|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = 2 \ln \left (x \right )^{2}+12 x \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.657 |
|
|
\[ {}\left (1+x \right )^{3} y^{\prime \prime }+3 \left (1+x \right )^{2} y^{\prime }+\left (1+x \right ) y = 6 \ln \left (1+x \right ) \] |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.355 |
|
|
\[ {}\left (-2+x \right )^{2} y^{\prime \prime }-3 \left (-2+x \right ) y^{\prime }+4 y = x \] |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.848 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime \prime }+\left (-2+4 x \right ) y^{\prime }-8 y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.714 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }+\left (2 x -3\right ) y^{\prime }-2 y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[_Jacobi] |
✓ |
✓ |
1.088 |
|
|
\[ {}\left (2 x^{2}+3 x \right ) y^{\prime \prime }-6 \left (1+x \right ) y^{\prime }+6 y = 6 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.181 |
|
|
\[ {}x^{2} \left (-1+\ln \left (x \right )\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.287 |
|
|
\[ {}y^{\prime \prime }+\left (\tan \left (x \right )-2 \cot \left (x \right )\right ) y^{\prime }+2 \cot \left (x \right )^{2} y = 0 \] |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.844 |
|
|
\[ {}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0 \] |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.562 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-y = 1 \] |
reduction_of_order |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.324 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-3 y = 5 x^{4} \] |
reduction_of_order |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.283 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = \left (-1+x \right )^{2} {\mathrm e}^{x} \] |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.352 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+{\mathrm e}^{-2 x} y = {\mathrm e}^{-3 x} \] |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.536 |
|
|
\[ {}\left (x^{4}-x^{3}\right ) y^{\prime \prime }+\left (2 x^{3}-2 x^{2}-x \right ) y^{\prime }-y = \frac {\left (-1+x \right )^{2}}{x} \] |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.374 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+{\mathrm e}^{2 x} y = {\mathrm e}^{2 x} x -1 \] |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.623 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+2 y = x^{2} \left (2 x -3\right ) \] |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.359 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\sin \left (x \right )} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.5 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = \frac {1}{1+{\mathrm e}^{x}} \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.326 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\cos \left (x \right )^{3}} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\sqrt {\sin \left (x \right )^{5} \cos \left (x \right )}} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.762 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{x}}{x^{2}+1} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.472 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = \frac {{\mathrm e}^{-x}}{\sin \left (x \right )} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.582 |
|
|
\[ {}y^{\prime \prime }+y = \frac {2}{\sin \left (x \right )^{3}} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.625 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = {\mathrm e}^{2 x} \cos \left ({\mathrm e}^{x}\right ) \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.529 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime } = \frac {-1+x}{x^{3}} \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.315 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime } = 4 x^{3} {\mathrm e}^{x^{2}} \] |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.488 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime } = 1 \] |
kovacic, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.468 |
|
|
\[ {}x \ln \left (x \right ) y^{\prime \prime }-y^{\prime } = \ln \left (x \right )^{2} \] |
second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.908 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x -1\right ) y^{\prime } = -4 x^{2} \] |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.764 |
|
|
\[ {}y^{\prime \prime }+\tan \left (x \right ) y^{\prime } = \cos \left (x \right ) \cot \left (x \right ) \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.911 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = 1 \] |
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, _with_linear_symmetries]] |
❇ |
N/A |
2.32 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = \frac {x +6}{x^{2}} \] |
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]] |
✗ |
N/A |
2.353 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime } = \frac {1}{x^{2}+1} \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.061 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (-1+x \right )^{2} {\mathrm e}^{x} \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.839 |
|
|
\[ {}2 x^{2} \left (2-\ln \left (x \right )\right ) y^{\prime \prime }+x \left (4-\ln \left (x \right )\right ) y^{\prime }-y = \frac {\left (2-\ln \left (x \right )\right )^{2}}{\sqrt {x}} \] |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.892 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-y = 4 \,{\mathrm e}^{x} \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
8.398 |
|
|
\[ {}x^{3} \left (-1+\ln \left (x \right )\right ) y^{\prime \prime }-x^{2} y^{\prime }+x y = 2 \ln \left (x \right ) \] |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.788 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-2 \left (1-x \right ) y = 2 x -2 \] |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.733 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.384 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+6 x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.363 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+x = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.253 |
|
|
\[ {}x^{\prime \prime }+{x^{\prime }}^{2}+x = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.035 |
|
|
\[ {}x^{\prime \prime }-2 {x^{\prime }}^{2}+x^{\prime }-2 x = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.378 |
|
|
