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) |
|
\[ {}y^{\prime } = 3-\sin \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.297 |
|
|
\[ {}y^{\prime } = 3-\sin \left (y\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.417 |
|
|
\[ {}y^{\prime }+4 y = {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.718 |
|
|
\[ {}x y^{\prime } = \arcsin \left (x^{2}\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.474 |
|
|
\[ {}y y^{\prime } = 2 x \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.46 |
|
|
\[ {}y^{\prime \prime } = \frac {1+x}{-1+x} \] |
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]] |
✓ |
✓ |
0.842 |
|
|
\[ {}x^{2} y^{\prime \prime } = 1 \] |
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]] |
✓ |
✓ |
0.615 |
|
|
\[ {}y^{2} y^{\prime \prime } = 8 x^{2} \] |
1 |
0 |
1 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.096 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+8 y = {\mathrm e}^{-x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.121 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.748 |
|
|
\[ {}y^{\prime } = 4 x^{3} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.131 |
|
|
\[ {}y^{\prime } = 20 \,{\mathrm e}^{-4 x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.154 |
|
|
\[ {}x y^{\prime }+\sqrt {x} = 2 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.186 |
|
|
\[ {}\sqrt {x +4}\, y^{\prime } = 1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.21 |
|
|
\[ {}y^{\prime } = x \cos \left (x^{2}\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.584 |
|
|
\[ {}y^{\prime } = x \cos \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.343 |
|
|
\[ {}x = \left (x^{2}-9\right ) y^{\prime } \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.208 |
|
|
\[ {}1 = \left (x^{2}-9\right ) y^{\prime } \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.185 |
|
|
\[ {}1 = x^{2}-9 y^{\prime } \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.128 |
|
|
\[ {}y^{\prime \prime } = \sin \left (2 x \right ) \] |
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]] |
✓ |
✓ |
1.073 |
|
|
\[ {}y^{\prime \prime }-3 = x \] |
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]] |
✓ |
✓ |
0.625 |
|
|
\[ {}y^{\prime \prime \prime \prime } = 1 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _quadrature]] |
✓ |
✓ |
0.167 |
|
|
\[ {}y^{\prime } = 40 \,{\mathrm e}^{2 x} x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.306 |
|
|
\[ {}\left (6+x \right )^{\frac {1}{3}} y^{\prime } = 1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.466 |
|
|
\[ {}y^{\prime } = \frac {-1+x}{1+x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.348 |
|
|
\[ {}x y^{\prime }+2 = \sqrt {x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.448 |
|
|
\[ {}\cos \left (x \right ) y^{\prime }-\sin \left (x \right ) = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.556 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = 1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.315 |
|
|
\[ {}x y^{\prime \prime }+2 = \sqrt {x} \] |
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]] |
✓ |
✓ |
1.524 |
|
|
\[ {}y^{\prime } = \sin \left (\frac {x}{2}\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.315 |
|
|
\[ {}y^{\prime } = \sin \left (\frac {x}{2}\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.232 |
|
|
\[ {}y^{\prime } = \sin \left (\frac {x}{2}\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.248 |
|
|
\[ {}y^{\prime } = 3 \sqrt {x +3} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.194 |
|
|
\[ {}y^{\prime } = 3 \sqrt {x +3} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.332 |
|
|
\[ {}y^{\prime } = 3 \sqrt {x +3} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.327 |
|
|
\[ {}y^{\prime } = 3 \sqrt {x +3} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.33 |
|
|
\[ {}y^{\prime } = x \,{\mathrm e}^{-x^{2}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.303 |
|
|
\[ {}y^{\prime } = \frac {x}{\sqrt {x^{2}+5}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.431 |
|
|
\[ {}y^{\prime } = \frac {1}{x^{2}+1} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.294 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{-9 x^{2}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.341 |
|
|
\[ {}x y^{\prime } = \sin \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.474 |
|
|
\[ {}x y^{\prime } = \sin \left (x^{2}\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{\prime } = \left \{\begin {array}{cc} 0 & x <0 \\ 1 & 0\le x \end {array}\right . \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.516 |
|
|
\[ {}y^{\prime } = \left \{\begin {array}{cc} 0 & x <1 \\ 1 & 1\le x \end {array}\right . \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.596 |
|
|
\[ {}y^{\prime } = \left \{\begin {array}{cc} 0 & x <1 \\ 1 & 1\le x <2 \\ 0 & 2\le x \end {array}\right . \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.693 |
|
|
\[ {}y^{\prime }+3 x y = 6 x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.275 |
|
|
\[ {}\sin \left (x +y\right )-y y^{\prime } = 0 \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.979 |
|
|
\[ {}y^{\prime }-y^{3} = 8 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.319 |
|
|
\[ {}x^{2} y^{\prime }+x y^{2} = x \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.619 |
|
|
\[ {}y^{\prime }-y^{2} = x \] |
1 |
1 |
1 |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
0.812 |
|
|
\[ {}y^{3}-25 y+y^{\prime } = 0 \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.358 |
|
|
\[ {}\left (-2+x \right ) y^{\prime } = 3+y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.36 |
|
|
\[ {}\left (y-2\right ) y^{\prime } = x -3 \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.321 |
|
|
\[ {}y^{\prime }+2 y-y^{2} = -2 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.35 |
|
|
\[ {}y^{\prime }+\left (8-x \right ) y-y^{2} = -8 x \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.43 |
|
|
\[ {}y^{\prime } = 2 \sqrt {y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.325 |
|
|
\[ {}y^{\prime } = 3 y^{2}-y^{2} \sin \left (x \right ) \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.945 |
|
|
\[ {}y^{\prime } = 3 x -y \sin \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.444 |
|
|
\[ {}x y^{\prime } = \left (x -y\right )^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.31 |
|
|
\[ {}y^{\prime } = \sqrt {x^{2}+1} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.291 |
|
|
\[ {}y^{\prime }+4 y = 8 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.28 |
|
|
\[ {}y^{\prime }+x y = 4 x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.996 |
|
|
\[ {}y^{\prime }+4 y = x^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime } = x y-3 x -2 y+6 \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.914 |
|
|
\[ {}y^{\prime } = \sin \left (x +y\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.269 |
|
|
\[ {}y y^{\prime } = {\mathrm e}^{x -3 y^{2}} \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.176 |
|
|
\[ {}y^{\prime } = \frac {x}{y} \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.408 |
|
|
\[ {}y^{\prime } = y^{2}+9 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.246 |
|
|
\[ {}x y y^{\prime } = y^{2}+9 \] |
1 |
1 |
2 |
exact, bernoulli, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.068 |
|
|
\[ {}y^{\prime } = \frac {1+y^{2}}{x^{2}+1} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.79 |
|
|
\[ {}\cos \left (y\right ) y^{\prime } = \sin \left (x \right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.252 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{2 x -3 y} \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.785 |
|
|
\[ {}y^{\prime } = \frac {x}{y} \] |
1 |
1 |
1 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.223 |
|
|
\[ {}y^{\prime } = 2 x -1+2 x y-y \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.231 |
|
|
\[ {}y y^{\prime } = x y^{2}+x \] |
1 |
1 |
1 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.479 |
|
|
\[ {}y y^{\prime } = 3 \sqrt {x y^{2}+9 x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.664 |
|
|
\[ {}y^{\prime } = x y-4 x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.863 |
|
|
\[ {}y^{\prime }-4 y = 2 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.276 |
|
|
\[ {}y y^{\prime } = x y^{2}-9 x \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.435 |
|
|
\[ {}y^{\prime } = \sin \left (y\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.402 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x +y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.653 |
|
|
\[ {}y^{\prime } = 200 y-2 y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.574 |
|
|
\[ {}y^{\prime } = x y-4 x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.678 |
|
|
\[ {}y^{\prime } = x y-3 x -2 y+6 \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.895 |
|
|
\[ {}y^{\prime } = 3 y^{2}-y^{2} \sin \left (x \right ) \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.82 |
|
|
\[ {}y^{\prime } = \tan \left (y\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.195 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.754 |
|
|
\[ {}y^{\prime } = \frac {6 x^{2}+4}{3 y^{2}-4 y} \] |
1 |
1 |
3 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
163.516 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = 1+y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.79 |
|
|
\[ {}\left (y^{2}-1\right ) y^{\prime } = 4 x y^{2} \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.783 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{-y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.171 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{-y}+1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.607 |
|
|
\[ {}y^{\prime } = 3 x y^{3} \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.864 |
|
|
\[ {}y^{\prime } = \frac {2+\sqrt {x}}{2+\sqrt {y}} \] |
1 |
1 |
1 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
99.218 |
|
|
\[ {}y^{\prime }-3 x^{2} y^{2} = -3 x^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.275 |
|
|
\[ {}y^{\prime }-3 x^{2} y^{2} = 3 x^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.945 |
|
|
\[ {}y^{\prime } = 200 y-2 y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.2 |
|
|
\[ {}y^{\prime }-2 y = -10 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.477 |
|
|
\[ {}y y^{\prime } = \sin \left (x \right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.444 |
|
|
\[ {}y^{\prime } = 2 x -1+2 x y-y \] |
1 |
0 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.051 |
|
|
\[ {}x y^{\prime } = y^{2}-y \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.388 |
|
|
\[ {}x y^{\prime } = y^{2}-y \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.352 |
|
|
\[ {}y^{\prime } = \frac {y^{2}-1}{x y} \] |
1 |
1 |
1 |
exact, bernoulli, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
7.687 |
|
|
\[ {}\left (y^{2}-1\right ) y^{\prime } = 4 x y \] |
1 |
0 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.379 |
|
|
\[ {}x^{2} y^{\prime }+3 x^{2} y = \sin \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.36 |
|
|
\[ {}y^{2} y^{\prime }+3 x^{2} y = \sin \left (x \right ) \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.335 |
|
|
\[ {}y^{\prime }-x y^{2} = \sqrt {x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.527 |
|
|
\[ {}y^{\prime } = 1+\left (x y+3 y\right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.67 |
|
|
\[ {}y^{\prime } = 1+x y+3 y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.843 |
|
|
\[ {}y^{\prime } = 4 y+8 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.277 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{2 x} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.123 |
