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) |
|
\[ {}y^{\prime \prime }-25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.777 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.183 |
|
|
\[ {}y^{\prime } = -y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.084 |
|
|
\[ {}y^{\prime } = \frac {y}{2 x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.488 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.327 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.619 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+3 y = 0 \] |
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]] |
✓ |
✓ |
1.514 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.855 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+13 y = 0 \] |
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]] |
✓ |
✓ |
1.283 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+y = 9 x^{2} \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.592 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x^{4} \sin \left (x \right ) \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.773 |
|
|
\[ {}y^{\prime \prime }-\left (a +b \right ) y^{\prime }+y a b = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.368 |
|
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+a^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.224 |
|
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+\left (a^{2}+b^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.247 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.178 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.242 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.474 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = 0 \] |
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]] |
✓ |
✓ |
0.839 |
|
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{x}-\sin \left (y\right )}{x \cos \left (y\right )} \] |
1 |
1 |
1 |
exact |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.505 |
|
|
\[ {}y^{\prime } = \frac {1-y^{2}}{2 x y+2} \] |
1 |
1 |
1 |
exact |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.023 |
|
|
\[ {}y^{\prime } = \frac {\left (1-y \,{\mathrm e}^{x y}\right ) {\mathrm e}^{-x y}}{x} \] |
1 |
1 |
1 |
exact |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.167 |
|
|
\[ {}y^{\prime } = \frac {x^{2} \left (1-y^{2}\right )+y \,{\mathrm e}^{\frac {y}{x}}}{x \left ({\mathrm e}^{\frac {y}{x}}+2 x^{2} y\right )} \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
37.297 |
|
|
\[ {}y^{\prime } = \frac {\cos \left (x \right )-2 x y^{2}}{2 x^{2} y} \] |
1 |
1 |
1 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
33.947 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.143 |
|
|
\[ {}y^{\prime } = \frac {1}{x^{\frac {2}{3}}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.094 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] |
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]] |
✓ |
✓ |
0.664 |
|
|
\[ {}y^{\prime \prime } = x^{n} \] |
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]] |
✓ |
✓ |
0.662 |
|
|
\[ {}y^{\prime } = \ln \left (x \right ) x^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.26 |
|
|
\[ {}y^{\prime \prime } = \cos \left (x \right ) \] |
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]] |
✓ |
✓ |
1.073 |
|
|
\[ {}y^{\prime \prime \prime } = 6 x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _quadrature]] |
✓ |
✓ |
0.235 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] |
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]] |
✓ |
✓ |
0.981 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.194 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-8 y = 0 \] |
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]] |
✓ |
✓ |
0.826 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = \ln \left (x \right ) x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.636 |
|
|
\[ {}y^{\prime } = 2 x y \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.128 |
|
|
\[ {}y^{\prime } = \frac {y^{2}}{x^{2}+1} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.11 |
|
|
\[ {}{\mathrm e}^{x +y} y^{\prime }-1 = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.105 |
|
|
\[ {}y^{\prime } = \frac {y}{x \ln \left (x \right )} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.1 |
|
|
\[ {}y-\left (-1+x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.119 |
|
|
\[ {}y^{\prime } = \frac {2 x \left (y-1\right )}{x^{2}+3} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.259 |
|
|
\[ {}y-x y^{\prime } = 3-2 x^{2} y^{\prime } \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.19 |
|
|
\[ {}y^{\prime } = \frac {\cos \left (x -y\right )}{\sin \left (x \right ) \sin \left (y\right )}-1 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.547 |
|
|
\[ {}y^{\prime } = \frac {x \left (y^{2}-1\right )}{2 \left (-2+x \right ) \left (-1+x \right )} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.187 |
|
|
\[ {}y^{\prime } = \frac {x^{2} y-32}{-x^{2}+16}+2 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.254 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }-y+c = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.269 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+y^{2} = -1 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.321 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+x y = x a \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.208 |
|
|
\[ {}y^{\prime } = 1-\frac {\sin \left (x +y\right )}{\sin \left (y\right ) \cos \left (x \right )} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.422 |
|
|
\[ {}y^{\prime } = y^{3} \sin \left (x \right ) \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.288 |
|
|
\[ {}y^{\prime } = \frac {2 \sqrt {y-1}}{3} \] |
1 |
1 |
1 |
separable |
[_quadrature] |
✓ |
✓ |
0.395 |
|
|
\[ {}m v^{\prime } = m g -k v^{2} \] |
1 |
1 |
1 |
separable |
[_quadrature] |
✓ |
✓ |
1.526 |
|
|
\[ {}y^{\prime }+y = 4 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
linear |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.184 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x} = 5 x^{2} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.155 |
