Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime } = {\mathrm e}^{3 x}+\sin \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.403 |
|
|
\[ {}y^{\prime \prime } = 2+x \] |
1 |
1 |
1 |
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 |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime \prime } = x^{2} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _quadrature]] |
✓ |
✓ |
0.185 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.06 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = \sin \left (x \right ) \cos \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.023 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.848 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.159 |
|
|
\[ {}y^{\prime \prime }+k^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.375 |
|
|
\[ {}y^{\prime }+5 y = 2 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.319 |
|
|
\[ {}y^{\prime \prime } = 1+3 x \] |
1 |
1 |
1 |
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 |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.64 |
|
|
\[ {}y^{\prime } = k y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.533 |
|
|
\[ {}y^{\prime }-2 y = 1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.277 |
|
|
\[ {}y^{\prime }+y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.675 |
|
|
\[ {}y^{\prime }-2 y = x^{2}+x \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.69 |
|
|
\[ {}3 y^{\prime }+y = 2 \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.788 |
|
|
\[ {}y^{\prime }+3 y = {\mathrm e}^{i x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.707 |
|
|
\[ {}y^{\prime }+i y = x \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.693 |
|
|
\[ {}L y^{\prime }+R y = E \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.819 |
|
|
\[ {}L y^{\prime }+R y = E \sin \left (\omega x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.498 |
|
|
\[ {}L y^{\prime }+R y = E \,{\mathrm e}^{i \omega x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.25 |
|
|
\[ {}y^{\prime }+a y = b \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.949 |
|
|
\[ {}y^{\prime }+2 x y = x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.325 |
|
|
\[ {}x y^{\prime }+y = 3 x^{3}-1 \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.842 |
|
|
\[ {}y^{\prime }+{\mathrm e}^{x} y = 3 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.077 |
|
|
\[ {}y^{\prime }-\tan \left (x \right ) y = {\mathrm e}^{\sin \left (x \right )} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.961 |
|
|
\[ {}y^{\prime }+2 x y = x \,{\mathrm e}^{-x^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.701 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = {\mathrm e}^{-\sin \left (x \right )} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.123 |
|
|
\[ {}x^{2} y^{\prime }+2 x y = 1 \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.868 |
|
|
\[ {}y^{\prime }+2 y = b \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.91 |
|
|
\[ {}y^{\prime } = y+1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.38 |
|
|
\[ {}y^{\prime } = 1+y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.312 |
|
|
\[ {}y^{\prime } = 1+y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.155 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.197 |
|
|
\[ {}3 y^{\prime \prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.757 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.253 |
|
|
\[ {}y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }+2 i y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.391 |
|
|
\[ {}y^{\prime \prime }+\left (-1+3 i\right ) y^{\prime }-3 i y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.33 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.95 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.652 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.816 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.593 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.473 |
|
|
\[ {}y^{\prime \prime }+\left (1+4 i\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.129 |
|
|
\[ {}y^{\prime \prime }+\left (-1+3 i\right ) y^{\prime }-3 i y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.533 |
|
|
\[ {}y^{\prime \prime }+10 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.344 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.658 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.788 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime \prime }+2 i y^{\prime }+y = x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 3 \,{\mathrm e}^{-x}+2 x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.666 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+6 y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.503 |
|
|
\[ {}y^{\prime \prime }+y = 2 \sin \left (x \right ) \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.153 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.475 |
|
|
\[ {}4 y^{\prime \prime }-y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.388 |
|
|
\[ {}6 y^{\prime \prime }+5 y^{\prime }-6 y = x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.44 |
|
|
\[ {}y^{\prime \prime }+\omega ^{2} y = A \cos \left (\omega x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.286 |
|
|
\[ {}y^{\prime \prime \prime }-8 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.482 |
|
|
\[ {}y^{\prime \prime \prime \prime }+16 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.957 |
|
|
\[ {}y^{\prime \prime \prime }-5 y^{\prime \prime }+6 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.22 |
|
|
\[ {}y^{\prime \prime \prime }-i y^{\prime \prime }+4 y^{\prime }-4 i y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.201 |
|
|
\[ {}y^{\prime \prime \prime \prime }+5 y^{\prime \prime }+4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.463 |
|
|
\[ {}y^{\prime \prime \prime \prime }-16 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.323 |
|
|
\[ {}y^{\prime \prime \prime }-3 y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.267 |
|
|
\[ {}y^{\prime \prime \prime }-3 i y^{\prime \prime }-3 y^{\prime }+i y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.172 |
|
|
\[ {}y^{\prime \prime \prime }-4 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.397 |
|
|
\[ {}y^{\left (5\right )}-y^{\prime \prime \prime \prime }-y^{\prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.951 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.71 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.634 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.299 |
|
|
\[ {}y^{\left (5\right )}+2 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.342 |
|