|
|
\[ {}y^{\prime } = y \sin \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.004 |
|
|
\[ {}y^{\prime }+4 y = y^{3} \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.278 |
|
|
\[ {}x y^{\prime }+\cos \left (x^{2}\right ) = 827 y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
36.53 |
|
|
\[ {}y^{\prime }+2 y = 6 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.285 |
|
|
\[ {}y^{\prime }+2 y = 20 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.733 |
|
|
\[ {}y^{\prime } = 4 y+16 x \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.691 |
|
|
\[ {}y^{\prime }-2 x y = x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.043 |
|
|
\[ {}x y^{\prime }+3 y-10 x^{2} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.754 |
|
|
\[ {}x^{2} y^{\prime }+2 x y = \sin \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.803 |
|
|
\[ {}x y^{\prime } = \sqrt {x}+3 y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.779 |
|
|
\[ {}\cos \left (x \right ) y^{\prime }+y \sin \left (x \right ) = \cos \left (x \right )^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.528 |
|
|
\[ {}x y^{\prime }+\left (5 x +2\right ) y = \frac {20}{x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.84 |
|
|
\[ {}2 \sqrt {x}\, y^{\prime }+y = 2 x \,{\mathrm e}^{-\sqrt {x}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.005 |
|
|
\[ {}y^{\prime }-3 y = 6 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.473 |
|
|
\[ {}y^{\prime }-3 y = 6 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.214 |
|
|
\[ {}y^{\prime }+5 y = {\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.95 |
|
|
\[ {}3 y+x y^{\prime } = 20 x^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.924 |
|
|
\[ {}x y^{\prime } = y+x^{2} \cos \left (x \right ) \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.349 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = x \left (3+3 x^{2}-y\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.303 |
|
|
\[ {}y^{\prime }+6 x y = \sin \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.661 |
|
|
\[ {}x^{2} y^{\prime }+x y = \sqrt {x}\, \sin \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
5.733 |
|
|
\[ {}-y+x y^{\prime } = x^{2} {\mathrm e}^{-x^{2}} \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.387 |
|
|
\[ {}y^{\prime } = \frac {1}{\left (3 x +3 y+2\right )^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.151 |
|
|
\[ {}y^{\prime } = \frac {\left (3 x -2 y\right )^{2}+1}{3 x -2 y}+\frac {3}{2} \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.398 |
|
|
\[ {}\cos \left (-4 y+8 x -3\right ) y^{\prime } = 2+2 \cos \left (-4 y+8 x -3\right ) \] |
1 |
1 |
1 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _dAlembert] |
✓ |
✓ |
42.072 |
|
|
\[ {}y^{\prime } = 1+\left (y-x \right )^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
1.104 |
|
|
\[ {}x^{2} y^{\prime }-x y = y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.936 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\frac {x}{y} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.075 |
|
|
\[ {}\cos \left (\frac {y}{x}\right ) \left (y^{\prime }-\frac {y}{x}\right ) = 1+\sin \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.826 |
|
|
\[ {}y^{\prime } = \frac {x -y}{x +y} \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.326 |
|
|
\[ {}y^{\prime }+3 y = 3 y^{3} \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.224 |
|
|
\[ {}y^{\prime }-\frac {3 y}{x} = \frac {y^{2}}{x^{2}} \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.058 |
|
|
\[ {}y^{\prime }+3 \cot \left (x \right ) y = 6 \cos \left (x \right ) y^{\frac {2}{3}} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
3.814 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = \frac {1}{y} \] |
1 |
1 |
1 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.112 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\frac {x^{2}}{y^{2}} \] |
1 |
1 |
3 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.426 |
|
|
\[ {}3 y^{\prime } = -2+\sqrt {2 x +3 y+4} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.576 |
|
|
\[ {}3 y^{\prime }+\frac {2 y}{x} = 4 \sqrt {y} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.432 |
|
|
\[ {}y^{\prime } = 4+\frac {1}{\sin \left (4 x -y\right )} \] |
1 |
1 |
1 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
3.558 |
|
|
\[ {}\left (y-x \right ) y^{\prime } = 1 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.053 |
|
|
\[ {}\left (x +y\right ) y^{\prime } = y \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.184 |
|
|
\[ {}\left (2 x y+2 x^{2}\right ) y^{\prime } = x^{2}+2 x y+2 y^{2} \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.078 |
|
|
\[ {}y^{\prime }+\frac {y}{x} = x^{2} y^{3} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.13 |
|
|
\[ {}y^{\prime } = 2 \sqrt {2 x +y-3}-2 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.504 |
|
|
\[ {}y^{\prime } = 2 \sqrt {2 x +y-3} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.576 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {x y+x^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.937 |
|
|
\[ {}y^{\prime }+3 y = \frac {28 \,{\mathrm e}^{2 x}}{y^{3}} \] |
1 |
1 |
4 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_1st_order, _with_linear_symmetries], _Bernoulli] |
✓ |
✓ |
1.609 |
|
|
\[ {}y^{\prime } = \left (x -y+3\right )^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
1.824 |
|
|
\[ {}y^{\prime }+2 x = 2 \sqrt {y+x^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.547 |
|
|
\[ {}\cos \left (y\right ) y^{\prime } = {\mathrm e}^{-x}-\sin \left (y\right ) \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.832 |
|
|
\[ {}y^{\prime } = x \left (1+\frac {2 y}{x^{2}}+\frac {y^{2}}{x^{4}}\right ) \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
1.08 |
|
|
\[ {}y^{\prime } = \frac {1}{y}-\frac {y}{2 x} \] |
1 |
1 |
2 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.885 |
|
|
\[ {}{\mathrm e}^{x y^{2}-x^{2}} \left (y^{2}-2 x \right )+2 \,{\mathrm e}^{x y^{2}-x^{2}} x y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.941 |
|
|
\[ {}2 x y+y^{2}+\left (2 x y+x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.411 |
|
|
\[ {}2 x y^{3}+4 x^{3}+3 x^{2} y^{2} y^{\prime } = 0 \] |
1 |
1 |
3 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
1.638 |
|
|
\[ {}2-2 x +3 y^{2} y^{\prime } = 0 \] |
1 |
1 |
3 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
12.222 |
|
|
\[ {}1+3 x^{2} y^{2}+\left (2 x^{3} y+6 y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[_exact, _rational, _Bernoulli] |
✓ |
✓ |
1.307 |
|
|
\[ {}4 x^{3} y+\left (x^{4}-y^{4}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
2.842 |
|
|
\[ {}1+\ln \left (x y\right )+\frac {x y^{\prime }}{y} = 0 \] |
1 |
1 |
1 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact] |
✓ |
✓ |
2.614 |
|
|
\[ {}1+{\mathrm e}^{y}+x \,{\mathrm e}^{y} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.158 |
|
|
\[ {}{\mathrm e}^{y}+\left (x \,{\mathrm e}^{y}+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_exponential_symmetries], _exact] |
✓ |
✓ |
0.989 |
|
|
\[ {}1+y^{4}+x y^{3} y^{\prime } = 0 \] |
1 |
1 |
4 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.348 |
|
|
\[ {}y+\left (y^{4}-3 x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.097 |
|
|
\[ {}\frac {2 y}{x}+\left (4 x^{2} y-3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
8.607 |
|
|
\[ {}1+\left (1-x \tan \left (y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.204 |
|
|
\[ {}3 y+3 y^{2}+\left (2 x +4 x y\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.772 |
|
|
\[ {}2 x \left (y+1\right )-y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.078 |
|
|
\[ {}2 y^{3}+\left (4 x^{3} y^{3}-3 x y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
6.408 |
|
|
\[ {}4 x y+\left (3 x^{2}+5 y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.204 |
|
|
\[ {}6+12 x^{2} y^{2}+\left (7 x^{3} y+\frac {x}{y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.278 |
|
|
\[ {}x y^{\prime } = 2 y-6 x^{3} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.721 |
|
|
\[ {}x y^{\prime } = 2 y^{2}-6 y \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.752 |
|
|
\[ {}4 y^{2}-x^{2} y^{2}+y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.845 |
|
|
\[ {}y^{\prime } = \sqrt {x +y} \] |
1 |
1 |
1 |
homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.669 |
|
|
\[ {}x^{2} y^{\prime }-\sqrt {x} = 3 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.18 |
|
|
\[ {}x y y^{\prime }-y^{2} = \sqrt {x^{4}+x^{2} y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.862 |
|
|
\[ {}y^{\prime } = x^{2}-2 x y+y^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
1.462 |
|
|
\[ {}4 x y-6+x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.74 |
|
|
\[ {}x y^{2}-6+x^{2} y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
1.003 |
|
|
\[ {}x^{3}+y^{3}+x y^{2} y^{\prime } = 0 \] |
1 |
1 |
3 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
2.127 |
|
|
\[ {}3 y-x^{3}+x y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.747 |
|
|
\[ {}1+2 x y^{2}+\left (2 x^{2} y+2 y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[_exact, _rational, _Bernoulli] |
✓ |
✓ |
1.183 |
|
|
\[ {}3 x y^{3}-y+x y^{\prime } = 0 \] |
1 |
2 |
2 |
bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.03 |
|
|
\[ {}2+2 x^{2}-2 x y+\left (x^{2}+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.832 |
|
|
\[ {}\left (y^{2}-4\right ) y^{\prime } = y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.355 |
|
|
\[ {}\left (x^{2}-4\right ) y^{\prime } = x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.19 |
|
|
\[ {}y^{\prime } = \frac {1}{x y-3 x} \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.931 |
|
|
\[ {}y^{\prime } = \frac {3 y}{1+x}-y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_1st_order, _with_linear_symmetries], _rational, _Bernoulli] |
✓ |
✓ |
0.89 |
|
|
\[ {}\sin \left (y\right )+\left (x +y\right ) \cos \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
10.493 |
|
|
\[ {}\sin \left (y\right )+\left (1+x \right ) \cos \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
10.951 |
|
|
\[ {}\sin \left (x \right )+2 \cos \left (x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.235 |
|
|
\[ {}x y y^{\prime } = 2 x^{2}+2 y^{2} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.588 |
|
|
\[ {}y^{\prime } = \frac {2 y+x}{x +2 y+3} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.162 |
|
|
\[ {}y^{\prime } = \frac {2 y+x}{2 x -y} \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.306 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\tan \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.951 |
|
|
\[ {}y^{\prime } = x y^{2}+3 y^{2}+x +3 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.981 |
|
|
\[ {}1-\left (2 y+x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.086 |
|
|
\[ {}\ln \left (y\right )+\left (\frac {x}{y}+3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.429 |
|
|
\[ {}y^{2}+1-y^{\prime } = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.144 |
|
|
\[ {}y^{\prime }-3 y = 12 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.749 |
|
|
\[ {}x y y^{\prime } = x^{2}+x y+y^{2} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.259 |
|
|
\[ {}\left (2+x \right ) y^{\prime }-x^{3} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.187 |
|
|
\[ {}x y^{3} y^{\prime } = y^{4}-x^{2} \] |
1 |
1 |
4 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.307 |
|
|
\[ {}y^{\prime } = 4 y-\frac {16 \,{\mathrm e}^{4 x}}{y^{2}} \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_1st_order, _with_linear_symmetries], _Bernoulli] |
