|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}x y^{\prime \prime }-\left (x +3\right ) y^{\prime }+3 y = 0 \] |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[_Laguerre] |
✓ |
✓ |
2.465 |
|
|
\[ {}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.355 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.87 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.629 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.741 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.934 |
|
|
\[ {}\left (2 x^{3}-1\right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 x y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.394 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = x^{3} \] |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.515 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 n x \left (1+x \right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.427 |
|
|
\[ {}x^{4} y^{\prime \prime }+2 x^{3} \left (1+x \right ) y^{\prime }+n^{2} y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.026 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.408 |
|
|
\[ {}\left (x y^{\prime \prime \prime }-y^{\prime \prime }\right )^{2} = {y^{\prime \prime \prime }}^{2}+1 \] |
unknown |
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime } = x \] |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.628 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] |
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]] |
✓ |
✓ |
0.692 |
|
|
\[ {}\left (y^{\prime }-x y^{\prime \prime }\right )^{2} = 1+{y^{\prime \prime }}^{2} \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.21 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2}-y^{2} y^{\prime } = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.53 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2}+1 = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
7.733 |
|
|
\[ {}2 y^{\prime \prime } = {\mathrm e}^{y} \] |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
3.118 |
|
|
\[ {}y y^{\prime \prime }+2 y^{\prime }-{y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.545 |
|
|
\[ {}\left (x^{2}-2 x +2\right ) y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.09 |
|
|
\[ {}x y^{\prime \prime \prime }-y^{\prime \prime }-x y^{\prime }+y = -x^{2}+1 \] |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.133 |
|
|
\[ {}\left (2+x \right )^{2} y^{\prime \prime \prime }+\left (2+x \right ) y^{\prime \prime }+y^{\prime } = 1 \] |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
1.587 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = x \] |
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 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.149 |
|
|
\[ {}\left (-1+x \right )^{2} y^{\prime \prime }+4 \left (-1+x \right ) y^{\prime }+2 y = \cos \left (x \right ) \] |
kovacic, 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_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.064 |
|
|
\[ {}\left (x^{3}-x \right ) y^{\prime \prime \prime }+\left (8 x^{2}-3\right ) y^{\prime \prime }+14 x y^{\prime }+4 y = 0 \] |
unknown |
[[_3rd_order, _fully, _exact, _linear]] |
✗ |
N/A |
0.408 |
|
|
\[ {}2 x^{3} y y^{\prime \prime \prime }+6 x^{3} y^{\prime } y^{\prime \prime }+18 x^{2} y y^{\prime \prime }+18 x^{2} {y^{\prime }}^{2}+36 x y y^{\prime }+6 y^{2} = 0 \] |
unknown |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}x^{5} y^{\prime \prime }+\left (2 x^{4}-x \right ) y^{\prime }-\left (2 x^{3}-1\right ) y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.261 |
|
|
\[ {}x^{2} \left (-x^{3}+1\right ) y^{\prime \prime }-x^{3} y^{\prime }-2 y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
73.061 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-5 x y^{\prime \prime }+\left (4 x^{4}+5\right ) y^{\prime }-8 x^{3} y = 0 \] |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.345 |
|
|
\[ {}y^{\prime \prime }+2 \cot \left (x \right ) y^{\prime }+2 \tan \left (x \right ) {y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.154 |
|
|
\[ {}x^{2} y y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.097 |
|
|
\[ {}x^{3} y^{\prime \prime }-\left (-y+x y^{\prime }\right )^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.108 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{2} \ln \left (y\right )-x^{2} y^{2} \] |
unknown |
[[_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.097 |
|
|
\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-2 y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.161 |
|
|
\[ {}y^{\prime \prime } = {y^{\prime }}^{2}+1 \] |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.786 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime } = 2 \] |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.341 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.943 |
|
|
\[ {}\left (x^{3}+1\right ) y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+18 x y^{\prime }+6 y = 0 \] |
unknown |
[[_3rd_order, _fully, _exact, _linear]] |
✗ |
N/A |
0.476 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }+\left (2+4 x \right ) y^{\prime }+2 y = 0 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.688 |
|
|
\[ {}y \left (1-\ln \left (y\right )\right ) y^{\prime \prime }+\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.129 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x} = 0 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.316 |
|
|
\[ {}x \left (2 y+x \right ) y^{\prime \prime }+2 x {y^{\prime }}^{2}+4 \left (x +y\right ) y^{\prime }+2 y+x^{2} = 0 \] |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.501 |
|
|
\[ {}y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.327 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-\frac {y^{\prime }}{x}+x^{2} = 0 \] |
kovacic, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.49 |
|
|
\[ {}4 x^{2} y^{\prime \prime \prime }+8 x y^{\prime \prime }+y^{\prime } = 0 \] |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.784 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }-\cos \left (x \right ) y^{\prime }+2 \sin \left (x \right ) y = 0 \] |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.695 |
