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) |
|
\[ {}y^{\prime } = y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.124 |
|
|
\[ {}x y^{\prime } = y \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.3 |
|
|
\[ {}x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime } = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.746 |
|
|
\[ {}\sin \left (x \right ) y^{\prime } = y \ln \left (y\right ) \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
2.048 |
|
|
\[ {}1+y^{2}+x y y^{\prime } = 0 \] |
1 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.519 |
|
|
\[ {}x y y^{\prime }-x y = y \] |
1 |
1 |
1 |
separable |
[_quadrature] |
✓ |
✓ |
0.375 |
|
|
\[ {}y^{\prime } = \frac {2 x y^{2}+x}{x^{2} y-y} \] |
1 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
1.046 |
|
|
\[ {}y y^{\prime }+x y^{2}-8 x = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.517 |
|
|
\[ {}y^{\prime }+2 x y^{2} = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.279 |
|
|
\[ {}\left (y+1\right ) y^{\prime } = y \] |
1 |
1 |
1 |
separable |
[_quadrature] |
✓ |
✓ |
0.387 |
|
|
\[ {}y^{\prime }-x y = x \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.321 |
|
|
\[ {}2 y^{\prime } = 3 \left (y-2\right )^{\frac {1}{3}} \] |
1 |
1 |
1 |
separable |
[_quadrature] |
✓ |
✓ |
0.264 |
|
|
\[ {}\left (x +x y\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
2.856 |
|
|
\[ {}y^{\prime }+y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
linear |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.201 |
|
|
\[ {}x^{2} y^{\prime }+3 x y = 1 \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.173 |
|
|
\[ {}y^{\prime }+2 x y-x \,{\mathrm e}^{-x^{2}} = 0 \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.192 |
|
|
\[ {}2 x y^{\prime }+y = 2 x^{\frac {5}{2}} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.275 |
|
|
\[ {}\cos \left (x \right ) y^{\prime }+y = \cos \left (x \right )^{2} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.316 |
|
|
\[ {}y^{\prime }+\frac {y}{\sqrt {x^{2}+1}} = \frac {1}{x +\sqrt {x^{2}+1}} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.214 |
|
|
\[ {}\left (1+{\mathrm e}^{x}\right ) y^{\prime }+2 \,{\mathrm e}^{x} y = \left (1+{\mathrm e}^{x}\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.2 |
|
|
\[ {}x \ln \left (x \right ) y^{\prime }+y = \ln \left (x \right ) \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.184 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime } = x y+2 x \sqrt {-x^{2}+1} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.214 |
|
|
\[ {}y^{\prime }+y \tanh \left (x \right ) = 2 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.206 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = \sin \left (2 x \right ) \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.217 |
|
|
\[ {}x^{\prime } = \cos \left (y \right )-x \tan \left (y \right ) \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.175 |
|
|
\[ {}x^{\prime }+x-{\mathrm e}^{y} = 0 \] |
1 |
1 |
1 |
linear |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.164 |
|
|
\[ {}x^{\prime } = \frac {3 y^{\frac {2}{3}}-x}{3 y} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.156 |
|
|
\[ {}y^{\prime }+y = x y^{\frac {2}{3}} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.57 |
|
|
\[ {}y^{\prime }+\frac {y}{x} = 2 x^{\frac {3}{2}} \sqrt {y} \] |
1 |
1 |
1 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
3.696 |
|
|
\[ {}3 x y^{2} y^{\prime }+3 y^{3} = 1 \] |
1 |
1 |
3 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.345 |
|
|
\[ {}2 x \,{\mathrm e}^{3 y}+{\mathrm e}^{x}+\left (3 x^{2} {\mathrm e}^{3 y}-y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
1.407 |
|
|
\[ {}\left (x -y\right ) y^{\prime }+1+x +y = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.817 |
|
|
\[ {}\cos \left (x \right ) \cos \left (y\right )+\sin \left (x \right )^{2}-\left (\sin \left (x \right ) \sin \left (y\right )+\cos \left (y\right )^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
unknown |
✓ |
✓ |
34.924 |
|
|
\[ {}x^{2} y^{\prime }+y^{2}-x y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.135 |
|
|
\[ {}y y^{\prime } = -x +\sqrt {x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.135 |
|
|
\[ {}x y+\left (-x^{2}+y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.023 |
|
|
\[ {}y^{2}-x y+\left (x y+x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.007 |
|
|
\[ {}y^{\prime } = \cos \left (x +y\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.887 |
|
|
\[ {}y^{\prime } = \frac {y}{x}-\tan \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.213 |
|
|
\[ {}\left (-1+x \right ) y^{\prime }+y-\frac {1}{x^{2}}+\frac {2}{x^{3}} = 0 \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.739 |
|
|
\[ {}y^{\prime } = x y^{2}-\frac {2 y}{x}-\frac {1}{x^{3}} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
1.128 |
|
|
\[ {}y^{\prime } = \frac {2 y^{2}}{x}+\frac {y}{x}-2 x \] |
1 |
1 |
1 |
riccati, exactByInspection, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.372 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{-x} y^{2}+y-{\mathrm e}^{x} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
0.921 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.303 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.319 |
|
|
\[ {}y^{\prime \prime }+9 y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
0.721 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.316 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.353 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.668 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.285 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
0.713 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.332 |
|
|
\[ {}2 y^{\prime \prime }+y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.287 |
|
|
\[ {}y^{\prime \prime }+\left (1+2 i\right ) y^{\prime }+\left (-1+i\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.311 |
|
|
\[ {}y^{\prime \prime }+\left (1+2 i\right ) y^{\prime }+\left (-1+i\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.29 |
|
|
