|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime \prime } = 0 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.694 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.812 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.736 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.279 |
|
|
\[ {}3 y^{\prime \prime \prime }+5 y^{\prime \prime }+y^{\prime }-y = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.64 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 4 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.433 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 12 \,{\mathrm e}^{x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.566 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{i x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.761 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.567 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \cos \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.55 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 8+6 \,{\mathrm e}^{x}+2 \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.806 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = x^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.218 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-8 y = 9 x \,{\mathrm e}^{x}+10 \,{\mathrm e}^{-x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.627 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = 2 \,{\mathrm e}^{2 x} \sin \left (x \right ) \] |
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]] |
✓ |
✓ |
3.536 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x^{2}+2 x \] |
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]] |
✓ |
✓ |
1.964 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x +\sin \left (2 x \right ) \] |
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]] |
✓ |
✓ |
4.503 |
|
|
\[ {}y^{\prime \prime }+y = 4 x \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.981 |
|
|
\[ {}y^{\prime \prime }+4 y = x \sin \left (2 x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.232 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = x^{2} {\mathrm e}^{-x} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.862 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{-2 x}+x^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.629 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = x \,{\mathrm e}^{-x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.501 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = x +{\mathrm e}^{2 x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.683 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )+{\mathrm e}^{-x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.292 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.139 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (2 x \right ) \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.018 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }-6 y = {\mathrm e}^{3 x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.899 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 5 \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.852 |
|
|
\[ {}y^{\prime \prime }+9 y = 8 \cos \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.328 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = {\mathrm e}^{x} \left (2 x -3\right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.792 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = {\mathrm e}^{-x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.731 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.731 |
|
|
\[ {}y^{\prime \prime }+y = \cot \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.792 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right )^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.793 |
|
|
\[ {}y^{\prime \prime }-y = \sin \left (x \right )^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.808 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 12 \,{\mathrm e}^{x} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.459 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = x^{2} {\mathrm e}^{-x} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.574 |
|
|
\[ {}y^{\prime \prime }+y = 4 x \sin \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = {\mathrm e}^{-x} \ln \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.709 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.672 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right )^{2} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.97 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = \frac {{\mathrm e}^{-x}}{x} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.62 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \csc \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.972 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = {\mathrm e}^{x} \ln \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.708 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \cos \left ({\mathrm e}^{-x}\right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.796 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = x \] |
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]] |
✓ |
✓ |
2.695 |
|
|
\[ {}y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\frac {2 y}{x^{2}} = x \ln \left (x \right ) \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.991 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = x^{3} \] |
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]] |
✓ |
✓ |
1.967 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = x^{2} {\mathrm e}^{-x} \] |
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_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.137 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y = \frac {1}{x} \] |
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, _nonhomogeneous]] |
✓ |
✓ |
2.844 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.987 |
|
|
\[ {}y^{3} y^{\prime \prime } = k \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.782 |
|
|
\[ {}y y^{\prime \prime } = {y^{\prime }}^{2}-1 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
8.385 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 1 \] |
kovacic, second_order_euler_ode, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.115 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = x^{2} \] |
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]] |
✓ |
✓ |
2.628 |
|
|
\[ {}\left (y+1\right ) y^{\prime \prime } = 3 {y^{\prime }}^{2} \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.61 |
|
|
\[ {}r^{\prime \prime } = -\frac {k}{r^{2}} \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
4.896 |
|
|
\[ {}y^{\prime \prime } = \frac {3 k y^{2}}{2} \] |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
14.666 |
|
|
\[ {}y^{\prime \prime } = 2 k y^{3} \] |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
7.368 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}-y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
4.184 |
|
|
\[ {}r^{\prime \prime } = \frac {h^{2}}{r^{3}}-\frac {k}{r^{2}} \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
8.885 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{3}-{y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
2.019 |
|
|
\[ {}y y^{\prime \prime }-3 {y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.004 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.483 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x \left (1+y^{\prime }\right ) = 0 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.169 |
|
|
\[ {}\left (y+1\right ) y^{\prime \prime } = 3 {y^{\prime }}^{2} \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.104 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } {\mathrm e}^{y} \] |
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_xy]] |
✓ |
✓ |
3.116 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.908 |
|
|
\[ {}2 y^{\prime \prime } = {\mathrm e}^{y} \] |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
63.523 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 1 \] |
kovacic, second_order_euler_ode, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.986 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = x^{2} \] |
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]] |
✓ |
✓ |
1.478 |
|
|
\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+y y^{\prime } = 0 \] |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.139 |
|
|
\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0 \] |
second_order_integrable_as_is, second_order_nonlinear_solved_by_mainardi_lioville_method |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.439 |
|
|
\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime } = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.03 |
|
|
\[ {}-a y^{3}-\frac {b}{x^{\frac {3}{2}}}+y^{\prime } = 0 \] |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Abel] |
✓ |
✓ |
0.852 |
|
|
\[ {}a x y^{3}+b y^{2}+y^{\prime } = 0 \] |
abelFirstKind, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
✓ |
4.519 |
|
|
\[ {}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1} = 0 \] |
abelFirstKind |
[_Abel] |
✓ |
✗ |
55.555 |
|
|
\[ {}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0 \] |
abelFirstKind |
[_Abel] |
✗ |
N/A |
4.111 |
|
|
\[ {}x^{2} y^{\prime }+x y^{3}+a y^{2} = 0 \] |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
1.022 |
|
|
\[ {}\left (x a +b \right )^{2} y^{\prime }+\left (x a +b \right ) y^{3}+c y^{2} = 0 \] |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
3.542 |
|
|
\[ {}y^{\prime }+y \tan \left (x \right ) = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.778 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
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_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 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.595 |
|
|
\[ {}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.514 |
|
|
\[ {}\left (-x^{2}+1\right ) {y^{\prime }}^{2}+1 = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.335 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x a}+a y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.368 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right )^{3} = a^{2} {y^{\prime \prime }}^{2} \] |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.842 |
|
|
\[ {}\left (1+x \right ) y+\left (1-y\right ) x y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.003 |
|
|
\[ {}y^{\prime } = a y^{2} x \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.365 |
|
|
\[ {}y^{2}+x y^{2}+\left (x^{2}-x^{2} y\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.09 |
|
|
\[ {}x y \left (x^{2}+1\right ) y^{\prime } = 1+y^{2} \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.289 |
|
|
\[ {}\frac {x}{y+1} = \frac {y y^{\prime }}{1+x} \] |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
177.621 |
|
|
\[ {}y^{\prime }+y^{2} b^{2} = a^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.294 |
|
|
\[ {}y^{\prime } = \frac {1+y^{2}}{x^{2}+1} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.748 |
|
|
\[ {}\sin \left (x \right ) \cos \left (y\right ) = \cos \left (x \right ) \sin \left (y\right ) y^{\prime } \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
34.78 |
|
|
\[ {}a x y^{\prime }+2 y = x y y^{\prime } \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.761 |
|
|
\[ {}x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.809 |
|
|
\[ {}y^{\prime \prime }+x y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.342 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2} \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.283 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+a^{3} x^{2} y = 2 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.596 |
|
|
\[ {}y^{\prime \prime }+a \,x^{2} y = 1+x \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.356 |
|
|
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