|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (1-5 x \right ) y^{\prime }-4 y = 0 \] |
second order series method. Regular singular point. Repeated root |
[_Jacobi] |
✓ |
✓ |
1.351 |
|
|
\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }-y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.297 |
|
|
\[ {}x y^{\prime \prime }+4 y^{\prime }-x y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.228 |
|
|
\[ {}2 x y^{\prime \prime }+\left (1+x \right ) y^{\prime }-k y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.607 |
|
|
\[ {}x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y = 0 \] |
second order series method. Irregular singular point |
[[_Emden, _Fowler]] |
❇ |
N/A |
0.255 |
|
|
\[ {}x^{2} y^{\prime \prime }+y^{\prime }-2 y = 0 \] |
second order series method. Irregular singular point |
[[_2nd_order, _exact, _linear, _homogeneous]] |
❇ |
N/A |
0.3 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.453 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }+3 x y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.608 |
|
|
\[ {}y^{\prime \prime }-x^{2} y = 0 \] |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.159 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+y = 0 \] |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.387 |
|
|
\[ {}x y^{\prime \prime }+\left (1+x \right )^{2} y = 0 \] |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.626 |
|
|
\[ {}y^{\prime \prime }+\alpha ^{2} y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.866 |
|
|
\[ {}y^{\prime \prime }-\alpha ^{2} y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.759 |
|
|
\[ {}y^{\prime \prime }+\beta y^{\prime }+\gamma y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.368 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0 \] |
unknown |
[_Gegenbauer] |
✗ |
N/A |
0.945 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-\nu ^{2}+x^{2}\right ) y = \sin \left (x \right ) \] |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
12.46 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.242 |
|
|
\[ {}{y^{\prime }}^{2}-y^{\prime }-x y^{\prime }+y = 0 \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.322 |
|
|
\[ {}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.759 |
|
|
\[ {}x y \left (1-{y^{\prime }}^{2}\right ) = \left (x^{2}-y^{2}-a^{2}\right ) y^{\prime } \] |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
71.363 |
|
|
\[ {}y^{\prime \prime \prime }+\frac {3 y^{\prime \prime }}{x} = 0 \] |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.71 |
|
|
\[ {}y^{\prime \prime }-2 k y^{\prime }+k^{2} y = {\mathrm e}^{x} \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.5 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }-a^{2} y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.716 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x} = 0 \] |
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]] |
✓ |
✓ |
1.6 |
|
|
\[ {}y-x y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.044 |
|
|
\[ {}\left (1+u \right ) v+\left (1-v\right ) u v^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.807 |
|
|
\[ {}1+y-\left (1-x \right ) y^{\prime } = 0 \] |
exact, linear, separable, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.373 |
|
|
\[ {}\left (t^{2}+x t^{2}\right ) x^{\prime }+x^{2}+t x^{2} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.522 |
|
|
\[ {}y-a +x^{2} y^{\prime } = 0 \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.454 |
|
|
\[ {}z-\left (-a^{2}+t^{2}\right ) z^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.063 |
|
|
\[ {}y^{\prime } = \frac {1+y^{2}}{x^{2}+1} \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.121 |
|
|
\[ {}1+s^{2}-\sqrt {t}\, s^{\prime } = 0 \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.293 |
|
|
\[ {}r^{\prime }+r \tan \left (t \right ) = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.293 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }-\sqrt {1-y^{2}} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.972 |
|
|
\[ {}\sqrt {-x^{2}+1}\, y^{\prime }-\sqrt {1-y^{2}} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.769 |