✓ |
✓ |
1.252 |
|
|
\[ {}2 y-6 x +\left (1+x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.564 |
|
|
\[ {}x y^{2}+\left (x^{2} y+10 y^{4}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
✓ |
2.127 |
|
|
\[ {}y y^{\prime }-x y^{2} = 6 x \,{\mathrm e}^{4 x^{2}} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
2.217 |
|
|
\[ {}\left (y-x +3\right )^{2} \left (y^{\prime }-1\right ) = 1 \] |
1 |
1 |
3 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
1.268 |
|
|
\[ {}x +y \,{\mathrm e}^{x y}+x \,{\mathrm e}^{x y} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.372 |
|
|
\[ {}y^{2}-y^{2} \cos \left (x \right )+y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.0 |
|
|
\[ {}y^{\prime }+2 y = \sin \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.923 |
|
|
\[ {}y^{\prime }+2 x = \sin \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.357 |
|
|
\[ {}y^{\prime } = y^{3}-y^{3} \cos \left (x \right ) \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.077 |
|
|
\[ {}y^{2} {\mathrm e}^{x y^{2}}-2 x +2 x y \,{\mathrm e}^{x y^{2}} y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.64 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{4 x +3 y} \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.825 |
|
|
\[ {}y^{\prime } = \tan \left (6 x +3 y+1\right )-2 \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
498.314 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{4 x +3 y} \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.732 |
|
|
\[ {}y^{\prime } = x \left (6 y+{\mathrm e}^{x^{2}}\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.832 |
|
|
\[ {}x \left (1-2 y\right )+\left (-x^{2}+y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.736 |
|
|
\[ {}x^{2} y^{\prime }+3 x y = 6 \,{\mathrm e}^{-x^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.908 |
|
|
\[ {}x y^{\prime \prime }+4 y^{\prime } = 18 x^{2} \] |
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]] |
✓ |
✓ |
2.438 |
|
|
\[ {}x y^{\prime \prime } = 2 y^{\prime } \] |
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]] |
✓ |
✓ |
2.237 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } \] |
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]] |
✓ |
✓ |
0.737 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 8 \,{\mathrm e}^{2 x} \] |
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]] |
✓ |
✓ |
1.638 |
|
|
\[ {}x y^{\prime \prime } = y^{\prime }-2 x^{2} y^{\prime } \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.319 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.174 |
|
|
\[ {}y^{\prime \prime } = 4 x \sqrt {y^{\prime }} \] |
2 |
1 |
3 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.045 |
|
|
\[ {}y^{\prime } y^{\prime \prime } = 1 \] |
1 |
2 |
2 |
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, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
2.463 |
|
|
\[ {}y y^{\prime \prime } = -{y^{\prime }}^{2} \] |
1 |
2 |
3 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.296 |
|
|
\[ {}x y^{\prime \prime } = {y^{\prime }}^{2}-y^{\prime } \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.948 |
|
|
\[ {}x y^{\prime \prime }-{y^{\prime }}^{2} = 6 x^{5} \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.695 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{\prime } \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.372 |
|
|
\[ {}y^{\prime \prime } = 2 y^{\prime }-6 \] |
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]] |
✓ |
✓ |
1.451 |
|
|
\[ {}\left (y-3\right ) y^{\prime \prime } = 2 {y^{\prime }}^{2} \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.821 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = 9 \,{\mathrm e}^{-3 x} \] |
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]] |
✓ |
✓ |
1.694 |
|
|
\[ {}y^{\prime \prime \prime } = y^{\prime \prime } \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.199 |
|
|
\[ {}x y^{\prime \prime \prime }+2 y^{\prime \prime } = 6 x \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.76 |
|
|
\[ {}y^{\prime \prime \prime } = 2 \sqrt {y^{\prime \prime }} \] |
1 |
1 |
2 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime \prime \prime } = -2 y^{\prime \prime \prime } \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.233 |
|
|
\[ {}y y^{\prime \prime } = {y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.182 |
|
|
\[ {}3 y y^{\prime \prime } = 2 {y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.884 |
|
|
\[ {}\sin \left (y\right ) y^{\prime \prime }+\cos \left (y\right ) {y^{\prime }}^{2} = 0 \] |
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], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.275 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } \] |
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]] |
✓ |
✓ |
0.634 |
|
|
\[ {}{y^{\prime }}^{2}+y y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
2 |
3 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode, second_order_nonlinear_solved_by_mainardi_lioville_method |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.186 |
|
|
\[ {}y^{2} y^{\prime \prime }+y^{\prime \prime }+2 y {y^{\prime }}^{2} = 0 \] |
1 |
3 |
5 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
602.552 |
|
|
\[ {}y^{\prime \prime } = 4 x \sqrt {y^{\prime }} \] |
2 |
1 |
3 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.939 |
|
|
\[ {}y^{\prime } y^{\prime \prime } = 1 \] |
1 |
2 |
2 |
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, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
1.559 |
|
|
\[ {}x y^{\prime \prime } = {y^{\prime }}^{2}-y^{\prime } \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.815 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = 6 x^{5} \] |
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]] |
✓ |
✓ |
2.281 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{\prime } \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.233 |
|
|
\[ {}y y^{\prime \prime } = 2 {y^{\prime }}^{2} \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.989 |
|
|
\[ {}\left (y-3\right ) y^{\prime \prime } = {y^{\prime }}^{2} \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.092 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = 9 \,{\mathrm e}^{-3 x} \] |
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]] |
✓ |
✓ |
1.542 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } \left (y^{\prime }-2\right ) \] |
1 |
1 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y, second_order_nonlinear_solved_by_mainardi_lioville_method |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.553 |
|
|
\[ {}x y^{\prime \prime }+4 y^{\prime } = 18 x^{2} \] |
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]] |
✓ |
✓ |
2.672 |
|
|
\[ {}x y^{\prime \prime } = 2 y^{\prime } \] |
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]] |
✓ |
✓ |
2.525 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } \] |
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]] |
✓ |
✓ |
1.114 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 8 \,{\mathrm e}^{2 x} \] |
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]] |
✓ |
✓ |
2.023 |
|
|
\[ {}y^{\prime \prime \prime } = y^{\prime \prime } \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.386 |
|
|
\[ {}x y^{\prime \prime \prime }+2 y^{\prime \prime } = 6 x \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.868 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime } = 6 \] |
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]] |
✓ |
✓ |
3.515 |
|
|
\[ {}2 x y^{\prime } y^{\prime \prime } = {y^{\prime }}^{2}-1 \] |
1 |
2 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
3.708 |
|
|
\[ {}3 y y^{\prime \prime } = 2 {y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.678 |
|
|
\[ {}y y^{\prime \prime }+2 {y^{\prime }}^{2} = 3 y y^{\prime } \] |
1 |
1 |
1 |
second_order_ode_missing_x, second_order_nonlinear_solved_by_mainardi_lioville_method |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.701 |
|
|
\[ {}y^{\prime \prime } = -y^{\prime } {\mathrm e}^{-y} \] |
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_xy]] |
✓ |
✓ |
1.977 |
|
|
\[ {}y^{\prime \prime } = -2 x {y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.859 |
|
|
\[ {}y^{\prime \prime } = -2 x {y^{\prime }}^{2} \] |
1 |
0 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.65 |
|
|
\[ {}y^{\prime \prime } = -2 x {y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.674 |
|
|
\[ {}y^{\prime \prime } = -2 x {y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.886 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
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]] |
✓ |
✓ |
1.579 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
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]] |
✓ |
✓ |
1.012 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
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]] |
✓ |
✓ |
0.766 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
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]] |
✓ |
✓ |
1.308 |
|
|
\[ {}y^{\prime \prime }+x^{2} y^{\prime }-4 y = x^{3} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
1.017 |
|
|
\[ {}y^{\prime \prime }+x^{2} y^{\prime }-4 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.801 |
|
|
\[ {}y^{\prime \prime }+x^{2} y^{\prime } = 4 y \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.806 |
|
|
\[ {}y^{\prime \prime }+x^{2} y^{\prime }+4 y = y^{3} \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.111 |
|
|
\[ {}3 y+x y^{\prime } = {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.904 |
|
|
\[ {}y^{\prime \prime \prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.535 |
|
|
\[ {}\left (y+1\right ) y^{\prime \prime } = {y^{\prime }}^{3} \] |
1 |
1 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.027 |
|
|
\[ {}y^{\prime \prime } = 2 y^{\prime }-5 y+30 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.893 |
|
|
\[ {}y^{\prime \prime \prime \prime }+6 y^{\prime \prime }+3 y^{\prime }-83 y-25 = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
66.16 |
|
|
\[ {}y y^{\prime \prime \prime }+6 y^{\prime \prime }+3 y^{\prime } = y \] |
1 |
1 |
2 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.22 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.215 |
|
|
\[ {}x^{2} y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.449 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.421 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.337 |
|
|
\[ {}y^{\prime \prime }-\left (4+\frac {2}{x}\right ) y^{\prime }+\left (4+\frac {4}{x}\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.615 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.625 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}-4 x^{2} y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.506 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.251 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x +2\right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.52 |
|
|
\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-2 \cos \left (x \right ) \sin \left (x \right ) y^{\prime }+\left (1+\cos \left (x \right )^{2}\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.813 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.589 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.575 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.577 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 9 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.366 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+8 y = {\mathrm e}^{4 x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.416 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = \sqrt {x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.429 |
