|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.818 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.169 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.121 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime }+y = x \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
67.554 |
|
|
\[ {}x^{4} y^{\prime \prime }+x^{3} y^{\prime }-4 x^{2} y = 1 \] |
kovacic, 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, _with_linear_symmetries]] |
✓ |
✓ |
1.953 |
|
|
\[ {}x^{4} y^{\prime \prime }+x^{3} y^{\prime }-4 x^{2} y = x \] |
kovacic, 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, _with_linear_symmetries]] |
✓ |
✓ |
1.736 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = x \] |
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 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.913 |
|
|
\[ {}x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+x y = 0 \] |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.77 |
|
|
\[ {}x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+x y = x \] |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.071 |
|
|
\[ {}5 x^{5} y^{\prime \prime \prime \prime }+4 x^{4} y^{\prime \prime \prime }+x^{2} y^{\prime }+x y = 0 \] |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
62.378 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.336 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = x \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✗ |
94.947 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+x {y^{\prime }}^{2} = 1 \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.063 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+y {y^{\prime }}^{2} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.071 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
2.509 |
|
|
\[ {}y^{\prime \prime }+\sin \left (y\right ) {y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.299 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{3} = 0 \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
2.088 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{-\frac {y}{x}} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.303 |
|
|
\[ {}y^{\prime } = 2 x^{2} \sin \left (\frac {y}{x}\right )^{2}+\frac {y}{x} \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
4.247 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 8 \sqrt {x}\, \left (1+\ln \left (x \right )\right ) \] |
kovacic, second_order_euler_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.844 |
|
|
\[ {}v v^{\prime } = \frac {2 v^{2}}{r^{3}}+\frac {\lambda r}{3} \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.752 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.487 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = 1 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.492 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = 1+x \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.358 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.158 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2}+x +1 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.42 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2} \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.279 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2}+1 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.556 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{4} \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.329 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = \sin \left (x \right ) \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.541 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = \sin \left (x \right )+1 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.593 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x \sin \left (x \right ) \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.678 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = \cos \left (x \right )+\sin \left (x \right ) \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.996 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (\cos \left (x \right )-1\right ) y^{\prime }+{\mathrm e}^{x} y = 0 \] |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
16.961 |
|
|
\[ {}\left (-2+x \right ) y^{\prime \prime }+\frac {y^{\prime }}{x}+y \left (1+x \right ) = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.357 |
|
|
\[ {}\left (-2+x \right ) y^{\prime \prime }+\frac {y^{\prime }}{x}+y \left (1+x \right ) = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.547 |
|
|
\[ {}\left (1+x \right ) \left (3 x -1\right ) y^{\prime \prime }+\cos \left (x \right ) y^{\prime }-3 x y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
17.183 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
1.501 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-x y = x^{2}+2 x \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.738 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = 1 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.22 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = 1 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
1.352 |
|
|
\[ {}y^{\prime \prime }+\left (x -6\right ) y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.036 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (3 x^{2}+2 x \right ) y^{\prime }-2 y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.627 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2}+\cos \left (x \right ) \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.145 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = \cos \left (x \right ) \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.615 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{3}+\cos \left (x \right ) \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.303 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{3} \cos \left (x \right ) \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.842 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{3} \cos \left (x \right )+\sin \left (x \right )^{2} \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
5.192 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = \ln \left (x \right ) \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.628 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.01 |
|
|
\[ {}x^{2} \left (x +3\right ) y^{\prime \prime }+5 x \left (1+x \right ) y^{\prime }-\left (1-4 x \right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.873 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.339 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2} = \sec \left (x \right )^{4} \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
12.38 |
|
|
\[ {}\left (y-2 x y^{\prime }\right )^{2} = {y^{\prime }}^{3} \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
276.817 |
|
|
\[ {}x^{2} y^{\prime \prime }+y = 0 \] |
second order series method. Regular singular point. Complex roots |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.102 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.472 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.714 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.36 |
|
|
\[ {}x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+2 y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.688 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }+3 x y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.53 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.018 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+y \left (1+x \right ) = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.253 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x y^{\prime }+\left (x -5\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.365 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = \sin \left (x \right ) \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.121 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = x \sin \left (x \right ) \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.728 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = \sin \left (x \right ) \cos \left (x \right ) \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.309 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y = x^{3}+x \sin \left (x \right ) \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.411 |
|
|
\[ {}\cos \left (x \right ) y^{\prime \prime }+2 x y^{\prime }-x y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
63.965 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.919 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-x y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.385 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.74 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.196 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}+6 x \right ) y^{\prime }+x y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.009 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+\left (x^{2}-8\right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.206 |
|
|
\[ {}x^{2} y^{\prime \prime }-9 x y^{\prime }+25 y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.118 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-\left (x^{2}+\frac {5}{4}\right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.864 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.55 |
|
|
\[ {}x y^{\prime \prime }+\left (2-x \right ) y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.73 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.329 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = 0 \] |
second order series method. Regular singular point. Complex roots |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.525 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+4 y x^{4} = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.769 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.93 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y = x \,{\mathrm e}^{x} \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
11.66 |
|
|
\[ {}y^{\prime } = y \left (1-y^{2}\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
2.602 |
|
|
\[ {}\frac {x y^{\prime \prime }}{1-x}+y = \frac {1}{1-x} \] |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.831 |
|
|
\[ {}\frac {x y^{\prime \prime }}{1-x}+x y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.949 |
|
|
\[ {}\frac {x y^{\prime \prime }}{1-x}+y = \cos \left (x \right ) \] |
second_order_bessel_ode, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
5.763 |
|
|
\[ {}\frac {x y^{\prime \prime }}{-x^{2}+1}+y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.057 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.413 |
|
|
\[ {}y^{\prime \prime }+\left (-1+x \right ) y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.198 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+2 y+2 t +1 \\ y^{\prime }=5 x+y+3 t -1 \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
2.147 |
|
|
\[ {}y^{\prime \prime }+20 y^{\prime }+500 y = 100000 \cos \left (100 x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.781 |
|
|
\[ {}y^{\prime \prime } \sin \left (2 x \right )^{2}+y^{\prime } \sin \left (4 x \right )-4 y = 0 \] |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
10.211 |
|
|
\[ {}y^{\prime \prime } = A y^{\frac {2}{3}} \] |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.175 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.978 |
|
|
\[ {}y^{\prime \prime }+2 \cot \left (x \right ) y^{\prime }-y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.969 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.065 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 4 \sqrt {x}\, {\mathrm e}^{x} \] |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.559 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 6 x^{3} {\mathrm e}^{x} \] |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.158 |
|
|
\[ {}y^{\prime }+y = \frac {1}{x} \] |
|
[[_linear, ‘class A‘]] |
❇ |
N/A |
0.693 |
|
|
|
||||||
|
|
||||||