|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}\left [\begin {array}{c} x^{\prime }=-y \\ y^{\prime }=x-y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.425 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-2 x+3 y \\ y^{\prime }=-6 x+4 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.66 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-11 x-2 y \\ y^{\prime }=13 x-9 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.72 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=7 x-5 y \\ y^{\prime }=10 x-3 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.666 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=5 x-4 y \\ y^{\prime }=x+y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.413 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-6 x+2 y \\ y^{\prime }=-2 x-2 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.428 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-3 x-y \\ y^{\prime }=x-5 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.427 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=13 x \\ y^{\prime }=13 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.308 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=7 x-4 y \\ y^{\prime }=x+3 y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.435 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-x+y \\ y^{\prime }=-x+y \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.297 |
|
|
\[ {}\tan \left (y\right )-\cot \left (x \right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.395 |
|
|
\[ {}12 x +6 y-9+\left (5 x +2 y-3\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
5.199 |
|
|
\[ {}x y^{\prime } = y+\sqrt {x^{2}+y^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
4.964 |
|
|
\[ {}x y^{\prime }+y = x^{3} \] |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.125 |
|
|
\[ {}y-x y^{\prime } = x^{2} y y^{\prime } \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.868 |
|
|
\[ {}x^{\prime }+3 x = {\mathrm e}^{2 t} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.984 |
|
|
\[ {}y \sin \left (x \right )+y^{\prime } \cos \left (x \right ) = 1 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.909 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x -y} \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.956 |
|
|
\[ {}x^{\prime } = x+\sin \left (t \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.2 |
|
|
\[ {}x \left (\ln \left (x \right )-\ln \left (y\right )\right ) y^{\prime }-y = 0 \] |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
5.086 |
|
|
\[ {}x y {y^{\prime }}^{2}-\left (x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
separable |
[_separable] |
✓ |
✓ |
1.464 |
|
|
\[ {}{y^{\prime }}^{2} = 9 y^{4} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.51 |
|
|
\[ {}x^{\prime } = {\mathrm e}^{\frac {x}{t}}+\frac {x}{t} \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.42 |
|
|
\[ {}x^{2}+{y^{\prime }}^{2} = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.323 |
|
|
\[ {}y = x y^{\prime }+\frac {1}{y} \] |
exact, bernoulli, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
6.814 |
|
|
\[ {}x = {y^{\prime }}^{3}-y^{\prime }+2 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.693 |
|
|
\[ {}y^{\prime } = \frac {y}{x +y^{3}} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.094 |
|
|
\[ {}y = {y^{\prime }}^{4}-{y^{\prime }}^{3}-2 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.596 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2} = 4 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.616 |
|
|
\[ {}y^{\prime } = \frac {2 y-x -4}{2 x -y+5} \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
5.189 |
|
|
\[ {}y^{\prime }-\frac {y}{1+x}+y^{2} = 0 \] |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_1st_order, _with_linear_symmetries], _rational, _Bernoulli] |
✓ |
✓ |
1.172 |
|
|
\[ {}y^{\prime } = x +y^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
3.212 |
|
|
\[ {}y^{\prime } = x y^{3}+x^{2} \] |
abelFirstKind |
[_Abel] |
❇ |
N/A |
0.744 |
|
|
\[ {}y^{\prime } = x^{2}-y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
1.406 |
|
|
\[ {}2 x +2 y-1+\left (x +y-2\right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.65 |
|
|
\[ {}{y^{\prime }}^{3}-y^{\prime } {\mathrm e}^{2 x} = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.772 |
|
|
\[ {}y = 5 x y^{\prime }-{y^{\prime }}^{2} \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.71 |
|
|
\[ {}y^{\prime } = x -y^{2} \] |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.542 |
|
|
\[ {}y^{\prime } = \left (x -5 y\right )^{\frac {1}{3}}+2 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
4.777 |
|
|
\[ {}\left (x -y\right ) y-x^{2} y^{\prime } = 0 \] |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.256 |
|
|
\[ {}x^{\prime }+5 x = 10 t +2 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
2.14 |
|
|
\[ {}x^{\prime } = \frac {x}{t}+\frac {x^{2}}{t^{3}} \] |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.663 |
|
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{2} \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.594 |
|
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{2} \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.797 |
|
|
\[ {}y^{\prime } = \frac {3 x -4 y-2}{3 x -4 y-3} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.773 |
|
|
\[ {}x^{\prime }-x \cot \left (t \right ) = 4 \sin \left (t \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.496 |
|
|
\[ {}y = x^{2}+2 x y^{\prime }+\frac {{y^{\prime }}^{2}}{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.135 |
|
|
\[ {}y^{\prime }-\frac {3 y}{x}+x^{3} y^{2} = 0 \] |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.12 |
|
|
\[ {}y \left ({y^{\prime }}^{2}+1\right ) = a \] |
quadrature |
[_quadrature] |
✓ |
✓ |
2.602 |
|
|
\[ {}x^{2}-y+\left (x^{2} y^{2}+x \right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.532 |
|
|
\[ {}3 y^{2}-x +2 y \left (y^{2}-3 x \right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
7.424 |
|
|
\[ {}\left (x -y\right ) y-x^{2} y^{\prime } = 0 \] |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.2 |
|
|