\[ {}y^{\prime \prime \prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.424 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }-6 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.254 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }-9 y^{\prime }-5 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
1.989 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.387 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime } = 10 \] |
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]] |
✓ |
✓ |
1.01 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 16 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.431 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 24 \,{\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.41 |
|
|
\[ {}y^{\prime \prime }+y = 2 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.409 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 12 \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 3 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.589 |
|
|
\[ {}y^{\prime \prime }-16 y = 40 \,{\mathrm e}^{4 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.582 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 2 \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.595 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 6 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.595 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+10 y = 100 \cos \left (4 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.845 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+12 y = 80 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.925 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.621 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+25 y = 120 \sin \left (5 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.773 |
|
|
\[ {}5 y^{\prime \prime }+12 y^{\prime }+20 y = 120 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.889 |
|
|
\[ {}y^{\prime \prime }+9 y = 30 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.637 |
|
|
\[ {}y^{\prime \prime }+16 y = 16 \cos \left (4 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.608 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+17 y = 60 \,{\mathrm e}^{-4 x} \sin \left (5 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.769 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+5 y = 40 \,{\mathrm e}^{-\frac {3 x}{2}} \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.737 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+8 y = 30 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {5 x}{2}\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.858 |
|
|
\[ {}5 y^{\prime \prime }+6 y^{\prime }+2 y = x^{2}+6 x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.678 |
|
|
\[ {}2 y^{\prime \prime }+y^{\prime } = 2 x \] |
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_y]] |
✓ |
✓ |
1.31 |
|
|
\[ {}y^{\prime \prime }+y = 2 x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.434 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 12 x \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.594 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 16 x^{2} {\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.565 |
|
|
\[ {}y^{\prime \prime }+y = 8 x \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.649 |
|
|
\[ {}y^{\prime \prime }+y = x^{3}-1+2 \cos \left (x \right )+\left (2-4 x \right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.52 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 2 \,{\mathrm e}^{x}+6 x -5 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.557 |
|
|
\[ {}y^{\prime \prime }-y = \sinh \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.751 |
|
|
\[ {}y^{\prime \prime }+y = 2 \sin \left (x \right )+4 x \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.742 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 4 \,{\mathrm e}^{x}+\left (1-x \right ) \left ({\mathrm e}^{2 x}-1\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.767 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = 9 x \,{\mathrm e}^{-x}-6 x^{2}+4 \,{\mathrm e}^{2 x} \] |
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_y]] |
✓ |
✓ |
2.269 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
1 |
0 |
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]] |
✓ |
✓ |
0.884 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
1 |
0 |
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]] |
✓ |
✓ |
0.857 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
1.142 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
0.98 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.459 |
|
|
\[ {}2 y y^{\prime \prime } = {y^{\prime }}^{2} \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.604 |
|
|
\[ {}x y^{\prime \prime } = y^{\prime }+{y^{\prime }}^{3} \] |
1 |
1 |
2 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.659 |
|
|
\[ {}{y^{\prime \prime }}^{2} = k^{2} \left (1+{y^{\prime }}^{2}\right ) \] |
2 |
4 |
3 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.845 |
|
|
\[ {}k = \frac {y^{\prime \prime }}{\left (1+y^{\prime }\right )^{\frac {3}{2}}} \] |
2 |
2 |
1 |
second_order_integrable_as_is, second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear]] |
✓ |
✓ |
1.687 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }-3 y = 0 \] |
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]] |
✓ |
✓ |
1.545 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
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)]‘]] |
✓ |
✓ |
0.941 |
|
|
\[ {}x^{2} y^{\prime \prime }+7 x y^{\prime }+9 y = 0 \] |
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]] |
✓ |
✓ |
1.026 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+6 y = 0 \] |
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]] |
✓ |
✓ |
1.272 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-16 y = 8 x^{4} \] |
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]] |
✓ |
✓ |
1.56 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = x -\frac {1}{x} \] |
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, _nonhomogeneous]] |
✓ |
✓ |
2.462 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = 2 x^{3} \] |
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]] |
✓ |
✓ |
1.783 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 6 \ln \left (x \right ) x^{2} \] |