|
|
\[ {}x^{2} y^{\prime \prime }-20 y = 27 x^{5} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.399 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x +2\right ) y^{\prime }+2 y = 8 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.468 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1+x \right )^{2} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.527 |
|
|
\[ {}y^{\prime \prime \prime }-9 y^{\prime \prime }+27 y^{\prime }-27 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.199 |
|
|
\[ {}y^{\prime \prime \prime }-9 y^{\prime \prime }+27 y^{\prime }-27 y = {\mathrm e}^{3 x} \sin \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.303 |
|
|
\[ {}y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+24 y^{\prime \prime }-32 y^{\prime }+16 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.204 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }-4 y^{\prime \prime }+10 y^{\prime }-12 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.159 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.091 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.599 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.513 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.624 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.694 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-y = 0 \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.521 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_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]] |
✓ |
✓ |
2.474 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.0 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.409 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
1 |
0 |
0 |
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 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
❇ |
N/A |
1.587 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \] |
1 |
0 |
0 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
❇ |
N/A |
1.619 |
|
|
\[ {}y^{\prime \prime \prime }+4 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.679 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.829 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.792 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.496 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime } = 0 \] |
1 |
0 |
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]] |
✓ |
✓ |
1.049 |
|
|
\[ {}y^{\prime \prime \prime }-9 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.233 |
|
|
\[ {}y^{\prime \prime \prime \prime }-10 y^{\prime \prime }+9 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.286 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.264 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-24 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.269 |
|
|
\[ {}y^{\prime \prime }-25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.19 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
0.973 |
|
|
\[ {}4 y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.026 |
|
|
\[ {}3 y^{\prime \prime }+7 y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.285 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+15 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.525 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+15 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.502 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+15 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.497 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.712 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.554 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.297 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.32 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.312 |
|
|
\[ {}4 y^{\prime \prime }-4 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.331 |
|
|
\[ {}25 y^{\prime \prime }-10 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.338 |
|
|
\[ {}16 y^{\prime \prime }-24 y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.348 |
|
|
\[ {}9 y^{\prime \prime }+12 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.342 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.618 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.664 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.621 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.694 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.648 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.699 |
|
|
\[ {}y^{\prime \prime }+25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.431 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.479 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.311 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+29 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.521 |
|
|
\[ {}9 y^{\prime \prime }+18 y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.526 |
|
|
\[ {}4 y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.109 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.506 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.408 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.925 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.753 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.563 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.59 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+\left (\frac {1}{4}+4 \pi ^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.89 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+\left (\frac {1}{4}+4 \pi ^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.817 |
|
|
\[ {}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.238 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.326 |
|
|
\[ {}y^{\prime \prime \prime \prime }-34 y^{\prime \prime }+225 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.294 |
|
|
\[ {}y^{\prime \prime \prime \prime }-81 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.357 |
|
|
\[ {}y^{\prime \prime \prime \prime }-18 y^{\prime \prime }+81 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.357 |
|
|
\[ {}y^{\left (5\right )}+18 y^{\prime \prime \prime }+81 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.159 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.303 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.239 |
|
|
\[ {}y^{\prime \prime \prime }-8 y^{\prime \prime }+37 y^{\prime }-50 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.369 |
|
|
\[ {}y^{\prime \prime \prime }-9 y^{\prime \prime }+31 y^{\prime }-39 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.366 |
|
|
\[ {}y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }+2 y^{\prime \prime }+4 y^{\prime }-8 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.396 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+10 y^{\prime \prime }+18 y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.426 |
|
|
\[ {}y^{\prime \prime \prime }+4 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.596 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.343 |
|
|
\[ {}y^{\prime \prime \prime \prime }+26 y^{\prime \prime }+25 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
1.199 |
|
|
\[ {}y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }+9 y^{\prime \prime }+9 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.816 |
|
|
\[ {}y^{\prime \prime \prime }-8 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.497 |
|
|
\[ {}y^{\prime \prime \prime }+216 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.546 |
|
|
\[ {}y^{\prime \prime \prime \prime }-3 y^{\prime \prime }-4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.346 |
|
|
\[ {}y^{\prime \prime \prime \prime }+13 y^{\prime \prime }+36 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.547 |
|
|
\[ {}y^{\left (6\right )}-3 y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.157 |
|
|
\[ {}y^{\left (6\right )}-2 y^{\prime \prime \prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.184 |
|
|
\[ {}16 y^{\prime \prime \prime \prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.383 |
|
|
\[ {}4 y^{\prime \prime \prime \prime }+15 y^{\prime \prime }-4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.362 |
|
|
\[ {}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+16 y^{\prime }-16 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.143 |
|
|
|
|||||||||
|
\[ {}y^{\left (6\right )}+16 y^{\prime \prime \prime }+64 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.181 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+8 y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.617 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.531 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.826 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.974 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = 0 \] |
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]] |
✓ |
✓ |
1.55 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = 0 \] |
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]] |
✓ |
✓ |
1.57 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.464 |
|
|
\[ {}x^{2} y^{\prime \prime }-19 x y^{\prime }+100 y = 0 \] |
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]] |
✓ |
✓ |
1.618 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+29 y = 0 \] |
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]] |
✓ |
✓ |
1.948 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+10 y = 0 \] |
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]] |
✓ |
✓ |
1.757 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+29 y = 0 \] |
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]] |
✓ |
✓ |
1.694 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_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)]‘]] |
✓ |
✓ |
1.405 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+y = 0 \] |
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 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.679 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+37 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.568 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.83 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-25 y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.443 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }+5 y = 0 \] |
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]] |
✓ |
✓ |
1.579 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-7 x y^{\prime }+3 y = 0 \] |
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 |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.578 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-10 y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.962 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-y = 0 \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.485 |
|
|
\[ {}x^{2} y^{\prime \prime }-11 x y^{\prime }+36 y = 0 \] |
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]] |
✓ |
✓ |
2.323 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_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]] |
✓ |
✓ |
2.377 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
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]] |
✓ |
✓ |
2.802 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+13 y = 0 \] |
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]] |
✓ |
✓ |