|
|
\[ {}x^{2} y^{\prime }-4 x y = x^{7} \sin \left (x \right ) \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.178 |
|
|
\[ {}y^{\prime }+2 x y = 2 x^{3} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.166 |
|
|
\[ {}y^{\prime }+\frac {2 x y}{-x^{2}+1} = 4 x \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.214 |
|
|
\[ {}y^{\prime }+\frac {2 x y}{x^{2}+1} = \frac {4}{\left (x^{2}+1\right )^{2}} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.195 |
|
|
\[ {}2 \cos \left (x \right )^{2} y^{\prime }+y \sin \left (2 x \right ) = 4 \cos \left (x \right )^{4} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.198 |
|
|
\[ {}y^{\prime }+\frac {y}{x \ln \left (x \right )} = 9 x^{2} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.165 |
|
|
\[ {}y^{\prime }-y \tan \left (x \right ) = 8 \sin \left (x \right )^{3} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.188 |
|
|
\[ {}t x^{\prime }+2 x = 4 \,{\mathrm e}^{t} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.174 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \left (y \sec \left (x \right )-2\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.619 |
|
|
\[ {}1-y \sin \left (x \right )-\cos \left (x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.188 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = 2 \ln \left (x \right ) x^{2} \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.169 |
|
|
\[ {}y^{\prime }+\alpha y = {\mathrm e}^{\beta x} \] |
1 |
1 |
1 |
linear |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.213 |
|
|
\[ {}y^{\prime }+\frac {m y}{x} = \ln \left (x \right ) \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.222 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x} = 4 x \] |
1 |
1 |
1 |
linear |
[_linear] |
✓ |
✓ |
0.383 |
|
|
\[ {}y^{\prime } \sin \left (x \right )-y \cos \left (x \right ) = \sin \left (2 x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.749 |
|
|
\[ {}x^{\prime }+\frac {2 x}{4-t} = 5 \] |
1 |
1 |
1 |
linear, homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.122 |
|
|
\[ {}y-{\mathrm e}^{x}+y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.675 |
|
|
\[
{}y^{\prime }-2 y = \left \{\begin {array}{cc} 1 & x \le 1 \\ 0 & 1 |
1 |
0 |
1 |
linear, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.357 |
|
|
\[ {}y^{\prime }-2 y = \left \{\begin {array}{cc} 1-x & x <1 \\ 0 & 1\le x \end {array}\right . \] |
1 |
0 |
1 |
linear, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.064 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x} = 9 x \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.858 |
|
|
\[ {}y^{\prime }+\frac {y}{x} = \cos \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.568 |
|
|
\[ {}y^{\prime }+y = {\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.488 |
|
|
\[ {}y^{\prime }+y \cot \left (x \right ) = 2 \cos \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.724 |
|
|
\[ {}-y+x y^{\prime } = \ln \left (x \right ) x^{2} \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.58 |
|
|
\[ {}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x^{2}} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.319 |
|
|
\[ {}\left (3 x -y\right ) y^{\prime } = 3 y \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.332 |
|
|
\[ {}y^{\prime } = \frac {\left (x +y\right )^{2}}{2 x^{2}} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.339 |
|
|
\[ {}\sin \left (\frac {y}{x}\right ) \left (-y+x y^{\prime }\right ) = x \cos \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.664 |
|
|
\[ {}x y^{\prime } = \sqrt {16 x^{2}-y^{2}}+y \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.809 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {9 x^{2}+y^{2}} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.483 |
|
|
\[ {}y \left (x^{2}-y^{2}\right )-x \left (x^{2}-y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
homogeneous |
[_separable] |
✓ |
✓ |
0.135 |
|
|
\[ {}x y^{\prime }+y \ln \left (x \right ) = y \ln \left (y\right ) \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+2 x y-2 x^{2}}{x^{2}-x y+y^{2}} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.586 |
|
|
\[ {}2 x y y^{\prime }-x^{2} {\mathrm e}^{-\frac {y^{2}}{x^{2}}}-2 y^{2} = 0 \] |
1 |
1 |
2 |
homogeneous |
[[_homogeneous, ‘class A‘]] |
✓ |
✓ |
0.325 |
|
|
\[ {}x^{2} y^{\prime } = y^{2}+3 x y+x^{2} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.296 |
|
|
\[ {}y y^{\prime } = \sqrt {x^{2}+y^{2}}-x \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.119 |
|
|
\[ {}2 x \left (y+2 x \right ) y^{\prime } = y \left (4 x -y\right ) \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.352 |
|
|
\[ {}x y^{\prime } = x \tan \left (\frac {y}{x}\right )+y \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.881 |
|
|
\[ {}y^{\prime } = \frac {x \sqrt {x^{2}+y^{2}}+y^{2}}{x y} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.543 |
|
|
\[ {}y^{\prime } = \frac {4 y-2 x}{x +y} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.953 |
|
|
\[ {}y^{\prime } = \frac {2 x -y}{x +4 y} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.803 |
|
|
\[ {}y^{\prime } = \frac {y-\sqrt {x^{2}+y^{2}}}{x} \] |
1 |
1 |
2 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.087 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {4 x^{2}-y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.148 |
|
|
\[ {}y^{\prime } = \frac {a y+x}{x a -y} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.393 |
|
|
\[ {}y^{\prime } = \frac {x +\frac {y}{2}}{\frac {x}{2}-y} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.944 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = \frac {4 x^{2} \cos \left (x \right )}{y} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _Bernoulli] |
✓ |
✓ |
2.286 |
|
|
\[ {}y^{\prime }+\frac {y \tan \left (x \right )}{2} = 2 y^{3} \sin \left (x \right ) \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
8.359 |
|
|
\[ {}y^{\prime }-\frac {3 y}{2 x} = 6 y^{\frac {1}{3}} x^{2} \ln \left (x \right ) \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