|
|
\[ {}3 \,{\mathrm e}^{x} \tan \left (y\right )+\left (-{\mathrm e}^{x}+1\right ) \sec \left (y\right )^{2} y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.153 |
|
|
\[ {}x -x y^{2}+\left (y-x^{2} y\right ) y^{\prime } = 0 \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.376 |
|
|
\[ {}y-x +\left (x +y\right ) y^{\prime } = 0 \] |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.889 |
|
|
\[ {}x +y+x y^{\prime } = 0 \] |
exact, linear, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.254 |
|
|
\[ {}x +y+\left (y-x \right ) y^{\prime } = 0 \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.426 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {x^{2}+y^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.636 |
|
|
\[ {}8 y+10 x +\left (5 y+7 x \right ) y^{\prime } = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.751 |
|
|
\[ {}2 \sqrt {s t}-s+t s^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.618 |
|
|
\[ {}t -s+t s^{\prime } = 0 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.084 |
|
|
\[ {}x y^{2} y^{\prime } = x^{3}+y^{3} \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.569 |
|
|
\[ {}x \cos \left (\frac {y}{x}\right ) \left (y+x y^{\prime }\right ) = y \sin \left (\frac {y}{x}\right ) \left (-y+x y^{\prime }\right ) \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.424 |
|
|
\[ {}3 y-7 x +7-\left (3 x -7 y-3\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.181 |
|
|
\[ {}x +2 y+1-\left (4 y+2 x +3\right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.838 |
|
|
\[ {}x +2 y+1-\left (2 x -3\right ) y^{\prime } = 0 \] |
linear, homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.298 |
|
|
\[ {}\frac {y-x y^{\prime }}{\sqrt {x^{2}+y^{2}}} = m \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
25.151 |
|
|
\[ {}\frac {x +y y^{\prime }}{\sqrt {x^{2}+y^{2}}} = m \] |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
✓ |
3.546 |
|
|
\[ {}y+\frac {x}{y^{\prime }} = \sqrt {x^{2}+y^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.977 |
|
|
\[ {}y y^{\prime } = -x +\sqrt {x^{2}+y^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.635 |
|
|
\[ {}y^{\prime }-\frac {2 y}{1+x} = \left (1+x \right )^{3} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.171 |
|
|
\[ {}y^{\prime }-\frac {a y}{x} = \frac {1+x}{x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.43 |
|
|
\[ {}\left (-x^{2}+x \right ) y^{\prime }+\left (2 x^{2}-1\right ) y-a \,x^{3} = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.608 |
|
|
\[ {}s^{\prime } \cos \left (t \right )+s \sin \left (t \right ) = 1 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.543 |
|
|
\[ {}s^{\prime }+s \cos \left (t \right ) = \frac {\sin \left (2 t \right )}{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.783 |
|
|
\[ {}y^{\prime }-\frac {n y}{x} = {\mathrm e}^{x} x^{n} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.383 |
|
|
\[ {}y^{\prime }+\frac {n y}{x} = a \,x^{-n} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.881 |
|
|
\[ {}y^{\prime }+y = {\mathrm e}^{-x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime }+\frac {\left (1-2 x \right ) y}{x^{2}}-1 = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.192 |
|
|
\[ {}y^{\prime }+x y = x^{3} y^{3} \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.788 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }-x y+a x y^{2} = 0 \] |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.735 |
|
|
\[ {}3 y^{2} y^{\prime }-a y^{3}-x -1 = 0 \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.909 |
|
|
\[ {}y^{\prime } \left (x^{2} y^{3}+x y\right ) = 1 \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✗ |
N/A |
1.272 |
|
|
\[ {}x y^{\prime } = \left (y \ln \left (x \right )-2\right ) y \] |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.277 |
|
|
\[ {}y-y^{\prime } \cos \left (x \right ) = y^{2} \cos \left (x \right ) \left (1-\sin \left (x \right )\right ) \] |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
4.242 |
|
|