3.096 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }-4 x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.46 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.517 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }-5 x^{2} y^{\prime \prime }+14 x y^{\prime }-18 y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.514 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+7 x y^{\prime }-8 y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.247 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+15 x^{2} y^{\prime \prime }+9 x y^{\prime }+16 y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.296 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }-9 x y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.585 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+2 x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.282 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+7 x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.642 |
|
|
\[ {}y^{\prime \prime }+4 y = 24 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.781 |
|
|
\[ {}y^{\prime \prime }+4 y = 24 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.721 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-8 y = 8 x^{2}-3 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.737 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-8 y = 8 x^{2}-3 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.677 |
|
|
\[ {}y^{\prime \prime }-9 y = 36 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.493 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -6 \,{\mathrm e}^{4 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.704 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = 7 \,{\mathrm e}^{5 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.885 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 169 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.215 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 10 x +12 \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.763 |
|
|
\[ {}y^{\prime \prime \prime \prime }+y^{\prime \prime } = 1 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.707 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = {\mathrm e}^{4 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.427 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = {\mathrm e}^{5 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.499 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -18 \,{\mathrm e}^{4 x}+14 \,{\mathrm e}^{5 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.657 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = 35 \,{\mathrm e}^{5 x}+12 \,{\mathrm e}^{4 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.65 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 1 \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.341 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.326 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 22 x +24 \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.438 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = x^{2} \] |
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]] |
✓ |
✓ |
1.988 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = x \] |
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]] |
✓ |
✓ |
1.911 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = 1 \] |
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]] |
✓ |
✓ |
1.905 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = 4 x^{2}+2 x +3 \] |
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]] |
✓ |
✓ |
2.025 |
|
|
\[ {}y^{\prime \prime }+9 y = 52 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.723 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 27 \,{\mathrm e}^{6 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.551 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-5 y = 30 \,{\mathrm e}^{-4 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.458 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = {\mathrm e}^{\frac {x}{2}} \] |
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]] |
✓ |
✓ |
1.729 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -5 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.777 |
|
|
\[ {}y^{\prime \prime }+9 y = 10 \cos \left (2 x \right )+15 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.36 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 25 \sin \left (6 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.85 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = 26 \cos \left (\frac {x}{3}\right )-12 \sin \left (\frac {x}{3}\right ) \] |
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]] |
✓ |
✓ |
3.344 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-5 y = \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.559 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -4 \cos \left (x \right )+7 \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.979 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -200 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-5 y = x^{3} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.469 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 18 x^{2}+3 x +4 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.549 |
|
|
\[ {}y^{\prime \prime }+9 y = 9 x^{4}-9 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.703 |
|
|
\[ {}y^{\prime \prime }+9 y = x^{3} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.901 |
|
|
\[ {}y^{\prime \prime }+9 y = 25 x \cos \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.119 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{2 x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.738 |
|
|
\[ {}y^{\prime \prime }+9 y = 54 x^{2} {\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.731 |
|
|
\[ {}y^{\prime \prime } = 6 x \,{\mathrm e}^{x} \sin \left (x \right ) \] |
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]] |
✓ |
✓ |
2.779 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \left (-6 x -8\right ) \cos \left (2 x \right )+\left (8 x -11\right ) \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.388 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \left (12 x -4\right ) {\mathrm e}^{-5 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{\prime \prime }+9 y = 39 \,{\mathrm e}^{2 x} x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.919 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -3 \,{\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.458 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = 20 \] |
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]] |
✓ |
✓ |
1.543 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = x^{2} \] |
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]] |
✓ |
✓ |
1.74 |
|
|
\[ {}y^{\prime \prime }+9 y = 3 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.773 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 10 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.523 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = \left (72 x^{2}-1\right ) {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.508 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = 4 x \,{\mathrm e}^{6 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.478 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 6 \,{\mathrm e}^{5 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.529 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 6 \,{\mathrm e}^{-5 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.536 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 24 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.914 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 8 \,{\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.559 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = {\mathrm e}^{2 x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.737 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = {\mathrm e}^{-x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.724 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 100 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.563 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = {\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.594 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 10 x^{2}+4 x +8 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.652 |
|
|
\[ {}y^{\prime \prime }+9 y = {\mathrm e}^{2 x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.944 |
|
|
\[ {}y^{\prime \prime }+y = 6 \cos \left (x \right )-3 \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.051 |
|
|
\[ {}y^{\prime \prime }+y = 6 \cos \left (2 x \right )-3 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.216 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = x^{3} {\mathrm e}^{-x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.536 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = x^{3} {\mathrm e}^{2 x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.343 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} {\mathrm e}^{-7 x}+2 \,{\mathrm e}^{-7 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.539 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.455 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 4 \,{\mathrm e}^{-8 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.444 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 4 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.446 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} {\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.484 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} \cos \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.149 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} {\mathrm e}^{3 x} \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.245 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = {\mathrm e}^{4 x} \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.263 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = {\mathrm e}^{2 x} \sin \left (4 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.74 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = x^{3} \sin \left (4 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.003 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 3 x^{2} {\mathrm e}^{5 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 3 x^{4} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.602 |
|
|
\[ {}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime } = 12 \,{\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.217 |
|
|
\[ {}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime } = 10 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.24 |
|
|
\[ {}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime } = 32 \,{\mathrm e}^{4 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.222 |
|
|
\[ {}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime } = 32 x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.215 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = x^{2} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.5 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = 30 \cos \left (2 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.134 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = 6 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.204 |
|
|
\[ {}y^{\left (5\right )}+18 y^{\prime \prime \prime }+81 y^{\prime } = x^{2} {\mathrm e}^{3 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.286 |
|
|
\[ {}y^{\left (5\right )}+18 y^{\prime \prime \prime }+81 y^{\prime } = x^{2} \sin \left (3 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
1.122 |
|
|