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, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.773 |
|
|
\[ {}x^{2} y^{\prime \prime }+y = 3 x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.57 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = 2 x \] |
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]] |
✓ |
✓ |
1.932 |
|
|
\[ {}x^{2} \left (2-x \right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.415 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.265 |
|
|
\[ {}x y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.429 |
|
|
\[ {}3 x y^{\prime \prime }-2 \left (3 x -1\right ) y^{\prime }+\left (3 x -2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.51 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.253 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.448 |
|
|
\[ {}x^{2} y^{\prime }-x y = \frac {1}{x} \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.643 |
|
|
\[ {}x \ln \left (y\right ) y^{\prime }-y \ln \left (x \right ) = 0 \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.005 |
|
|
\[ {}y^{\prime \prime \prime }+2 y^{\prime \prime }+2 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.309 |
|
|
\[ {}r^{\prime \prime }-6 r^{\prime }+9 r = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.317 |
|
|
\[ {}2 x -y \sin \left (2 x \right ) = \left (\sin \left (x \right )^{2}-2 y\right ) y^{\prime } \] |
1 |
1 |
2 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.685 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 10 \,{\mathrm e}^{x}+6 \,{\mathrm e}^{-x} \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.824 |
|
|
\[ {}3 x^{3} y^{2} y^{\prime }-y^{3} x^{2} = 1 \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.158 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = x \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.896 |
|
|
\[ {}y^{\prime }-2 y-y^{2} {\mathrm e}^{3 x} = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_1st_order, _with_linear_symmetries], _Bernoulli] |
✓ |
✓ |
0.631 |
|
|
\[ {}u \left (-v +1\right )+v^{2} \left (1-u\right ) u^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.009 |
|
|
\[ {}y+2 x -x y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.604 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 4 x \] |
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]] |
✓ |
✓ |
1.035 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 26 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.477 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 2 \,{\mathrm e}^{-2 x} \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.508 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 6 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.397 |
|
|
\[ {}\left (y+2 x \right ) y^{\prime }-x +2 y = 0 \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.436 |
|
|
\[ {}\left (x \cos \left (y\right )-{\mathrm e}^{-\sin \left (y\right )}\right ) y^{\prime }+1 = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.866 |
|
|
\[ {}\sin \left (x \right )^{2} y^{\prime }+\sin \left (x \right )^{2}+\left (x +y\right ) \sin \left (2 x \right ) = 0 \] |
1 |
1 |
1 |
exact, linear, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.256 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 5 x +4 \,{\mathrm e}^{x} \left (1+\sin \left (2 x \right )\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.926 |
|
|
\[ {}y^{\prime }+x y = \frac {x}{y} \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.196 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+13 y^{\prime \prime }-18 y^{\prime }+36 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.404 |
|
|
\[ {}\sin \left (\theta \right ) \cos \left (\theta \right ) r^{\prime }-\sin \left (\theta \right )^{2} = r \cos \left (\theta \right )^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.109 |
|
|
\[ {}x \left (y y^{\prime \prime }+{y^{\prime }}^{2}\right ) = y y^{\prime } \] |
1 |
1 |
3 |
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]] |
✓ |
✓ |
0.622 |
|
|
\[ {}3 x^{2} y+x^{3} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.992 |
|
|
\[ {}-y+x y^{\prime } = x^{2} \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.856 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 6 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.501 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}+4 = 0 \] |
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], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.543 |
|
|
\[ {}x y^{\prime } = x y+y \] |
1 |
1 |
1 |
first order ode series method. Regular singular point |
[_separable] |
✓ |
✓ |
0.33 |
|
|
\[ {}x y^{\prime } = x y+y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.536 |
|
|
\[ {}y^{\prime } = 3 x^{2} y \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.328 |
|
|
\[ {}y^{\prime } = 3 x^{2} y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.51 |
|
|
\[ {}x y^{\prime } = y \] |
1 |
1 |
1 |
first order ode series method. Regular singular point |
[_separable] |
✓ |
✓ |
0.296 |
|
|
\[ {}x y^{\prime } = y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.454 |
|
|
\[ {}y^{\prime \prime } = -4 y \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime } = -4 y \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.68 |
|
|
\[ {}y^{\prime \prime } = y \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.356 |
|
|
\[ {}y^{\prime \prime } = y \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.48 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.475 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.25 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.815 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y = 0 \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.131 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.043 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
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]] |
✓ |
✓ |
1.088 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.326 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
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]] |
✓ |
✓ |
0.998 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.569 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.391 |
|
|
|
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