|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime \prime }+2 y = x^{3}+x^{2}+{\mathrm e}^{-2 x}+\cos \left (3 x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.331 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-y = {\mathrm e}^{x} \cos \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.578 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \frac {{\mathrm e}^{2 x}}{x^{2}} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.534 |
|
|
\[ {}y^{\prime \prime }-y = x \,{\mathrm e}^{3 x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.404 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = {\mathrm e}^{-2 x} \sec \left (x \right )^{2} \left (1+2 \tan \left (x \right )\right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.136 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = x +\ln \left (x \right ) x^{2} \] |
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 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.908 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = \ln \left (x \right )^{2}-\ln \left (x^{2}\right ) \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
11.81 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime } = x +\sin \left (\ln \left (x \right )\right ) \] |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.463 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+x y^{\prime }-y = 3 x^{4} \] |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.336 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }-y = \ln \left (1+x \right )^{2}+x -1 \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.375 |
|
|
\[ {}\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }-12 y = 6 x \] |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.28 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }+2 y = 0 \] |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[_Laguerre] |
✓ |
✓ |
1.691 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 2 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.546 |
|
|
\[ {}\left (x^{2}+4\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 8 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.884 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+\left (2+x \right ) y = \left (x^{2}+2 x +1\right ) {\mathrm e}^{2 x} \] |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.653 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-10 y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.606 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2 x +3\right ) y^{\prime }+\left (x^{2}+3 x +3\right ) y = \left (-x^{2}+6\right ) {\mathrm e}^{x} \] |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.617 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right )^{2} y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.719 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-4 x^{2}+x \right ) y^{\prime }+\left (4 x^{2}-2 x +1\right ) y = \left (x^{2}-x +1\right ) {\mathrm e}^{x} \] |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.564 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.25 |
|
|
\[ {}x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+y = \frac {1+x}{x} \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.923 |
|
|
\[ {}x^{8} y^{\prime \prime }+4 x^{7} y^{\prime }+y = \frac {1}{x^{3}} \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.064 |
|
|
\[ {}\left (x \sin \left (x \right )+\cos \left (x \right )\right ) y^{\prime \prime }-x \cos \left (x \right ) y^{\prime }+y \cos \left (x \right ) = x \] |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.556 |
|
|
\[ {}x y^{\prime \prime }-3 y^{\prime }+\frac {3 y}{x} = 2+x \] |
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]] |
✓ |
✓ |
1.818 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (3 x +4\right ) y^{\prime }+3 y = \left (3 x +2\right ) {\mathrm e}^{3 x} \] |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.077 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (9 x^{2}+6\right ) y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 x y = 4 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.108 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = \frac {-x^{2}+1}{x} \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.576 |
|
|
\[ {}y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.191 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime } = \frac {2}{x^{3}} \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.133 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = -\frac {2}{x}-\ln \left (x \right ) \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.816 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime } = x^{2} \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.174 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{3} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.63 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 0 \] |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.932 |
|
|
\[ {}y y^{\prime \prime } = {y^{\prime }}^{2} \left (1-y^{\prime } \cos \left (y\right )+y y^{\prime } \sin \left (y\right )\right ) \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
3.074 |
|
|
\[ {}\left (2 x -3\right ) y^{\prime \prime \prime }-\left (6 x -7\right ) y^{\prime \prime }+4 x y^{\prime }-4 y = 8 \] |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.174 |
|
|
\[ {}\left (2 x^{3}-1\right ) y^{\prime \prime \prime }-6 x^{2} y^{\prime \prime }+6 x y^{\prime } = 0 \] |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
3.004 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{2} \ln \left (y\right ) \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.854 |
|
|
\[ {}\left (2 y+x \right ) y^{\prime \prime }+2 {y^{\prime }}^{2}+2 y^{\prime } = 2 \] |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.543 |
|
|
\[ {}\left (1+2 y+3 y^{2}\right ) y^{\prime \prime \prime }+6 y^{\prime } \left (y^{\prime \prime }+{y^{\prime }}^{2}+3 y y^{\prime \prime }\right ) = x \] |
unknown |
[[_3rd_order, _exact, _nonlinear]] |
✗ |
N/A |
0.0 |
|
|
\[ {}3 x \left (y^{2} y^{\prime \prime \prime }+6 y y^{\prime } y^{\prime \prime }+2 {y^{\prime }}^{3}\right )-3 y \left (y y^{\prime \prime }+2 {y^{\prime }}^{2}\right ) = -\frac {2}{x} \] |
unknown |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y y^{\prime \prime \prime }+3 y^{\prime } y^{\prime \prime }-2 y y^{\prime \prime }-2 {y^{\prime }}^{2}+y y^{\prime } = {\mathrm e}^{2 x} \] |