\[ {}y^{\left (5\right )}+18 y^{\prime \prime \prime }+81 y^{\prime } = x^{2} {\mathrm e}^{3 x} \sin \left (3 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.657 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = 30 x \cos \left (2 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.895 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = 3 x \cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.313 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = 3 x \,{\mathrm e}^{x} \cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.799 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = 5 x^{5} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.487 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 27 \,{\mathrm e}^{6 x}+25 \sin \left (6 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.159 |
|
|
\[ {}y^{\prime \prime }+9 y = 25 x \cos \left (2 x \right )+3 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.902 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 5 \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.822 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 20 \sinh \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.024 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+8 y = \frac {5}{x^{3}} \] |
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]] |
✓ |
✓ |
2.225 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+y = \frac {50}{x^{3}} \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.514 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+y = 85 \cos \left (2 \ln \left (x \right )\right ) \] |
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 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
14.775 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = 15 \cos \left (3 \ln \left (x \right )\right )-10 \sin \left (3 \ln \left (x \right )\right ) \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.625 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-7 x y^{\prime }+3 y = 4 x^{3} \] |
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]] |
✓ |
✓ |
3.026 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+y = \frac {10}{x} \] |
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 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.096 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = 6 x^{3} \] |
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]] |
✓ |
✓ |
2.621 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = 64 \ln \left (x \right ) x^{2} \] |
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]] |
✓ |
✓ |
3.189 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 3 \sqrt {x} \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.726 |
|
|
\[ {}y^{\prime \prime }+y = \cot \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.758 |
|
|
\[ {}y^{\prime \prime }+4 y = \csc \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.926 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+10 y = 6 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.468 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \left (24 x^{2}+2\right ) {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.661 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = \frac {{\mathrm e}^{-2 x}}{x^{2}+1} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.649 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = \sqrt {x} \] |
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]] |
✓ |
✓ |
3.602 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 12 x^{3} \] |
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]] |
✓ |
✓ |
1.957 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = x^{2} \] |
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]] |
✓ |
✓ |
2.657 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = \ln \left (x \right ) \] |
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]] |
✓ |
✓ |
3.113 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = \frac {1}{-2+x} \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.904 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }-4 x^{3} y = x^{3} {\mathrm e}^{x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.109 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x +2\right ) y^{\prime }+2 y = 8 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.189 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1+x \right )^{2} \] |
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]] |
✓ |
✓ |
2.115 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-4 y = \frac {10}{x} \] |
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_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.388 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 12 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.742 |
|
|
\[ {}y^{\prime \prime \prime }-4 y^{\prime } = 30 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.567 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 x y^{\prime }-6 y = x^{3} \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.396 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 x y^{\prime }-6 y = {\mathrm e}^{-x^{2}} \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.522 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = \tan \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
6.913 |
|
|
\[ {}y^{\prime \prime \prime \prime }-81 y = \sinh \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
5.175 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }-9 x y^{\prime }+9 y = 12 x \sin \left (x^{2}\right ) \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.773 |
|
|
\[ {}y^{\prime \prime }+36 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.648 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+36 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.351 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 0 \] |
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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.517 |
|
|
\[ {}y^{\prime \prime }-36 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.254 |
|
|
\[ {}y^{\prime \prime }-9 y^{\prime }+14 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.289 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+16 y = 0 \] |
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]] |
✓ |
✓ |
1.699 |
|
|
\[ {}2 x y^{\prime \prime }+y^{\prime } = \sqrt {x} \] |
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]] |
✓ |
✓ |
2.704 |
|
|
\[ {}y^{\prime \prime \prime \prime }-8 y^{\prime \prime }+16 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.375 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.334 |
|
|
\[ {}y^{\prime \prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.802 |
|
|
\[ {}x^{2} y^{\prime \prime }+7 x y^{\prime }+9 y = 0 \] |
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]] |
✓ |
✓ |
1.795 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {5 y}{2} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.704 |
|
|
\[ {}y^{\left (5\right )}-6 y^{\prime \prime \prime \prime }+13 y^{\prime \prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.478 |
|
|
\[ {}x^{2} y^{\prime \prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.476 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.576 |
|
|
\[ {}y^{\prime \prime } = {y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.581 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+9 y = 0 \] |
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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.559 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.513 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-30 y = 0 \] |
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]] |
✓ |
✓ |
1.296 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-30 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.28 |
|
|
\[ {}16 y^{\prime \prime }-8 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.347 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }+y = 0 \] |
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]] |
✓ |
✓ |
1.596 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime } = 8 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
1.616 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-3 x y^{\prime }+2 y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.631 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x y^{\prime }+y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.632 |
|
|
\[ {}y^{\prime \prime \prime \prime }-16 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.348 |
|
|
\[ {}2 y^{\prime \prime }-7 y^{\prime }+3 = 0 \] |
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]] |
✓ |
✓ |
1.789 |
|
|
\[ {}y^{\prime \prime }+20 y^{\prime }+100 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.345 |
|
|
\[ {}x y^{\prime \prime } = 3 y^{\prime } \] |
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]] |
✓ |
✓ |
2.601 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
1.03 |
|
|
\[ {}y^{\prime \prime }-9 y^{\prime }+14 y = 98 x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.49 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+36 y = 25 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.83 |
|
|
\[ {}y^{\prime \prime }-9 y^{\prime }+14 y = 576 x^{2} {\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+36 y = 81 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.594 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 3 \sqrt {x} \] |
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]] |
✓ |
✓ |
1.888 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+36 y = 3 x \,{\mathrm e}^{6 x}-2 \,{\mathrm e}^{6 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.596 |
|
|
\[ {}y^{\prime \prime }+36 y = 6 \sec \left (6 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.027 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-6 y = 18 \ln \left (x \right ) \] |
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 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.647 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 10 \,{\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.547 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }-2 y = 10 x^{2} \] |
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 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.115 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 2 \cos \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.775 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = -3 x {y^{\prime }}^{3} \] |
1 |
2 |
2 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.964 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+2 y = 6 \] |
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]] |
✓ |
✓ |
6.336 |
|
|
|
|||||||||
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = \frac {1}{x^{2}+1} \] |
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]] |
✓ |
✓ |
3.786 |
|
|
\[ {}4 y^{\prime \prime }-12 y^{\prime }+9 y = x \,{\mathrm e}^{\frac {3 x}{2}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.587 |
|
|
\[ {}3 y^{\prime \prime }+8 y^{\prime }-3 y = 123 x \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.035 |
|
|
\[ {}y^{\prime \prime \prime }+8 y = {\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.844 |
|
|
\[ {}y^{\left (6\right )}-64 y = {\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
68.905 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \frac {1}{\left (1+x \right )^{2}} \] |
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 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.776 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \frac {1}{x} \] |
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 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.302 |
|
|
\[ {}y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
first_order_laplace |
[_quadrature] |
✓ |
✓ |
0.461 |
|
|
\[ {}y^{\prime }-2 y = t^{3} \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.671 |
|
|
\[ {}y^{\prime }+3 y = \operatorname {Heaviside}\left (t -4\right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.297 |
|
|