\[ {}y^{\prime } = \frac {x +y-3}{-x +y+1} \] |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.433 |
|
|
\[ {}x y^{\prime }-y^{2} \ln \left (x \right )+y = 0 \] |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.475 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+2 x y-\cos \left (x \right ) = 0 \] |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.432 |
|
|
\[ {}\left (4 y+2 x +3\right ) y^{\prime }-2 y-x -1 = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.665 |
|
|
\[ {}\left (y^{2}-x \right ) y^{\prime }-y+x^{2} = 0 \] |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
13.914 |
|
|
\[ {}\left (-x^{2}+y^{2}\right ) y^{\prime }+2 x y = 0 \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
4.112 |
|
|
\[ {}3 x y^{2} y^{\prime }+y^{3}-2 x = 0 \] |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
2.188 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x +a \right ) y^{\prime }-y = 0 \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.456 |
|
|
\[ {}{y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.609 |
|
|
\[ {}{y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right )-y^{2} = 0 \] |
separable |
[_separable] |
✓ |
✓ |
7.544 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+10 y = 100 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.149 |
|
|
\[ {}x^{\prime \prime }+x = \sin \left (t \right )-\cos \left (2 t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.955 |
|
|
\[ {}y^{\prime }+y^{\prime \prime \prime }-3 y^{\prime \prime } = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.354 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\sin \left (x \right )^{3}} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.08 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 2 \] |
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 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.027 |
|
|
\[ {}y^{\prime \prime }+y = \cosh \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.199 |
|
|
\[ {}y^{\prime \prime }+\frac {2 {y^{\prime }}^{2}}{1-y} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.143 |
|
|
\[ {}x^{\prime \prime }-4 x^{\prime }+4 x = {\mathrm e}^{t}+{\mathrm e}^{2 t}+1 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.817 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.81 |
|
|
\[ {}x^{3} x^{\prime \prime }+1 = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
3.418 |
|
|
\[ {}y^{\prime \prime \prime \prime }-16 y = x^{2}-{\mathrm e}^{x} \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.098 |
|
|
\[ {}{y^{\prime \prime \prime }}^{2}+{y^{\prime \prime }}^{2} = 1 \] |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✗ |
N/A |
0.0 |
|
|
\[ {}x^{\left (6\right )}-x^{\prime \prime \prime \prime } = 1 \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.289 |
|
|
\[ {}x^{\prime \prime \prime \prime }-2 x^{\prime \prime }+x = t^{2}-3 \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.293 |
|
|
\[ {}y^{\prime \prime }+4 x y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.817 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (9 x^{2}-\frac {1}{25}\right ) y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.268 |
|
|
\[ {}y^{\prime \prime }+{y^{\prime }}^{2} = 1 \] |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.799 |
|
|
\[ {}y^{\prime \prime } = 3 \sqrt {y} \] |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
38.673 |
|
|
\[ {}y^{\prime \prime }+y = 1-\frac {1}{\sin \left (x \right )} \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.99 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{r} = 0 \] |
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]] |
✓ |
✓ |
3.149 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = \frac {y y^{\prime }}{\sqrt {x^{2}+1}} \] |
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]] |
✓ |
✓ |
0.667 |
|
|
\[ {}y y^{\prime } y^{\prime \prime } = {y^{\prime }}^{3}+{y^{\prime \prime }}^{2} \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.189 |
|
|
\[ {}x^{\prime \prime }+9 x = t \sin \left (3 t \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.362 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = \sinh \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.047 |
|
|
\[ {}y^{\prime \prime \prime }-y = {\mathrm e}^{x} \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.431 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = x \,{\mathrm e}^{x} \cos \left (x \right ) \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.105 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 y = 1 \] |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.853 |
|
|
\[ {}m x^{\prime \prime } = f \left (x\right ) \] |
unknown |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✗ |
N/A |
0.407 |
|
|
\[ {}m x^{\prime \prime } = f \left (x^{\prime }\right ) \] |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.896 |
|
|
\[ {}y^{\left (6\right )}-3 y^{\left (5\right )}+3 y^{\prime \prime \prime \prime }-y^{\prime \prime \prime } = x \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.307 |
|
|
\[ {}x^{\prime \prime \prime \prime }+2 x^{\prime \prime }+x = \cos \left (t \right ) \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.273 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }+y = 2 \cos \left (\ln \left (1+x \right )\right ) \] |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
83.548 |
|
|
\[ {}x^{3} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.25 |
|
|
\[ {}x^{\prime \prime \prime \prime }+x = t^{3} \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.079 |
|
|
\[ {}{y^{\prime \prime }}^{3}+y^{\prime \prime }+1 = x \] |
second_order_ode_high_degree, second_order_ode_missing_y |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
3.099 |
|
|
\[ {}x^{\prime \prime }+10 x^{\prime }+25 x = 2^{t}+t \,{\mathrm e}^{-5 t} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.064 |
|
|
\[ {}x y y^{\prime \prime }-x {y^{\prime }}^{2}-y y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
0.255 |
|
|
\[ {}y^{\left (6\right )}-y = {\mathrm e}^{2 x} \] |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
68.467 |
|
|
|
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