|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime } = {\mathrm e}^{-t^{2}}+y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
0.775 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{-t^{2}}+y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
0.78 |
|
|
\[ {}y^{\prime } = y+{\mathrm e}^{-y}+{\mathrm e}^{-t} \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.07 |
|
|
\[ {}y^{\prime } = y^{3}+{\mathrm e}^{-5 t} \] |
abelFirstKind |
[_Abel] |
❇ |
N/A |
1.05 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{\left (-t +y\right )^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.563 |
|
|
\[ {}y^{\prime } = \left (4 y+{\mathrm e}^{-t^{2}}\right ) {\mathrm e}^{2 y} \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.27 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{-t}+\ln \left (1+y^{2}\right ) \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.707 |
|
|
\[ {}y^{\prime } = \frac {\left (1+\cos \left (4 t \right )\right ) y}{4}-\frac {\left (1-\cos \left (4 t \right )\right ) y^{2}}{800} \] |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
14.143 |
|
|
\[ {}y^{\prime } = t^{2}+y^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.753 |
|
|
\[ {}y^{\prime } = t \left (1+y\right ) \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.227 |
|
|
\[ {}y^{\prime } = t \sqrt {1-y^{2}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.516 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+3 t y^{\prime }-y = 0 \] |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.391 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+3 t y^{\prime }-y = 0 \] |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
4.299 |
|
|
\[ {}y^{\prime \prime }+t y^{\prime }+y = 0 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.205 |
|
|
\[ {}y^{\prime \prime }+t y^{\prime }+y = 0 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.416 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.351 |
|
|
\[ {}6 y^{\prime \prime }-7 y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.434 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.503 |
|
|
\[ {}3 y^{\prime \prime }+6 y^{\prime }+3 y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.544 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-4 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.84 |
|
|
\[ {}2 y^{\prime \prime }+y^{\prime }-10 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.158 |
|
|
\[ {}5 y^{\prime \prime }+5 y^{\prime }-y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.638 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.933 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.744 |
|
|
\[ {}t^{2} y^{\prime \prime }+\alpha t y^{\prime }+\beta y = 0 \] |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.751 |
|
|
\[ {}t^{2} y^{\prime \prime }+5 t y^{\prime }-5 y = 0 \] |
kovacic, second_order_euler_ode, 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 |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.38 |
|
|
\[ {}t^{2} y^{\prime \prime }-t y^{\prime }-2 y = 0 \] |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler]] |
✓ |
✓ |
6.679 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+4 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.17 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.845 |
|
|
\[ {}2 y^{\prime \prime }+3 y^{\prime }+4 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.901 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+3 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.82 |
|
|
\[ {}4 y^{\prime \prime }-y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.888 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+2 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.162 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.177 |
|
|
\[ {}2 y^{\prime \prime }-y^{\prime }+3 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.005 |
|
|
\[ {}3 y^{\prime \prime }-2 y^{\prime }+4 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.797 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+y = 0 \] |
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)]‘]] |
✓ |
✓ |
2.204 |
|
|
\[ {}t^{2} y^{\prime \prime }+2 t y^{\prime }+2 y = 0 \] |
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]] |
✓ |
✓ |
2.766 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.486 |
|
|
\[ {}4 y^{\prime \prime }-12 y^{\prime }+9 y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.532 |
|
|
\[ {}9 y^{\prime \prime }+6 y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.116 |
|
|
\[ {}4 y^{\prime \prime }-4 y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.977 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.093 |
|
|
\[ {}9 y^{\prime \prime }-12 y^{\prime }+4 y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.296 |
|
|
\[ {}y^{\prime \prime }-\frac {2 \left (t +1\right ) y^{\prime }}{t^{2}+2 t -1}+\frac {2 y}{t^{2}+2 t -1} = 0 \] |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.399 |
|
|
\[ {}y^{\prime \prime }-4 t y^{\prime }+\left (4 t^{2}-2\right ) y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.315 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[_Gegenbauer] |
✓ |
✓ |
2.775 |
|
|
\[ {}\left (t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.386 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+6 y = 0 \] |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
1.505 |
|
|
\[ {}\left (1+2 t \right ) y^{\prime \prime }-4 \left (t +1\right ) y^{\prime }+4 y = 0 \] |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.343 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.497 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = 0 \] |
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]] |
✓ |
✓ |
3.682 |
|
|
\[ {}t^{2} y^{\prime \prime }-t y^{\prime }+y = 0 \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.765 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.048 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = {\mathrm e}^{2 t} t \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.849 |
|
|