\[ {}y^{\prime \prime }-4 y = t^{3} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.528 |
|
|
\[ {}y^{\prime \prime }+4 y = 20 \,{\mathrm e}^{4 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.656 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (2 t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.783 |
|
|
\[ {}y^{\prime \prime }+4 y = 3 \operatorname {Heaviside}\left (t -2\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.299 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = {\mathrm e}^{4 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.568 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = t^{2} {\mathrm e}^{4 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.594 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 7 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.512 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = {\mathrm e}^{2 t} \sin \left (3 t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.938 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+13 y = 4 t +2 \,{\mathrm e}^{2 t} \sin \left (3 t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.158 |
|
|
\[ {}y^{\prime \prime \prime }-27 y = {\mathrm e}^{-3 t} \] |
1 |
1 |
1 |
higher_order_laplace |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.95 |
|
|
\[ {}t y^{\prime \prime }+y^{\prime }+t y = 0 \] |
1 |
1 |
1 |
unknown |
[_Lienard] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.419 |
|
|
\[ {}y^{\prime \prime }+9 y = 27 t^{3} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.719 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+7 y = 165 \,{\mathrm e}^{4 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.57 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+17 y = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.472 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{3 t} t^{2} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.522 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+13 y = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.504 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+17 y = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.457 |
|
|
\[ {}y^{\prime \prime } = {\mathrm e}^{t} \sin \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.699 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+40 y = 122 \,{\mathrm e}^{-3 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.789 |
|
|
\[ {}y^{\prime \prime }-9 y = 24 \,{\mathrm e}^{-3 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.559 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = {\mathrm e}^{2 t} \sin \left (3 t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.546 |
|
|
\[ {}y^{\prime \prime }+4 y = 1 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.49 |
|
|
\[ {}y^{\prime \prime }+4 y = t \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.644 |
|
|
\[ {}y^{\prime \prime }+4 y = {\mathrm e}^{3 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.594 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (2 t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.449 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.926 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 1 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.493 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = t \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.581 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{3 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.477 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{-3 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.564 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.558 |
|
|
\[ {}y^{\prime } = \operatorname {Heaviside}\left (t -3\right ) \] |
1 |
1 |
1 |
first_order_laplace |
[_quadrature] |
✓ |
✓ |
0.345 |
|
|
\[ {}y^{\prime } = \operatorname {Heaviside}\left (t -3\right ) \] |
1 |
1 |
1 |
first_order_laplace |
[_quadrature] |
✓ |
✓ |
0.336 |
|
|
\[ {}y^{\prime \prime } = \operatorname {Heaviside}\left (t -2\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.45 |
|
|
\[ {}y^{\prime \prime } = \operatorname {Heaviside}\left (t -2\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.489 |
|
|
\[ {}y^{\prime \prime }+9 y = \operatorname {Heaviside}\left (t -10\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.137 |
|
|
\[
{}y^{\prime } = \left \{\begin {array}{cc} 0 & t <1 \\ 1 & 1 |
1 |
1 |
1 |
first_order_laplace |
[_quadrature] |
✓ |
✓ |
0.509 |
|
|
\[
{}y^{\prime \prime } = \left \{\begin {array}{cc} 0 & t <1 \\ 1 & 1 |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.521 |
|
|
\[
{}y^{\prime \prime }+9 y = \left \{\begin {array}{cc} 0 & t <1 \\ 1 & 1 |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.169 |
|
|
\[ {}y^{\prime } = 3 \delta \left (t -2\right ) \] |
1 |
1 |
1 |
first_order_laplace |
[_quadrature] |
✓ |
✓ |
0.337 |
|
|
\[ {}y^{\prime } = \delta \left (t -2\right )-\delta \left (t -4\right ) \] |
1 |
1 |
1 |
first_order_laplace |
[_quadrature] |
✓ |
✓ |
0.356 |
|
|
\[ {}y^{\prime \prime } = \delta \left (t -3\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.433 |
|
|
\[ {}y^{\prime \prime } = \delta \left (-1+t \right )-\delta \left (t -4\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.514 |
|
|
\[ {}y^{\prime }+2 y = 4 \delta \left (-1+t \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.068 |
|
|
\[ {}y^{\prime \prime }+y = \delta \left (t \right )+\delta \left (t -\pi \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.799 |
|
|
\[ {}y^{\prime \prime }+y = -2 \delta \left (t -\frac {\pi }{2}\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.83 |
|
|
\[ {}y^{\prime }+3 y = \delta \left (t -2\right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.099 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = \delta \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.398 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = \delta \left (-1+t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.04 |
|
|
\[ {}y^{\prime \prime }+16 y = \delta \left (t -2\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.061 |
|
|
\[ {}y^{\prime \prime }-16 y = \delta \left (t -10\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.053 |
|
|
\[ {}y^{\prime \prime }+y = \delta \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
0.733 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-12 y = \delta \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
0.936 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-12 y = \delta \left (t -3\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.296 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = \delta \left (t -4\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.979 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+45 y = \delta \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
0.967 |
|
|
\[ {}y^{\prime \prime \prime }+9 y^{\prime } = \delta \left (-1+t \right ) \] |
1 |
1 |
1 |
higher_order_laplace |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
1.954 |
|
|
\[ {}y^{\prime \prime \prime \prime }-16 y = \delta \left (t \right ) \] |
1 |
1 |
1 |
higher_order_laplace |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
3.119 |
|
|
\[ {}y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_quadrature] |
✓ |
✓ |
0.687 |
|
|
\[ {}y^{\prime }-2 x y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.564 |
|
|
\[ {}y^{\prime }+\frac {2 y}{2 x -1} = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.599 |
|
|
\[ {}\left (x -3\right ) y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.537 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }-2 x y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime }+\frac {y}{-1+x} = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.584 |
|
|
\[ {}y^{\prime }+\frac {y}{-1+x} = 0 \] |
1 |
1 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.532 |
|
|
\[ {}\left (1-x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.664 |
|
|
\[ {}\left (-x^{3}+2\right ) y^{\prime }-3 x^{2} y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.634 |
|
|
\[ {}\left (-x^{3}+2\right ) y^{\prime }+3 x^{2} y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.55 |
|
|
\[ {}\left (1+x \right ) y^{\prime }-x y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.624 |
|
|
\[ {}\left (1+x \right ) y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.684 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.634 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.757 |
|
|
\[ {}\left (x^{2}+4\right ) y^{\prime \prime }+2 x y^{\prime } = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.624 |
|
|
\[ {}y^{\prime \prime }-3 x^{2} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.682 |
|
|
\[ {}\left (-x^{2}+4\right ) y^{\prime \prime }-5 x y^{\prime }-3 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.056 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.883 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+6 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.042 |
|
|
\[ {}\left (x^{2}-6 x \right ) y^{\prime \prime }+4 \left (x -3\right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.237 |
|
|
\[ {}y^{\prime \prime }+\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.145 |
|
|
\[ {}\left (x^{2}-2 x +2\right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.981 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.984 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-2 x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.004 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\lambda y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.175 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\lambda y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
1.141 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.535 |
|
|
\[ {}y^{\prime \prime }-x^{2} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.559 |
|
|
\[ {}y^{\prime \prime }+{\mathrm e}^{2 x} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.727 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.102 |
|
|
\[ {}y^{\prime \prime }+x y = \sin \left (x \right ) \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.025 |
|
|
\[ {}y^{\prime \prime }-\sin \left (x \right ) y^{\prime }-x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.483 |
|
|
\[ {}y^{\prime \prime }-y^{2} = 0 \] |
1 |
1 |
1 |
second order series method. Taylor series method |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
8.054 |
|
|
\[ {}y^{\prime }+\cos \left (y\right ) = 0 \] |
1 |
1 |
1 |
first order ode series method. Taylor series method |
[_quadrature] |
✓ |
✓ |
0.87 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.733 |
|
|
\[ {}y^{\prime }-\tan \left (x \right ) y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.992 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-{\mathrm e}^{x} y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
87.506 |
|
|
\[ {}\sinh \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-{\mathrm e}^{x} y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
16.42 |
|
|
\[ {}\sinh \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \sin \left (x \right ) = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
172.314 |
|
|
\[ {}{\mathrm e}^{3 x} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+\frac {2 y}{x^{2}+4} = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
28.862 |
|
|
\[ {}y^{\prime \prime }+\frac {\left (1+{\mathrm e}^{x}\right ) y}{-{\mathrm e}^{x}+1} = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.645 |
|
|
\[ {}\left (x^{2}-4\right ) y^{\prime \prime }+\left (x^{2}+x -6\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.214 |
|
|