2.007 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x} = 6 \sqrt {x^{2}+1}\, \sqrt {y} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.845 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x} = 6 x^{4} y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.689 |
|
|
\[ {}2 x \left (y^{\prime }+x^{2} y^{3}\right )+y = 0 \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.687 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) \left (y^{\prime }-\sqrt {y}\right ) = 2 \left (-a +b \right ) y \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
2.076 |
|
|
\[ {}y^{\prime }+\frac {6 y}{x} = \frac {3 y^{\frac {2}{3}} \cos \left (x \right )}{x} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
82.877 |
|
|
\[ {}y^{\prime }+4 x y = 4 x^{3} \sqrt {y} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.561 |
|
|
\[ {}y^{\prime }-\frac {y}{2 x \ln \left (x \right )} = 2 x y^{3} \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.931 |
|
|
\[ {}y^{\prime }-\frac {y}{\left (\pi -1\right ) x} = \frac {3 x y^{\pi }}{1-\pi } \] |
1 |
1 |
1 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.887 |
|
|
\[ {}2 y^{\prime }+y \cot \left (x \right ) = \frac {8 \cos \left (x \right )^{3}}{y} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
14.635 |
|
|
\[ {}\left (1-\sqrt {3}\right ) y^{\prime }+y \sec \left (x \right ) = y^{\sqrt {3}} \sec \left (x \right ) \] |
1 |
1 |
1 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
187.417 |
|
|
\[ {}y^{\prime }+\frac {2 x y}{x^{2}+1} = x y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.042 |
|
|
\[ {}y^{\prime }+y \cot \left (x \right ) = y^{3} \sin \left (x \right )^{3} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.64 |
|
|
\[ {}y^{\prime } = \left (9 x -y\right )^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
1.585 |
|
|
\[ {}y^{\prime } = \left (4 x +y+2\right )^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.72 |
|
|
\[ {}y^{\prime } = \sin \left (3 x -3 y+1\right )^{2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
6.854 |
|
|
\[ {}y^{\prime } = \frac {y \left (\ln \left (x y\right )-1\right )}{x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.297 |
|
|
\[ {}y^{\prime } = 2 x \left (x +y\right )^{2}-1 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
1.493 |
|
|
\[ {}y^{\prime } = \frac {x +2 y-1}{2 x -y+3} \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.549 |
|
|
\[ {}y^{\prime }+p \left (x \right ) y+q \left (x \right ) y^{2} = r \left (x \right ) \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
0.925 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x}-y^{2} = -\frac {2}{x^{2}} \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
1.568 |
|
|
\[ {}y^{\prime }+\frac {7 y}{x}-3 y^{2} = \frac {3}{x^{2}} \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
0.924 |
|
|
\[ {}\frac {y^{\prime }}{y}+p \left (x \right ) \ln \left (y\right ) = q \left (x \right ) \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.563 |
|
|
\[ {}\frac {y^{\prime }}{y}-\frac {2 \ln \left (y\right )}{x} = \frac {1-2 \ln \left (x \right )}{x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.614 |
|
|
\[ {}\sec \left (y\right )^{2} y^{\prime }+\frac {\tan \left (y\right )}{2 \sqrt {1+x}} = \frac {1}{2 \sqrt {1+x}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
32.49 |
|
|
\[ {}y \,{\mathrm e}^{x y}+\left (2 y-x \,{\mathrm e}^{x y}\right ) y^{\prime } = 0 \] |
1 |
0 |
0 |
unknown |
[‘x=_G(y,y’)‘] |
❇ |
N/A |
0.589 |
|
|
\[ {}\cos \left (x y\right )-x y \sin \left (x y\right )-x^{2} \sin \left (x y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class G‘], _exact] |
✓ |
✓ |
0.332 |
|
|
\[ {}y+3 x^{2}+x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_linear] |
✓ |
✓ |
0.166 |
|
|
\[ {}2 x \,{\mathrm e}^{y}+\left (3 y^{2}+x^{2} {\mathrm e}^{y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
0.252 |
|
|
\[ {}2 x y+\left (x^{2}+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_separable] |
✓ |
✓ |
0.228 |
|
|
\[ {}y^{2}-2 x +2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.207 |
|
|
\[ {}4 \,{\mathrm e}^{2 x}+2 x y-y^{2}+\left (x -y\right )^{2} y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
0.302 |
|
|
\[ {}\frac {1}{x}-\frac {y}{x^{2}+y^{2}}+\frac {x y^{\prime }}{x^{2}+y^{2}} = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, _Riccati] |
✓ |
✓ |
0.394 |
|
|
\[ {}y \cos \left (x y\right )-\sin \left (x \right )+x \cos \left (x y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
0.261 |
|
|
\[ {}2 y^{2} {\mathrm e}^{2 x}+3 x^{2}+2 y \,{\mathrm e}^{2 x} y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _Bernoulli] |
✓ |
✓ |
0.25 |
|
|
\[ {}y^{2}+\cos \left (x \right )+\left (2 x y+\sin \left (y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.282 |
|
|
\[ {}\sin \left (y\right )+y \cos \left (x \right )+\left (x \cos \left (y\right )+\sin \left (x \right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.319 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.189 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.2 |
|
|
\[ {}y^{\prime \prime }-36 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.67 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
0.61 |
|
|
\[ {}y^{\prime \prime \prime }-3 y^{\prime \prime }-y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.168 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }-4 y^{\prime }-12 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.188 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }-18 y^{\prime }-40 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.188 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }-2 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.162 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }-10 y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.179 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }-y^{\prime \prime }+2 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.203 |
|
|