\[ {}x^{\prime \prime }-x \,{\mathrm e}^{x^{\prime }} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
0.585 |
|
|
\[ {}x^{\prime \prime }+{\mathrm e}^{-x^{\prime }}-x = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.542 |
|
|
\[ {}x^{\prime \prime }+x {x^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.524 |
|
|
\[ {}x^{\prime \prime }+\left (x+2\right ) x^{\prime } = 0 \] |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.703 |
|
|
\[ {}x^{\prime \prime }-x^{\prime }+x-x^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.385 |
|
|
\[ {}y^{\prime \prime }+\lambda y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.363 |
|
|
\[ {}y^{\prime \prime }+\lambda y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.679 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.101 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
1.0 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
8.125 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.275 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.601 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.421 |
|
|
\[ {}y^{\prime \prime }+\alpha y^{\prime } = 0 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.14 |
|
|
\[ {}y^{\prime \prime }+\alpha ^{2} y = 1 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
111.799 |
|
|
\[ {}y^{\prime \prime }+y = 1 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.641 |
|
|
\[ {}y^{\prime \prime }+\lambda ^{2} y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.908 |
|
|
\[ {}y^{\prime \prime }+\lambda ^{2} y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.921 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }-y^{\prime }-y = 0 \] |
unknown |
[[_3rd_order, _missing_x]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime \prime \prime }-\lambda ^{4} y = 0 \] |
unknown |
[[_high_order, _missing_x]] |
✗ |
N/A |
0.0 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
0.819 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }+4 x y^{\prime \prime \prime }+2 y^{\prime \prime } = 0 \] |
higher_order_missing_y |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.757 |
|
|
\[ {}x^{3} y^{\prime \prime \prime \prime }+6 x^{2} y^{\prime \prime \prime }+6 x y^{\prime \prime } = 0 \] |
higher_order_missing_y |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.667 |
|
|
\[ {}y^{\prime } = 1-x y \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
1.467 |
|
|
\[ {}y^{\prime } = \frac {y-x}{x +y} \] |
first order ode series method. Taylor series method |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.197 |
|
|
\[ {}y^{\prime } = y \sin \left (x \right ) \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
1.372 |
|
|
\[ {}y^{\prime \prime }+x y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.148 |
|
|
\[ {}y^{\prime \prime }-y^{\prime } \sin \left (x \right ) = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.501 |
|
|
\[ {}x y^{\prime \prime }+y \sin \left (x \right ) = x \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.568 |
|
|
\[ {}\ln \left (x \right ) y^{\prime \prime }-y \sin \left (x \right ) = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
40.846 |
|
|
\[ {}y^{\prime \prime \prime }+x \sin \left (y\right ) = 0 \] |
unknown |
[NONE] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime }-2 x y = 0 \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
1.181 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.511 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+y = 1 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.207 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2}+1\right ) y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.238 |
|
|
\[ {}y^{\prime \prime } = x^{2} y-y^{\prime } \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.415 |
|
|
\[ {}y^{\prime \prime }-{\mathrm e}^{x} y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.629 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{y}+x y \] |
first order ode series method. Taylor series method |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.368 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.831 |
|
|
\[ {}\left (1+x \right ) y^{\prime }-n y = 0 \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.543 |
|
|
\[ {}9 x \left (1-x \right ) y^{\prime \prime }-12 y^{\prime }+4 y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[_Jacobi] |
✓ |
✓ |
1.122 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (4 x^{2}-\frac {1}{9}\right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.582 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.802 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+\frac {y}{9} = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.517 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+4 y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.52 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+4 \left (x^{4}-1\right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.675 |
|
|
\[ {}x y^{\prime \prime }+\frac {y^{\prime }}{2}+\frac {y}{4} = 0 \] |
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)]‘]] |
✓ |
✓ |
0.894 |
|
|
\[ {}y^{\prime \prime }+\frac {5 y^{\prime }}{x}+y = 0 \] |
second_order_bessel_ode |
[_Lienard] |
✓ |
✓ |
0.542 |
|
|
\[ {}y^{\prime \prime }+\frac {3 y^{\prime }}{x}+4 y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.556 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (x \right )^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.67 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \pi ^{2}-x^{2} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.481 |
|
|
\[ {}y^{\prime \prime }-4 y = \cos \left (\pi x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.514 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \arcsin \left (\sin \left (x \right )\right ) \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.8 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (x \right )^{3} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.135 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime }=-2 t x_{1}^{2} \\ x_{2}^{\prime }=\frac {x_{2}+t}{t} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.119 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime }={\mathrm e}^{t -x_{1}} \\ x_{2}^{\prime }=2 \,{\mathrm e}^{x_{1}} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.246 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=\frac {y^{2}}{x} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.122 |
|
|
|
||||||
|
|
||||||