\[ {}x^{2}+y+\left (x -2 y\right ) y^{\prime } = 0 \] |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.257 |
|
|
\[ {}y-3 x^{2}-\left (4 y-x \right ) y^{\prime } = 0 \] |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.303 |
|
|
\[ {}\left (y^{3}-x \right ) y^{\prime } = y \] |
exact, differentialType, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
✓ |
2.472 |
|
|
\[ {}\frac {y^{2}}{\left (x -y\right )^{2}}-\frac {1}{x}+\left (\frac {1}{y}-\frac {x^{2}}{\left (x -y\right )^{2}}\right ) y^{\prime } = 0 \] |
exact |
[_exact, _rational] |
✓ |
✓ |
2.048 |
|
|
\[ {}6 x y^{2}+4 x^{3}+3 \left (2 x^{2} y+y^{2}\right ) y^{\prime } = 0 \] |
exact |
[_exact, _rational] |
✓ |
✓ |
1.201 |
|
|
\[ {}\frac {x}{\left (x +y\right )^{2}}+\frac {\left (y+2 x \right ) y^{\prime }}{\left (x +y\right )^{2}} = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.196 |
|
|
\[ {}\frac {1}{x^{2}}+\frac {3 y^{2}}{x^{4}} = \frac {2 y y^{\prime }}{x^{3}} \] |
exact, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
1.155 |
|
|
\[ {}\frac {x^{2} y^{\prime }}{\left (x -y\right )^{2}}-\frac {y^{2}}{\left (x -y\right )^{2}} = 0 \] |
exact, riccati, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.786 |
|
|
\[ {}x +y y^{\prime } = \frac {y}{x^{2}+y^{2}}-\frac {x y^{\prime }}{x^{2}+y^{2}} \] |
exact, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
✓ |
1.104 |
|
|
\[ {}y = 2 x y^{\prime }+{y^{\prime }}^{2} \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.43 |
|
|
\[ {}y = x {y^{\prime }}^{2}+{y^{\prime }}^{2} \] |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.406 |
|
|
\[ {}y = x \left (1+y^{\prime }\right )+{y^{\prime }}^{2} \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.448 |
|
|
\[ {}y = y {y^{\prime }}^{2}+2 x y^{\prime } \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.701 |
|
|
\[ {}y = y y^{\prime }+y^{\prime }-{y^{\prime }}^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.45 |
|
|
\[ {}y = x y^{\prime }+\sqrt {1-{y^{\prime }}^{2}} \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
1.396 |
|
|
\[ {}y = x y^{\prime }+y^{\prime } \] |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.766 |
|
|
\[ {}y = x y^{\prime }+\frac {1}{y^{\prime }} \] |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.372 |
|
|
\[ {}y = x y^{\prime }-\frac {1}{{y^{\prime }}^{2}} \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.758 |
|
|
\[ {}y^{\prime } = \frac {2 y}{x}-\sqrt {3} \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.802 |
|
|
\[ {}y^{\prime \prime \prime }-2 y^{\prime \prime }-y^{\prime }+2 y = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.243 |
|
|
\[ {}y^{\prime \prime } = \frac {1}{2 y^{\prime }} \] |
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]] |
✓ |
✓ |
1.533 |
|
|
\[ {}x y^{\prime \prime \prime } = 2 \] |
higher_order_missing_y |
[[_3rd_order, _quadrature]] |
✓ |
✓ |
0.278 |
|
|
\[ {}y^{\prime \prime } = a^{2} y \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.788 |
|
|
\[ {}y^{\prime \prime } = \frac {a}{y^{3}} \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.326 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = x^{2} {\mathrm e}^{x} \] |
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]] |
✓ |
✓ |
1.897 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2}+{y^{\prime }}^{3} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.005 |
|
|
\[ {}y^{\prime \prime }+\tan \left (x \right ) y^{\prime } = \sin \left (2 x \right ) \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.233 |
|
|
\[ {}{y^{\prime \prime }}^{2}+{y^{\prime }}^{2} = a^{2} \] |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.263 |
|
|
\[ {}y^{\prime \prime } = \frac {1}{2 y^{\prime }} \] |
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]] |
✓ |
✓ |
1.049 |
|
|
\[ {}y^{\prime \prime \prime } = {y^{\prime \prime }}^{2} \] |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime } y^{\prime \prime \prime }-3 {y^{\prime \prime }}^{2} = 0 \] |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime } = 9 y \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.745 |
|
|
|
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