\[ {}y^{\prime \prime \prime \prime }-13 y^{\prime \prime }+36 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.23 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }-8 y = 0 \] |
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)]‘]] |
✓ |
✓ |
0.77 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+y = 0 \] |
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]] |
✓ |
✓ |
1.52 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.342 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+3 x^{2} y^{\prime \prime }-6 x y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.375 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 18 \,{\mathrm e}^{5 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.35 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 4 x^{2}+5 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.355 |
|
|
\[ {}y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-2 y = 4 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.471 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }-10 y^{\prime }+8 y = 24 \,{\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.55 |
|
|
\[ {}y^{\prime \prime \prime }+5 y^{\prime \prime }+6 y^{\prime } = 6 \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.427 |
|
|
\[ {}y^{\prime \prime }+y = 6 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.398 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 5 x \,{\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.452 |
|
|
\[ {}y^{\prime \prime }+4 y = 8 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.515 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 5 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.413 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 3 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime \prime }+2 y^{\prime \prime }-5 y^{\prime }-6 y = 4 x^{2} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.568 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = 9 \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.941 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y = 2 \,{\mathrm e}^{-x}+3 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.175 |
|
|
\[ {}y^{\prime \prime }+9 y = 5 \cos \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.797 |
|
|
\[ {}y^{\prime \prime }-y = 9 \,{\mathrm e}^{2 x} x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = -10 \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.627 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 4 \cos \left (x \right )-2 \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.716 |
|
|
\[ {}y^{\prime \prime }+\omega ^{2} y = \frac {F_{0} \cos \left (\omega t \right )}{m} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.726 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+6 y = 7 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.643 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y = 4 x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.939 |
|
|
\[ {}y^{\prime \prime \prime \prime }+104 y^{\prime \prime \prime }+2740 y^{\prime \prime } = 5 \,{\mathrm e}^{-2 x} \cos \left (3 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.227 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-3 y = \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime \prime }+6 y = \sin \left (x \right )^{2} \cos \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.912 |
|
|
\[ {}y^{\prime \prime }-16 y = 20 \cos \left (4 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.481 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 50 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.598 |
|
|
\[ {}y^{\prime \prime }-y = 10 \,{\mathrm e}^{2 x} \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.487 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 169 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.605 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 40 \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.558 |
|
|
\[ {}y^{\prime \prime }+y = 3 \,{\mathrm e}^{x} \cos \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.644 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 2 \,{\mathrm e}^{-x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.513 |
|
|
\[ {}y^{\prime \prime }-4 y = 100 x \,{\mathrm e}^{x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.662 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 4 \,{\mathrm e}^{-x} \cos \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.531 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+10 y = 24 \,{\mathrm e}^{x} \cos \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.64 |
|
|
\[ {}y^{\prime \prime }+16 y = 34 \,{\mathrm e}^{x}+16 \cos \left (4 x \right )-8 \sin \left (4 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.276 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 4 \,{\mathrm e}^{3 x} \ln \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.565 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = \frac {{\mathrm e}^{-2 x}}{x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.536 |
|
|
\[ {}y^{\prime \prime }+9 y = 18 \sec \left (3 x \right )^{3} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.716 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = \frac {2 \,{\mathrm e}^{-3 x}}{x^{2}+1} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.597 |
|
|
\[ {}y^{\prime \prime }-4 y = \frac {8}{{\mathrm e}^{2 x}+1} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.544 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = {\mathrm e}^{2 x} \tan \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.62 |
|
|
\[ {}y^{\prime \prime }+9 y = \frac {36}{4-\cos \left (3 x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.895 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = \frac {2 \,{\mathrm e}^{5 x}}{x^{2}+4} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+13 y = 4 \,{\mathrm e}^{3 x} \sec \left (2 x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.822 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right )+4 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.698 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right )+2 x^{2}+5 x +1 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.716 |