\[ {}x y^{\prime \prime }+\left (-{\mathrm e}^{x}+1\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.217 |
|
|
\[ {}\sin \left (\pi \,x^{2}\right ) y^{\prime \prime }+x^{2} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.764 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.621 |
|
|
\[ {}y^{\prime }+{\mathrm e}^{2 x} y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.686 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.691 |
|
|
\[ {}y^{\prime }+y \ln \left (x \right ) = 0 \] |
1 |
1 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
1.01 |
|
|
\[ {}y^{\prime \prime }-{\mathrm e}^{x} y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.3 |
|
|
\[ {}y^{\prime \prime }+3 x y^{\prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.239 |
|
|
\[ {}x y^{\prime \prime }-3 x y^{\prime }+y \sin \left (x \right ) = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.786 |
|
|
\[ {}y^{\prime \prime }+y \ln \left (x \right ) = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Titchmarsh] |
✓ |
✓ |
1.066 |
|
|
\[ {}\sqrt {x}\, y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.191 |
|
|
\[ {}y^{\prime \prime }+\left (6 x^{2}+2 x +1\right ) y^{\prime }+\left (2+12 x \right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.326 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.969 |
|
|
\[ {}y^{\prime }+y \sqrt {x^{2}+1} = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
1.208 |
|
|
\[ {}\cos \left (x \right ) y^{\prime }+y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
1.845 |
|
|
\[ {}y^{\prime }+\sqrt {2 x^{2}+1}\, y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
1.302 |
|
|
\[ {}y^{\prime \prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.108 |
|
|
\[ {}y^{\prime \prime }+y \cos \left (x \right ) = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.992 |
|
|
\[ {}y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+y \cos \left (x \right ) = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.44 |
|
|
\[ {}\sqrt {x}\, y^{\prime \prime }+y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.071 |
|
|
\[ {}\left (x -3\right )^{2} y^{\prime \prime }-2 \left (x -3\right ) y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.005 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.0 |
|
|
\[ {}\left (-1+x \right )^{2} y^{\prime \prime }-5 \left (-1+x \right ) y^{\prime }+9 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.19 |
|
|
\[ {}\left (2+x \right )^{2} y^{\prime \prime }+\left (2+x \right ) y^{\prime } = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.795 |
|
|
\[ {}3 \left (-2+x \right )^{2} y^{\prime \prime }-4 \left (x -5\right ) y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.511 |
|
|
\[ {}\left (x -5\right )^{2} y^{\prime \prime }+\left (x -5\right ) y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.296 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {x y^{\prime }}{-2+x}+\frac {2 y}{2+x} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.737 |
|
|
\[ {}x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.245 |
|
|
\[ {}\left (-x^{4}+x^{3}\right ) y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+827 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.955 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x -3}+\frac {y}{x -4} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.136 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{\left (x -3\right )^{2}}+\frac {y}{\left (x -4\right )^{2}} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.844 |
|
|
\[ {}y^{\prime \prime }+\left (\frac {1}{x}-\frac {1}{3}\right ) y^{\prime }+\left (\frac {1}{x}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.148 |
|
|
\[ {}\left (4 x^{2}-1\right ) y^{\prime \prime }+\left (4-\frac {2}{x}\right ) y^{\prime }+\frac {\left (-x^{2}+1\right ) y}{x^{2}+1} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.801 |
|
|
\[ {}\left (x^{2}+4\right )^{2} y^{\prime \prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.651 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.18 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (1-4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.11 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-4+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.79 |
|
|
\[ {}\left (-9 x^{4}+x^{2}\right ) y^{\prime \prime }-6 x y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.319 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+\frac {y}{1-x} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.257 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Lienard] |
✓ |
✓ |
1.086 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{x^{2}}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Bessel] |
✓ |
✓ |
2.846 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+\left (-2 x^{3}+5 x \right ) y^{\prime }+\left (-x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.73 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 x^{2}+5 x \right ) y^{\prime }+\left (9+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.265 |
|
|
\[ {}\left (-3 x^{3}+3 x^{2}\right ) y^{\prime \prime }-\left (5 x^{2}+4 x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.589 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}+x \right ) y^{\prime }+4 x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
3.245 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x^{2} y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.282 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-x^{4}+x \right ) y^{\prime }+3 x^{3} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.655 |
|
|
\[ {}\left (9 x^{3}+9 x^{2}\right ) y^{\prime \prime }+\left (27 x^{2}+9 x \right ) y^{\prime }+\left (8 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.417 |
|
|
\[ {}\left (x -3\right ) y^{\prime \prime }+\left (x -3\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.125 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{2+x}+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.444 |
|
|
\[ {}4 y^{\prime \prime }+\frac {\left (4 x -3\right ) y}{\left (-1+x \right )^{2}} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.477 |
|
|
\[ {}\left (x -3\right )^{2} y^{\prime \prime }+\left (x^{2}-3 x \right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.636 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.257 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x^{2} y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.482 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Lienard] |
✓ |
✓ |
0.997 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+\left (4 x^{2}+5 x \right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.926 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 x^{2}+5 x \right ) y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.208 |
|
|
\[ {}x^{2} \left (2 x +1\right ) y^{\prime \prime }+x y^{\prime }+\left (4 x^{3}-4\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.866 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }+\left (1-4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.211 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.799 |
|
|
\[ {}x y^{\prime \prime }+4 y^{\prime }+\frac {12 y}{\left (2+x \right )^{2}} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.595 |
|
|
\[ {}x y^{\prime \prime }+4 y^{\prime }+\frac {12 y}{\left (2+x \right )^{2}} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.553 |
|
|
\[ {}\left (x -3\right ) y^{\prime \prime }+\left (x -3\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.032 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.484 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (1-4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.082 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Lienard] |
✓ |
✓ |
1.54 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}+x \right ) y^{\prime }+4 x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre] |
✓ |
✓ |
4.856 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-4+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.403 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 y \\ y^{\prime }=1-2 x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.827 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=4 x-3 y \\ y^{\prime }=6 x-7 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.719 |
|
|
\[ {}\left [\begin {array}{c} t x^{\prime }+2 x=15 y \\ t y^{\prime }=x \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.042 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+2 y \\ y^{\prime }=5 x-2 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.712 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=5 x+4 y \\ y^{\prime }=8 x+y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.627 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=4 x+2 y \\ y^{\prime }=3 x-y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.734 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+2 y \\ y^{\prime }=5 x-2 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.668 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 y \\ y^{\prime }=2 x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.605 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 y \\ y^{\prime }=-2 x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.75 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-2 y \\ y^{\prime }=8 x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.816 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=4 x-13 y \\ y^{\prime }=x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.793 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=3 x+2 y \\ y^{\prime }=-2 x+3 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.695 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=8 x+2 y-17 \\ y^{\prime }=4 x+y-13 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.019 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=8 x+2 y+7 \,{\mathrm e}^{2 t} \\ y^{\prime }=4 x+y-7 \,{\mathrm e}^{2 t} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.035 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=4 x+3 y-6 \,{\mathrm e}^{3 t} \\ y^{\prime }=x+6 y+2 \,{\mathrm e}^{3 t} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.15 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-y \\ y^{\prime }=4 x+24 t \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.218 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=4 x-13 y \\ y^{\prime }=x+19 \cos \left (4 t \right )-13 \sin \left (4 t \right ) \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
3.619 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=4 x+3 y+5 \operatorname {Heaviside}\left (t -2\right ) \\ y^{\prime }=x+6 y+17 \operatorname {Heaviside}\left (t -2\right ) \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.465 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=5 x+4 y \\ y^{\prime }=8 x+y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.64 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x-5 y \\ y^{\prime }=3 x-7 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.059 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x-5 y+4 \\ y^{\prime }=3 x-7 y+5 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
2.147 |
|
|
|
|||||||||
|
\[ {}\left [\begin {array}{c} x^{\prime }=3 x+y \\ y^{\prime }=6 x+2 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.67 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x y-6 y \\ y^{\prime }=x-y-5 \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.257 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-x+2 y \\ y^{\prime }=2 x-y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.6 |
|
|
|
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|
|
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|
|