|
|
\[ {}\left [\begin {array}{c} 3 x^{\prime }+3 x+2 y={\mathrm e}^{t} \\ 4 x-3 y^{\prime }+3 y=3 t \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.904 |
|
|
\[ {}x^{\prime } = \frac {2 x}{t} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.427 |
|
|
\[ {}x^{\prime } = -\frac {t}{x} \] |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.038 |
|
|
\[ {}x^{\prime } = -x^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.141 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+2 x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.426 |
|
|
\[ {}x^{\prime } = {\mathrm e}^{-x} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.162 |
|
|
\[ {}x^{\prime }+2 x = t^{2}+4 t +7 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.774 |
|
|
\[ {}2 t x^{\prime } = x \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.954 |
|
|
\[ {}t^{2} x^{\prime \prime }-6 x = 0 \] |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.434 |
|
|
\[ {}2 x^{\prime \prime }-5 x^{\prime }-3 x = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.263 |
|
|
\[ {}x^{\prime } = x \left (1-\frac {x}{4}\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.826 |
|
|
\[ {}x^{\prime } = x^{2}+t^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.108 |
|
|
\[ {}x^{\prime } = t \cos \left (t^{2}\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.882 |
|
|
\[ {}x^{\prime } = \frac {t +1}{\sqrt {t}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.402 |
|
|
\[ {}x^{\prime \prime } = -3 \sqrt {t} \] |
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]] |
✓ |
✓ |
1.354 |
|
|
\[ {}x^{\prime } = t \,{\mathrm e}^{-2 t} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.18 |
|
|
\[ {}x^{\prime } = \frac {1}{\ln \left (t \right ) t} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.158 |
|
|
\[ {}\sqrt {t}\, x^{\prime } = \cos \left (\sqrt {t}\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.291 |
|
|
\[ {}x^{\prime } = \frac {{\mathrm e}^{-t}}{\sqrt {t}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.479 |
|
|
\[ {}x^{\prime }+t x^{\prime \prime } = 1 \] |
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]] |
✓ |
✓ |
2.283 |
|
|
\[ {}x^{\prime } = \sqrt {x} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.613 |
|
|
\[ {}x^{\prime } = {\mathrm e}^{-2 x} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.347 |
|
|
\[ {}y^{\prime } = 1+y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.296 |
|
|
\[ {}u^{\prime } = \frac {1}{5-2 u} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.48 |
|
|
\[ {}x^{\prime } = a x+b \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.579 |
|
|
\[ {}Q^{\prime } = \frac {Q}{4+Q^{2}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.431 |
|
|
\[ {}x^{\prime } = {\mathrm e}^{x^{2}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.164 |
|
|
\[ {}y^{\prime } = r \left (a -y\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.803 |
|
|
\[ {}x^{\prime } = \frac {2 x}{t +1} \] |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.369 |
|
|
\[ {}\theta ^{\prime } = t \sqrt {t^{2}+1}\, \sec \left (\theta \right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.33 |
|
|
\[ {}\left (2 u+1\right ) u^{\prime }-t -1 = 0 \] |
exact, separable, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.651 |
|
|
\[ {}R^{\prime } = \left (t +1\right ) \left (1+R^{2}\right ) \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.952 |
|
|
\[ {}y^{\prime }+y+\frac {1}{y} = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.637 |
|
|
\[ {}\left (t +1\right ) x^{\prime }+x^{2} = 0 \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.79 |
|
|
\[ {}y^{\prime } = \frac {1}{2 y+1} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.281 |
|
|
\[ {}x^{\prime } = \left (4 t -x\right )^{2} \] |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
2.657 |
|
|
\[ {}x^{\prime } = 2 t x^{2} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.859 |
|
|
\[ {}x^{\prime } = t^{2} {\mathrm e}^{-x} \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.192 |
|
|
\[ {}x^{\prime } = x \left (4+x\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.192 |
|
|
\[ {}x^{\prime } = {\mathrm e}^{t +x} \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.159 |
|
|
\[ {}T^{\prime } = 2 a t \left (T^{2}-a^{2}\right ) \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.982 |
|
|
\[ {}y^{\prime } = t^{2} \tan \left (y\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.928 |
|
|
\[ {}x^{\prime } = \frac {\left (4+2 t \right ) x}{\ln \left (x\right )} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.101 |
|
|
\[ {}y^{\prime } = \frac {2 t y^{2}}{t^{2}+1} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.32 |
|
|
\[ {}x^{\prime } = \frac {t^{2}}{1-x^{2}} \] |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
34.553 |
|
|
\[ {}x^{\prime } = 6 t \left (x-1\right )^{\frac {2}{3}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.516 |
|
|
\[ {}x^{\prime } = \frac {4 t^{2}+3 x^{2}}{2 x t} \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.906 |
|
|
\[ {}x^{\prime } {\mathrm e}^{2 t}+2 x \,{\mathrm e}^{2 t} = {\mathrm e}^{-t} \] |
exact, linear, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.14 |
|
|
\[ {}\frac {x^{\prime }+t x^{\prime \prime }}{t} = -2 \] |
kovacic, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.619 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+2 t y}{t^{2}} \] |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.068 |
|
|
\[ {}y^{\prime } = -y^{2} {\mathrm e}^{-t^{2}} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.162 |
|
|
\[ {}x^{\prime } = 2 t^{3} x-6 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.665 |
|
|
\[ {}\cos \left (t \right ) x^{\prime }-2 x \sin \left (x\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.375 |
|
|
\[ {}x^{\prime } = t -x^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
0.973 |
|
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