|
|
\[ {}y^{\prime \prime }-y = 2 \tanh \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime \prime }-2 m y^{\prime }+m^{2} y = \frac {{\mathrm e}^{m x}}{x^{2}+1} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.572 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {4 \,{\mathrm e}^{x} \ln \left (x \right )}{x^{3}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = \frac {{\mathrm e}^{-x}}{\sqrt {-x^{2}+4}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.606 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+17 y = \frac {64 \,{\mathrm e}^{-x}}{3+\sin \left (4 x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.911 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = \frac {4 \,{\mathrm e}^{-2 x}}{x^{2}+1}+2 x^{2}-1 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.776 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 15 \,{\mathrm e}^{-2 x} \ln \left (x \right )+25 \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.893 |
|
|
\[ {}y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y = \frac {2 \,{\mathrm e}^{x}}{x^{2}} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.275 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y = 36 \,{\mathrm e}^{2 x} \ln \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.336 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y = \frac {2 \,{\mathrm e}^{-x}}{x^{2}+1} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.32 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime } = 12 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.151 |
|
|
\[ {}y^{\prime \prime }-9 y = F \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.629 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+4 y = F \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.639 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = F \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.649 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-12 y = F \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.686 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 5 \,{\mathrm e}^{2 x} x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.798 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.906 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = 4 \ln \left (x \right ) \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.421 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = \cos \left (x \right ) \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.505 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+9 y = 9 \ln \left (x \right ) \] |
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]] |
✓ |
✓ |
5.307 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+5 y = 8 x \ln \left (x \right )^{2} \] |
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]] |
✓ |
✓ |
7.878 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x^{4} \sin \left (x \right ) \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.737 |
|
|
\[ {}x^{2} y^{\prime \prime }+6 x y^{\prime }+6 y = 4 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.735 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = \frac {x^{2}}{\ln \left (x \right )} \] |
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, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.603 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 m -1\right ) x y^{\prime }+m^{2} y = x^{m} \ln \left (x \right )^{k} \] |
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, _linear, _nonhomogeneous]] |
✓ |
✓ |
9.447 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+5 y = 0 \] |
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]] |
✓ |
✓ |
2.285 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+25 y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.694 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.26 |
|
|
\[ {}x y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.375 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.418 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[_Gegenbauer] |
✓ |
✓ |
0.384 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}+4 x^{2} y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.375 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.425 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right ) \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.385 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+2 y = 8 x^{2} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.377 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 8 x^{4} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.298 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 15 \,{\mathrm e}^{3 x} \sqrt {x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.332 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 4 \,{\mathrm e}^{2 x} \ln \left (x \right ) \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.333 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = \sqrt {x}\, \ln \left (x \right ) \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.283 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }-4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.226 |
|
|
\[ {}y^{\prime \prime \prime }+11 y^{\prime \prime }+36 y^{\prime }+26 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.273 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 4 \,{\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.435 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 4 \,{\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.418 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+25 y^{\prime } = x^{2} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
1.04 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+25 y^{\prime } = \sin \left (4 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
1.569 |
|
|
\[ {}y^{\prime \prime \prime }+9 y^{\prime \prime }+24 y^{\prime }+16 y = 8 \,{\mathrm e}^{-x}+1 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.529 |
|
|