|
\[ {}y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.28 |
|
|
\[ {}y^{\prime \prime \prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.982 |
|
|
\[ {}y^{\prime \prime \prime }-i y^{\prime \prime }+y^{\prime }-i y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.171 |
|
|
\[ {}y^{\prime \prime }-2 i y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.365 |
|
|
\[ {}y^{\prime \prime \prime \prime }-k^{4} y = 0 \] |
1 |
1 |
1 |
unknown |
[[_high_order, _missing_x]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime \prime }-y = x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.712 |
|
|
\[ {}y^{\prime \prime \prime }-8 y = {\mathrm e}^{i x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.415 |
|
|
\[ {}y^{\prime \prime \prime \prime }+16 y = \cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.905 |
|
|
\[ {}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.226 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y = \cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.013 |
|
|
\[ {}y^{\prime \prime }-2 i y^{\prime }-y = {\mathrm e}^{i x}-2 \,{\mathrm e}^{-i x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.722 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.48 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.589 |
|
|
\[ {}y^{\prime \prime }-4 y = 3 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.568 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = x^{2}+\cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.651 |
|
|
\[ {}y^{\prime \prime }+9 y = x^{2} {\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.714 |
|
|
\[ {}y^{\prime \prime }+y = x \,{\mathrm e}^{x} \cos \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.089 |
|
|
\[ {}y^{\prime \prime }+i y^{\prime }+2 y = 2 \cosh \left (2 x \right )+{\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.957 |
|
|
\[ {}y^{\prime \prime \prime } = x^{2}+{\mathrm e}^{-x} \sin \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _quadrature]] |
✓ |
✓ |
0.697 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y = x^{2} {\mathrm e}^{-x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.233 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}} = 0 \] |
1 |
1 |
1 |
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, _homogeneous]] |
✓ |
✓ |
3.038 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}} = 0 \] |
1 |
1 |
1 |
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, _homogeneous]] |
✓ |
✓ |
2.88 |
|
|
\[ {}\left (3 x -1\right )^{2} y^{\prime \prime }+\left (9 x -3\right ) y^{\prime }-9 y = 0 \] |
1 |
1 |
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, _homogeneous]] |
✓ |
✓ |
2.356 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.408 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.406 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Laguerre] |
✓ |
✓ |
0.429 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Gegenbauer] |
✓ |
✓ |
0.493 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.453 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 x y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.467 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.325 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.373 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.83 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Hermite] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime \prime }+3 x^{2} y^{\prime }-x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.015 |
|
|
\[ {}y^{\prime \prime }-x^{2} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.46 |
|
|
\[ {}y^{\prime \prime }+x^{3} y^{\prime }+x^{2} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.029 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.444 |
|
|
\[ {}y^{\prime \prime }+\left (-1+x \right )^{2} y^{\prime }-\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.578 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.714 |
|
|
\[ {}y^{\prime \prime }+{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.763 |
|
|
\[ {}y^{\prime \prime \prime }-x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.27 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\alpha \left (\alpha +1\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
1.148 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\alpha ^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.593 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+2 \alpha y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.876 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.141 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
1.513 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
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)]‘]] |
✓ |
✓ |
1.326 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = x^{2} \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
2.587 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.454 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+4 y = 1 \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
4.092 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
1.629 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-2-i\right ) x y^{\prime }+3 i y = 0 \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
2.288 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 \pi y = x \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
2.379 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.196 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x^{6} y^{\prime }+2 x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.418 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 y^{\prime }+3 x^{2} y = 0 \] |
1 |
0 |
0 |
second order series method. Irregular singular point |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.068 |
|
|
\[ {}x y^{\prime \prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
4.078 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Gegenbauer] |
✓ |
✓ |
1.394 |
|
|
\[ {}\left (x^{2}+x -2\right )^{2} y^{\prime \prime }+3 \left (2+x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.83 |
|
|
\[ {}x^{2} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+y \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.051 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.151 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (4 x^{4}-5 x \right ) y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.125 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-3 x^{2}+x \right ) y^{\prime }+{\mathrm e}^{x} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.393 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+5 x y^{\prime }+3 x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.337 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+x^{2} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Lienard] |