\[ {}2 y^{\prime \prime }-3 y^{\prime }+y = \left (t^{2}+1\right ) {\mathrm e}^{t} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.963 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = t \,{\mathrm e}^{3 t}+1 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.8 |
|
|
\[ {}3 y^{\prime \prime }+4 y^{\prime }+y = \sin \left (t \right ) {\mathrm e}^{-t} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.539 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = t^{\frac {5}{2}} {\mathrm e}^{-2 t} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.86 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \sqrt {t +1} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.749 |
|
|
\[ {}y^{\prime \prime }-y = f \left (t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.052 |
|
|
\[ {}y^{\prime \prime }+\frac {y t^{2}}{4} = f \cos \left (t \right ) \] |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.624 |
|
|
\[ {}y^{\prime \prime }-\frac {2 t y^{\prime }}{t^{2}+1}+\frac {2 y}{t^{2}+1} = t^{2}+1 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.929 |
|
|
\[ {}m y^{\prime \prime }+c y^{\prime }+k y = F_{0} \cos \left (\omega t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.87 |
|
|
\[ {}y^{\prime \prime }+t y^{\prime }+y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.953 |
|
|
\[ {}y^{\prime \prime }-t y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.784 |
|
|
\[ {}\left (t^{2}+2\right ) y^{\prime \prime }-t y^{\prime }-3 y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.174 |
|
|
\[ {}y^{\prime \prime }-t^{3} y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.09 |
|
|
\[ {}t \left (2-t \right ) y^{\prime \prime }-6 \left (-1+t \right ) y^{\prime }-4 y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
4.537 |
|
|
\[ {}y^{\prime \prime }+y t^{2} = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.493 |
|
|
\[ {}y^{\prime \prime }-t^{3} y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.54 |
|
|
\[ {}y^{\prime \prime }+\left (t^{2}+2 t +1\right ) y^{\prime }-\left (4+4 t \right ) y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.217 |
|
|
\[ {}y^{\prime \prime }-2 t y^{\prime }+\lambda y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.348 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+\alpha \left (\alpha +1\right ) y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
1.877 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-t y^{\prime }+\alpha ^{2} y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.632 |
|
|
\[ {}y^{\prime \prime }+t^{3} y^{\prime }+3 y t^{2} = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.387 |
|
|
\[ {}y^{\prime \prime }+t^{3} y^{\prime }+3 y t^{2} = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.655 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }+y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.183 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+t y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.14 |
|
|
\[ {}y^{\prime \prime }+t y^{\prime }+y \,{\mathrm e}^{t} = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.386 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y \,{\mathrm e}^{t} = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.13 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+{\mathrm e}^{-t} y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.431 |
|
|
\[ {}t^{2} y^{\prime \prime }-5 t y^{\prime }+9 y = 0 \] |
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]] |
✓ |
✓ |
2.365 |
|
|
\[ {}t^{2} y^{\prime \prime }+5 t y^{\prime }-5 y = 0 \] |
kovacic, second_order_euler_ode, 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 |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.343 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+3 t y^{\prime }-y = 0 \] |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.304 |
|
|
\[ {}\left (-1+t \right )^{2} y^{\prime \prime }-2 \left (-1+t \right ) y^{\prime }+2 y = 0 \] |
kovacic, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.893 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = 0 \] |
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]] |
✓ |
✓ |
3.58 |
|
|
\[ {}t^{2} y^{\prime \prime }-t y^{\prime }+y = 0 \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.729 |
|
|
\[ {}\left (t -2\right )^{2} y^{\prime \prime }+5 \left (t -2\right ) y^{\prime }+4 y = 0 \] |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.547 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+y = 0 \] |
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)]‘]] |
✓ |
✓ |
2.097 |
|
|
\[ {}t^{2} y^{\prime \prime }-t y^{\prime }+2 y = 0 \] |
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]] |
✓ |
✓ |
4.398 |
|
|
\[ {}t^{2} y^{\prime \prime }-3 t y^{\prime }+4 y = 0 \] |
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)]‘]] |
✓ |
✓ |
3.281 |
|
|
\[ {}t \left (t -2\right )^{2} y^{\prime \prime }+t y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.911 |
|
|
\[ {}t \left (t -2\right )^{2} y^{\prime \prime }+t y^{\prime }+y = 0 \] |
second order series method. Irregular singular point |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.993 |
|
|
\[ {}\sin \left (t \right ) y^{\prime \prime }+\cos \left (t \right ) y^{\prime }+\frac {y}{t} = 0 \] |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
22.459 |
|
|
\[ {}\left ({\mathrm e}^{t}-1\right ) y^{\prime \prime }+{\mathrm e}^{t} y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.695 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }+\frac {y^{\prime }}{\sin \left (t +1\right )}+y = 0 \] |
second order series method. Irregular singular point |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
28.076 |
|
|
\[ {}t^{3} y^{\prime \prime }+\sin \left (t^{3}\right ) y^{\prime }+t y = 0 \] |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
10.746 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+3 t y^{\prime }-\left (t +1\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.634 |
|
|
\[ {}2 t y^{\prime \prime }+\left (1-2 t \right ) y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[_Laguerre] |
✓ |
✓ |
2.509 |
|
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