\[ {}y^{\prime \prime }-4 y = 5 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.35 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 2 x \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.457 |
|
|
\[ {}y^{\prime \prime }-y = 4 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.389 |
|
|
\[ {}y^{\prime \prime }+x y = \sin \left (x \right ) \] |
1 |
1 |
1 |
second_order_airy, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.121 |
|
|
\[ {}y^{\prime \prime }+4 y = \ln \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.698 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-3 y = 5 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.412 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime }+y = 4 \cos \left (2 x \right )+3 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.806 |
|
|
\[ {}y^{\prime }-2 y = 6 \,{\mathrm e}^{5 t} \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.441 |
|
|
\[ {}y^{\prime }+y = 8 \,{\mathrm e}^{3 t} \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.39 |
|
|
\[ {}y^{\prime }+3 y = 2 \,{\mathrm e}^{-t} \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.418 |
|
|
\[ {}y^{\prime }+2 y = 4 t \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.377 |
|
|
\[ {}y^{\prime }-y = 6 \cos \left (t \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.501 |
|
|
\[ {}y^{\prime }-y = 5 \sin \left (2 t \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.529 |
|
|
\[ {}y^{\prime }+y = 5 \,{\mathrm e}^{t} \sin \left (t \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.362 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.313 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 4 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.366 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-12 y = 36 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.35 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 10 \,{\mathrm e}^{-t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.391 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 4 \,{\mathrm e}^{3 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.377 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = 30 \,{\mathrm e}^{-3 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.413 |
|
|
\[ {}y^{\prime \prime }-y = 12 \,{\mathrm e}^{2 t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.38 |
|
|
\[ {}y^{\prime \prime }+4 y = 10 \,{\mathrm e}^{-t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.425 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 12-6 \,{\mathrm e}^{t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.469 |
|
|
\[ {}y^{\prime \prime }-y = 6 \cos \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.478 |
|
|
\[ {}y^{\prime \prime }-9 y = 13 \sin \left (2 t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.551 |
|
|
\[ {}y^{\prime \prime }-y = 8 \sin \left (t \right )-6 \cos \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.747 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 10 \cos \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.477 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+4 y = 20 \sin \left (2 t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.529 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+4 y = 20 \sin \left (2 t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.438 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 3 \cos \left (t \right )+\sin \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.553 |
|
|
\[ {}y^{\prime \prime }+4 y = 9 \sin \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.64 |
|
|
\[ {}y^{\prime \prime }+y = 6 \cos \left (2 t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.542 |
|
|
\[ {}y^{\prime \prime }+9 y = 7 \sin \left (4 t \right )+14 \cos \left (4 t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.015 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.282 |
|
|
\[ {}y^{\prime }+2 y = 2 \operatorname {Heaviside}\left (-1+t \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.896 |
|
|
\[ {}y^{\prime }-2 y = \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{t -2} \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.053 |
|
|
\[ {}y^{\prime }-y = 4 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \sin \left (t +\frac {\pi }{4}\right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.749 |
|
|
\[ {}y^{\prime }+2 y = \operatorname {Heaviside}\left (t -\pi \right ) \sin \left (2 t \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.399 |
|
|
\[ {}y^{\prime }+3 y = \left \{\begin {array}{cc} 1 & 0\le t <1 \\ 0 & 1\le t \end {array}\right . \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.232 |
|
|
\[ {}y^{\prime }-3 y = \left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\frac {\pi }{2} \\ 1 & \frac {\pi }{2}\le t \end {array}\right . \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.909 |
|
|
\[ {}y^{\prime }-3 y = -10 \,{\mathrm e}^{-t +a} \sin \left (-2 t +2 a \right ) \operatorname {Heaviside}\left (t -a \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
13.737 |
|
|
\[ {}y^{\prime \prime }-y = \operatorname {Heaviside}\left (-1+t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.826 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 1-3 \operatorname {Heaviside}\left (t -2\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.349 |
|
|
\[ {}y^{\prime \prime }-4 y = \operatorname {Heaviside}\left (-1+t \right )-\operatorname {Heaviside}\left (t -2\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.668 |
|
|
\[ {}y^{\prime \prime }+y = t -\operatorname {Heaviside}\left (-1+t \right ) \left (-1+t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.913 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = -10 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (t +\frac {\pi }{4}\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.341 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 30 \operatorname {Heaviside}\left (-1+t \right ) {\mathrm e}^{1-t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.366 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 5 \operatorname {Heaviside}\left (t -3\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.24 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 2 \sin \left (t \right )+\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (1+\cos \left (t \right )\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.093 |