unknown |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}2 \left (y+1\right ) y^{\prime \prime }+2 {y^{\prime }}^{2}+y^{2}+2 y = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.474 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-y^{\prime }+y=-{\mathrm e}^{t} \\ x+y^{\prime }-y={\mathrm e}^{2 t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.679 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+2 x+y^{\prime }+y=t \\ 5 x+y^{\prime }+3 y=t^{2} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.152 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+x+2 y^{\prime }+7 y={\mathrm e}^{t}+2 \\ -2 x+y^{\prime }+3 y={\mathrm e}^{t}-1 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.399 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-x+y^{\prime }+3 y={\mathrm e}^{-t}-1 \\ x^{\prime }+2 x+y^{\prime }+3 y={\mathrm e}^{2 t}+1 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.407 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-x+y^{\prime }+2 y=1+{\mathrm e}^{t} \\ y^{\prime }+2 y+z^{\prime }+z={\mathrm e}^{t}+2 \\ x^{\prime }-x+z^{\prime }+z=3+{\mathrm e}^{t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.734 |
|
|
\[ {}\left (1-x \right ) y^{\prime } = x^{2}-y \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
0.369 |
|
|
\[ {}x y^{\prime } = 1-x +2 y \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
0.432 |
|
|
\[ {}x y^{\prime } = 1-x +2 y \] |
linear, homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.968 |
|
|
\[ {}y^{\prime } = 2 x^{2}+3 y \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.52 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = x^{2}-2 x +y \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
0.385 |
|
|
\[ {}y^{\prime \prime }+x y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.401 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.416 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+x^{2} y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.938 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+p \left (p +1\right ) y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.976 |
|
|
\[ {}y^{\prime \prime }+x^{2} y = x^{2}+x +1 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.563 |
|
|
\[ {}2 \left (x^{3}+x^{2}\right ) y^{\prime \prime }-\left (-3 x^{2}+x \right ) y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.044 |
|
|
\[ {}4 x y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.099 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.872 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+x y = 0 \] |
second order series method. Regular singular point. Repeated root |
[_Lienard] |
✓ |
✓ |
0.77 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.845 |
|
|
\[ {}x y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.696 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.826 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.016 |
|
|
\[ {}2 x y^{\prime \prime }+y^{\prime }-y = 1+x \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.929 |
|
|
\[ {}2 x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.925 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.901 |
|
|
\[ {}z^{\prime \prime }+t z^{\prime }+\left (t^{2}-\frac {1}{9}\right ) z = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.991 |
|
|
\[ {}x \left (-x^{2}+2\right ) y^{\prime \prime }-\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right ) = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.262 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-\left (2 x +1\right ) \left (-y+x y^{\prime }\right ) = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.035 |
|
|
\[ {}x^{3} \left (1+x \right ) y^{\prime \prime \prime }-\left (4 x +2\right ) x^{2} y^{\prime \prime }+\left (4+10 x \right ) x y^{\prime }-\left (4+12 x \right ) y = 0 \] |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}x^{3} \left (x^{2}+1\right ) y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-\left (12 x^{2}+4\right ) y = 0 \] |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}2 \left (2-x \right ) x^{2} y^{\prime \prime }-\left (4-x \right ) x y^{\prime }+\left (-x +3\right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.068 |
|
|
\[ {}\left (1-x \right ) x^{2} y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.832 |
|
|
\[ {}x y^{\prime \prime }+\left (4 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.019 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 \left (x +a \right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.258 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{3}+1\right ) y^{\prime }+b x y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.65 |
|
|
\[ {}\left (-1+x \right ) \left (-2+x \right ) y^{\prime \prime }+\left (4 x -6\right ) y^{\prime }+2 y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.69 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.019 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.874 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
1.562 |
|
|
\[ {}y^{\prime \prime } = \left (-1+x \right ) y \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.369 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+2 \left (1+x \right ) y^{\prime }-2 y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.984 |
|
|
\[ {}x y^{\prime \prime }+y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.671 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{x}-1\right ) y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.612 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }-3 x y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.03 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+x^{2} y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.816 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }-y \sin \left (x \right ) = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.325 |
|
|
\[ {}y^{\prime \prime }-x^{2} y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.468 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }-4 y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.049 |
|
|
\[ {}x y^{\prime \prime }+\left (\frac {1}{2}-x \right ) y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.929 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}+\frac {1}{4}\right ) y = 0 \] |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.945 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}+\frac {9}{4}\right ) y = 0 \] |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.938 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}+\frac {25}{4}\right ) y = 0 \] |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.287 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime }+x y = \cos \left (x \right ) \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
0.548 |
|
|
\[ {}y^{\prime }+x y = \frac {1}{x^{3}} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.727 |
|
|
\[ {}x^{3} y^{\prime \prime }+y = \frac {1}{x^{4}} \] |
second order series method. Irregular singular point |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
0.241 |
|
|
|
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