✓ |
✓ |
0.958 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \,{\mathrm e}^{x} y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.501 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+\left (x^{2}+5 x \right ) y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.977 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \,{\mathrm e}^{x} y^{\prime }+3 y \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
56.942 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+3 \left (x^{2}+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.635 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.151 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.217 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (-x^{3}+3\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.227 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.293 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Bessel] |
✓ |
✓ |
4.095 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x^{2} y^{\prime }+\left (-2+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
4.155 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.834 |
|
|
\[ {}y^{\prime } = x^{2} y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.765 |
|
|
\[ {}y y^{\prime } = x \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.651 |
|
|
\[ {}y^{\prime } = \frac {x^{2}+x}{y-y^{2}} \] |
1 |
1 |
3 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
172.495 |
|
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{x -y}}{1+{\mathrm e}^{x}} \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.982 |
|
|
\[ {}y^{\prime } = x^{2} y^{2}-4 x^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.866 |
|
|
\[ {}y^{\prime } = y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.291 |
|
|
\[ {}y^{\prime } = 2 \sqrt {y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.634 |
|
|
\[ {}y^{\prime } = 2 \sqrt {y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.323 |
|
|
\[ {}y^{\prime } = \frac {x +y}{x -y} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.71 |
|
|
\[ {}y^{\prime } = \frac {y^{2}}{x y+x^{2}} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.534 |
|
|
\[ {}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x^{2}} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.552 |
|
|
\[ {}y^{\prime } = \frac {y+x \,{\mathrm e}^{-\frac {2 y}{x}}}{x} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.418 |
|
|
\[ {}y^{\prime } = \frac {x -y+2}{x +y-1} \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.474 |
|
|
\[ {}y^{\prime } = \frac {2 x +3 y+1}{x -2 y-1} \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.199 |
|
|
\[ {}y^{\prime } = \frac {1+x +y}{2 x +2 y-1} \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.244 |
|
|
\[ {}y^{\prime } = \frac {\left (x +y-1\right )^{2}}{2 \left (2+x \right )^{2}} \] |
1 |
1 |
1 |
riccati, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
✓ |
2.06 |
|
|
\[ {}2 x y+\left (x^{2}+3 y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
0.279 |
|
|
\[ {}x^{2}+x y+\left (x +y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[_quadrature] |
✓ |
✓ |
0.166 |
|
|
\[ {}{\mathrm e}^{x}+{\mathrm e}^{y} \left (y+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_separable] |
✓ |
✓ |
0.303 |
|
|
\[ {}\cos \left (x \right ) \cos \left (y\right )^{2}-\sin \left (x \right ) \sin \left (2 y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[_separable] |
✓ |
✓ |
1.139 |
|
|
\[ {}x^{2} y^{3}-x^{3} y^{2} y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[_separable] |
✓ |
✓ |
0.329 |
|
|
\[ {}x +y+\left (x -y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.236 |
|
|
\[ {}2 y \,{\mathrm e}^{2 x}+2 x \cos \left (y\right )+\left ({\mathrm e}^{2 x}-x^{2} \sin \left (y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.428 |
|
|
\[ {}3 \ln \left (x \right ) x^{2}+x^{2}+y+x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_linear] |
✓ |
✓ |
0.294 |
|
|
\[ {}2 y^{3}+2+3 x y^{2} y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_separable] |
✓ |
✓ |
1.077 |
|
|
\[ {}\cos \left (x \right ) \cos \left (y\right )-2 \sin \left (x \right ) \sin \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[_separable] |
✓ |
✓ |
0.922 |
|
|
\[ {}5 x^{3} y^{2}+2 y+\left (3 x^{4} y+2 x \right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.348 |
|
|
\[ {}{\mathrm e}^{y}+x \,{\mathrm e}^{y}+x \,{\mathrm e}^{y} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_quadrature] |
✓ |
✓ |
0.261 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = 1 \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
1.278 |
|
|
\[ {}y^{\prime \prime }+{\mathrm e}^{x} y^{\prime } = {\mathrm e}^{x} \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.932 |
|
|
\[ {}y y^{\prime \prime }+4 {y^{\prime }}^{2} = 0 \] |
1 |
5 |
6 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
4.504 |
|
|
\[ {}y^{\prime \prime }+k^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.828 |
|
|
\[ {}y^{\prime \prime } = y y^{\prime } \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
0.914 |
|
|
\[ {}x y^{\prime \prime }-2 y^{\prime } = x^{3} \] |
1 |
1 |
1 |
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]] |
✓ |
✓ |
2.16 |
|
|
\[ {}y^{\prime \prime } = 1+{y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.777 |
|
|
\[ {}y^{\prime \prime } = -\frac {1}{2 {y^{\prime }}^{2}} \] |
1 |
1 |
1 |
second_order_integrable_as_is, second_order_ode_missing_x, second_order_ode_missing_y, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
8.194 |
|
|
\[ {}y^{\prime \prime }+\sin \left (y\right ) = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
51.869 |
|
|
\[ {}y^{\prime \prime }+\sin \left (y\right ) = 0 \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
133.487 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{1} \\ y_{2}^{\prime }=y_{1}+y_{2} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.253 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{2} \\ y_{2}^{\prime }=6 y_{1}+y_{2} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.39 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{1}+y_{2} \\ y_{2}^{\prime }=y_{1}+y_{2}+{\mathrm e}^{3 x} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.546 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=3 y_{1}+x y_{3} \\ y_{2}^{\prime }=y_{2}+x^{3} y_{3} \\ y_{3}^{\prime }=2 x y_{2}-y_{2}+{\mathrm e}^{x} y_{3} \end {array}\right ] \] |
1 |
0 |
3 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.033 |
|
|
|
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