|
|
\[ {}y^{\prime }-y = \left \{\begin {array}{cc} 2 & 0\le t <1 \\ -1 & 1\le t \end {array}\right . \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.273 |
|
|
\[ {}y^{\prime }-y = \left \{\begin {array}{cc} 2 & 0\le t <1 \\ -1 & 1\le t \end {array}\right . \] |
1 |
0 |
1 |
linear, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
2.247 |
|
|
\[ {}y^{\prime }+y = \delta \left (t -5\right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime }-2 y = \delta \left (t -2\right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.748 |
|
|
\[ {}y^{\prime }+4 y = 3 \delta \left (-1+t \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.836 |
|
|
\[ {}y^{\prime }-5 y = 2 \,{\mathrm e}^{-t}+\delta \left (t -3\right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.939 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \delta \left (-1+t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.901 |
|
|
\[ {}y^{\prime \prime }-4 y = \delta \left (t -3\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.839 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = \delta \left (t -\frac {\pi }{2}\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.927 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = \delta \left (t -\frac {\pi }{4}\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.868 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = \delta \left (t -2\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.972 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+13 y = \delta \left (t -\frac {\pi }{4}\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.152 |
|
|
\[ {}y^{\prime \prime }+9 y = 15 \sin \left (2 t \right )+\delta \left (t -\frac {\pi }{6}\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.599 |
|
|
\[ {}y^{\prime \prime }+16 y = 4 \cos \left (3 t \right )+\delta \left (t -\frac {\pi }{3}\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.829 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 4 \sin \left (t \right )+\delta \left (t -\frac {\pi }{6}\right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.188 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.402 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_erf] |
✓ |
✓ |
0.655 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }-2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-2 x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.652 |
|
|
\[ {}y^{\prime \prime }+x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.431 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+3 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.774 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.692 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y^{\prime }+2 x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.713 |
|
|
\[ {}\left (x^{2}-3\right ) y^{\prime \prime }-3 x y^{\prime }-5 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.647 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+4 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.608 |
|
|
\[ {}\left (-4 x^{2}+1\right ) y^{\prime \prime }-20 x y^{\prime }-16 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.921 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.691 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+4 x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.723 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.855 |
|
|
\[ {}y^{\prime \prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.671 |
|
|
\[ {}x y^{\prime \prime }-\left (-1+x \right ) y^{\prime }-x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.005 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.718 |
|
|
\[ {}4 y^{\prime \prime }+x y^{\prime }+4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Lienard] |
✓ |
✓ |
1.668 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y^{\prime }+x y = 2 \cos \left (x \right ) \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.144 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-4 y = 6 \,{\mathrm e}^{x} \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.201 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{1-x}+x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.216 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {x y^{\prime }}{\left (-x^{2}+1\right )^{2}}+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.875 |
|
|
\[ {}\left (-2+x \right )^{2} y^{\prime \prime }+\left (-2+x \right ) {\mathrm e}^{x} y^{\prime }+\frac {4 y}{x} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.556 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x \left (x -3\right )}-\frac {y}{x^{3} \left (x +3\right )} = 0 \] |
1 |
0 |
0 |
second order series method. Irregular singular point |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.008 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-7 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.538 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+x \,{\mathrm e}^{x} y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.398 |
|
|
\[ {}4 x y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.968 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \cos \left (x \right ) y^{\prime }+5 \,{\mathrm e}^{2 x} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
9.15 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+3 x y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.045 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (1+12 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.265 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.2 |
|
|
