 |
|
|
\[ {}z y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y = 0 \] |
second order series method. Regular singular point. Repeated root |
[_Laguerre] |
✓ |
✓ |
1.276 |
|
|
\[ {}\left (-z^{2}+1\right ) y^{\prime \prime }-z y^{\prime }+m^{2} y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.903 |
|
|
\[ {}y^{\prime } = 2 x y \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.505 |
|
|
\[ {}y^{\prime } = \frac {y^{2}}{x^{2}+1} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.513 |
|
|
\[ {}{\mathrm e}^{x +y} y^{\prime }-1 = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.407 |
|
|
\[ {}y^{\prime } = \frac {y}{x \ln \left (x \right )} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.674 |
|
|
\[ {}y-\left (-2+x \right ) y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.615 |
|
|
\[ {}y^{\prime } = \frac {2 x \left (y-1\right )}{x^{2}+3} \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.744 |
|
|
\[ {}y-x y^{\prime } = 3-2 x^{2} y^{\prime } \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.659 |
|
|
\[ {}y^{\prime } = \frac {\cos \left (x -y\right )}{\sin \left (x \right ) \sin \left (y\right )}-1 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
36.244 |
|
|
\[ {}y^{\prime } = \frac {x \left (y^{2}-1\right )}{2 \left (-2+x \right ) \left (-1+x \right )} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.073 |
|
|
\[ {}y^{\prime } = \frac {x^{2} y-32}{-x^{2}+16}+32 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.852 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }-y+c = 0 \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.233 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+y^{2} = -1 \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.069 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+x y = x a \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.399 |
|
|
\[ {}y^{\prime } = 1-\frac {\sin \left (x +y\right )}{\sin \left (y\right ) \cos \left (x \right )} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
36.764 |
|
|
\[ {}y^{\prime } = y^{3} \sin \left (x \right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.524 |
|
|
\[ {}y^{\prime }-y = {\mathrm e}^{2 x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.464 |
|
|
\[ {}x^{2} y^{\prime }-4 x y = x^{7} \sin \left (x \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.71 |
|
|
\[ {}y^{\prime }+2 x y = 2 x^{3} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.51 |
|
|
\[ {}y^{\prime }+\frac {2 x y}{x^{2}+1} = 4 x \] |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.722 |
|
|
\[ {}y^{\prime }+\frac {2 x y}{x^{2}+1} = \frac {4}{\left (x^{2}+1\right )^{2}} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.65 |
|
|
\[ {}2 \cos \left (x \right )^{2} y^{\prime }+y \sin \left (2 x \right ) = 4 \cos \left (x \right )^{4} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.946 |
|
|
\[ {}y^{\prime }+\frac {y}{x \ln \left (x \right )} = 9 x^{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.669 |
|
|
\[ {}y^{\prime }-y \tan \left (x \right ) = 8 \sin \left (x \right )^{3} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.744 |
|
|
\[ {}t x^{\prime }+2 x = 4 \,{\mathrm e}^{t} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.585 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \left (y \sec \left (x \right )-2\right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.757 |
|
|
\[ {}1-y \sin \left (x \right )-\cos \left (x \right ) y^{\prime } = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.935 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = 2 \ln \left (x \right ) x^{2} \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.595 |
|
|
\[ {}y^{\prime }+\alpha y = {\mathrm e}^{\beta x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime }+\frac {m}{x} = \ln \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.135 |
|
|
\[ {}\left (3 x -y\right ) y^{\prime } = 3 y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.94 |
|
|
\[ {}y^{\prime } = \frac {\left (x +y\right )^{2}}{2 x^{2}} \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.915 |
|
|
\[ {}\sin \left (\frac {y}{x}\right ) \left (-y+x y^{\prime }\right ) = x \cos \left (\frac {y}{x}\right ) \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.482 |
|
|
\[ {}x y^{\prime } = \sqrt {16 x^{2}-y^{2}}+y \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.104 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {9 x^{2}+y^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.286 |
|
|
\[ {}x \left (x^{2}-y^{2}\right )-x \left (x^{2}+y^{2}\right ) y^{\prime } = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.763 |
|
|
\[ {}x y^{\prime }+y \ln \left (x \right ) = y \ln \left (y\right ) \] |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.011 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+2 x y-2 x^{2}}{x^{2}-x y+y^{2}} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.046 |
|
|
\[ {}2 x y y^{\prime }-x^{2} {\mathrm e}^{-\frac {y^{2}}{x^{2}}}-2 y^{2} = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘]] |
✓ |
✓ |
0.896 |
|
|
\[ {}x^{2} y^{\prime } = y^{2}+3 x y+x^{2} \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.714 |
|
|
\[ {}y y^{\prime } = \sqrt {x^{2}+y^{2}}-x \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.407 |
|
|
\[ {}2 x \left (y+2 x \right ) y^{\prime } = y \left (4 x -y\right ) \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.148 |
|
|
\[ {}x y^{\prime } = x \tan \left (\frac {y}{x}\right )+y \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.953 |
|
|
\[ {}y^{\prime } = \frac {x \sqrt {x^{2}+y^{2}}+y^{2}}{x y} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.997 |
|
|
\[ {}y^{\prime \prime }-25 y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.777 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.183 |
|
|
\[ {}y^{\prime } = -y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.084 |
|
|
\[ {}y^{\prime } = \frac {y}{2 x} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.488 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.327 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.619 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+3 y = 0 \] |
kovacic, second_order_euler_ode, 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, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.514 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] |
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 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.855 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+13 y = 0 \] |
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 |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.283 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+y = 9 x^{2} \] |
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]] |
✓ |
✓ |
1.592 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x^{4} \sin \left (x \right ) \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.773 |
|
|
\[ {}y^{\prime \prime }-\left (a +b \right ) y^{\prime }+y a b = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.368 |
|
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+a^{2} y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.224 |
|
|
|
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