\[ {}2 x y^{\prime \prime }+y^{\prime }-2 x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.897 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-x \left (x +8\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.338 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+2 \left (4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.486 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-\left (x +5\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.489 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (7+3 x \right ) y^{\prime }+\left (6 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.223 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.365 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (3 x^{2}+1\right ) y^{\prime }-2 x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.42 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.07 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (1-2 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.071 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.055 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-x +3\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.967 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x +4\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.833 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (-x^{2}+x \right ) y^{\prime }+\left (x^{3}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.437 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.73 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-2 x^{5}+9 x \right ) y^{\prime }+\left (10 x^{4}+5 x^{2}+25\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Complex roots |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
13.138 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (4 x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}\right ) y^{\prime }-\frac {7 y}{4} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.686 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.934 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.956 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.979 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \cos \left (x \right ) y^{\prime }-2 \,{\mathrm e}^{x} y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.797 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.025 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x -\frac {3}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.122 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (2 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.931 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{3} y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.369 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+7 x \,{\mathrm e}^{x} y^{\prime }+9 \left (1+\tan \left (x \right )\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
244.654 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.188 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.96 |
|
|
\[ {}x y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.803 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}+6\right ) y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.296 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.032 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (1-4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.892 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.918 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.977 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.974 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.04 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-\left (3 x +2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.197 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.053 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.212 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (2+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.135 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.027 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (2 x +1\right ) y^{\prime }+\left (4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.223 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-\left (3+4 x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.931 |
|
|
\[ {}x y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Laguerre, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.821 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.071 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.127 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
1.889 |
|
|
\[ {}y^{\prime \prime }+x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.419 |
|
|
\[ {}y^{\prime \prime }-x^{2} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.498 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-6 x y^{\prime }-4 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.911 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.875 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.98 |
|
|
\[ {}2 x y^{\prime \prime }+5 \left (1-2 x \right ) y^{\prime }-5 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.448 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Lienard] |
✓ |
✓ |
0.845 |
|
|
\[ {}\left (4 x^{2}+1\right ) y^{\prime \prime }-8 y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.516 |
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\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.024 |
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\[ {}4 x y^{\prime \prime }+3 y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler]] |
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✓ |
1.069 |
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\[ {}x^{2} y^{\prime \prime }+\frac {3 x y^{\prime }}{2}-\frac {\left (1+x \right ) y}{2} = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.934 |
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\[ {}x^{2} y^{\prime \prime }-x \left (2-x \right ) y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.198 |
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\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.93 |
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\[ {}y^{\prime \prime }+\left (1-\frac {3}{4 x^{2}}\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.905 |
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