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) |
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] |
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]] |
✓ |
✓ |
1.733 |
|
|
\[ {}-y+x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.635 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y = 0 \] |
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_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.487 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.287 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.987 |
|
|
\[ {}y^{\prime }+\frac {1}{2 y} = 0 \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.3 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = 1 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.638 |
|
|
\[ {}y^{\prime }-2 \sqrt {{| y|}} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.026 |
|
|
\[ {}x^{2} y^{\prime }+2 x y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.812 |
|
|
\[ {}y^{\prime }-y^{2} = 1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.214 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.102 |
|
|
\[ {}x y^{\prime }-\sin \left (x \right ) = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.306 |
|
|
\[ {}y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.274 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.276 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.337 |
|
|
\[ {}y^{\prime \prime \prime }-7 y^{\prime \prime }+12 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.266 |
|
|
\[ {}2 x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.7 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.579 |
|
|
\[ {}x^{2} y^{\prime \prime }+6 x y^{\prime }+4 y = 0 \] |
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]] |
✓ |
✓ |
1.641 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = 0 \] |
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]] |
✓ |
✓ |
1.111 |
|
|
\[ {}{y^{\prime }}^{2}-4 y = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.437 |
|
|
\[ {}{y^{\prime }}^{2}-9 x y = 0 \] |
2 |
2 |
5 |
first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
0.653 |
|
|
\[ {}{y^{\prime }}^{2} = x^{6} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.417 |
|
|
\[ {}y^{\prime }-2 x y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.699 |
|
|
\[ {}y^{\prime }+y = x^{2}+2 x -1 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.628 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.291 |
|
|
\[ {}y^{\prime } = x \sqrt {y} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.069 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.558 |
|
|
\[ {}y^{\prime } = 3 y^{\frac {2}{3}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.273 |
|
|
\[ {}x \ln \left (x \right ) y^{\prime }-\left (1+\ln \left (x \right )\right ) y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.98 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.462 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.498 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.596 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.645 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 x y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.439 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.969 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.894 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.881 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.908 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.438 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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 |
1.063 |
|
|
\[ {}y^{\prime } = 1-x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.184 |
|
|
\[ {}y^{\prime } = -1+x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.171 |
|
|
\[ {}y^{\prime } = 1-y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.281 |
|
|
\[ {}y^{\prime } = y+1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.254 |
|
|
\[ {}y^{\prime } = y^{2}-4 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.426 |
|
|
\[ {}y^{\prime } = 4-y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.404 |
|
|
\[ {}y^{\prime } = x y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.642 |
|
|
\[ {}y^{\prime } = -x y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.712 |
|
|
\[ {}y^{\prime } = x^{2}-y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
0.895 |
|
|
\[ {}y^{\prime } = -x^{2}+y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
0.799 |
|
|
\[ {}y^{\prime } = x +y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.621 |
|
|
\[ {}y^{\prime } = x y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.557 |
|
|
\[ {}y^{\prime } = \frac {x}{y} \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.197 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.602 |
|
|
\[ {}y^{\prime } = 1+y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.211 |
|
|
\[ {}y^{\prime } = y^{2}-3 y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.434 |
|
|
\[ {}y^{\prime } = x^{3}+y^{3} \] |
1 |
0 |
0 |
abelFirstKind |
[_Abel] |
❇ |
N/A |
0.439 |
|
|
\[ {}y^{\prime } = {| y|} \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x -y} \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.626 |
|
|
\[ {}y^{\prime } = \ln \left (x +y\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.801 |
|
|
\[ {}y^{\prime } = \frac {2 x -y}{x +3 y} \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.579 |
|
|
\[ {}y^{\prime } = \frac {1}{\sqrt {15-x^{2}-y^{2}}} \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
0.855 |
|
|
\[ {}y^{\prime } = \frac {3 y}{\left (x -5\right ) \left (x +3\right )}+{\mathrm e}^{-x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.183 |
|
|
\[ {}y^{\prime } = \frac {x y}{x^{2}+y^{2}} \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.0 |
|
|
\[ {}y^{\prime } = \frac {1}{x y} \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.677 |
|
|
\[ {}y^{\prime } = \ln \left (y-1\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.23 |
|
|
\[ {}y^{\prime } = \sqrt {\left (y+2\right ) \left (y-1\right )} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.705 |
|
|
\[ {}y^{\prime } = \frac {y}{y-x} \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.806 |
|
|
\[ {}y^{\prime } = \frac {x}{y^{2}} \] |
1 |
1 |
3 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
5.332 |
|
|
\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.188 |
|
|
\[ {}y^{\prime } = \frac {x y}{1-y} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.788 |
|
|
\[ {}y^{\prime } = \left (x y\right )^{\frac {1}{3}} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
96.204 |
|
|
\[ {}y^{\prime } = \sqrt {\frac {y-4}{x}} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
3.463 |
|
|
\[ {}y^{\prime } = -\frac {y}{x}+y^{\frac {1}{4}} \] |
1 |
1 |
1 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
2.401 |
|
|
\[ {}y^{\prime } = 4 y-5 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.545 |
|
|
\[ {}y^{\prime }+3 y = 1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.487 |
|
|
\[ {}y^{\prime } = a y+b \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.597 |
|
|
\[ {}y^{\prime } = x^{2}+{\mathrm e}^{x}-\sin \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.977 |
|
|
\[ {}y^{\prime } = x y+\frac {1}{x^{2}+1} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.97 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\cos \left (x \right ) \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.202 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\tan \left (x \right ) \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
5.545 |
|
|
\[ {}y^{\prime } = \frac {y}{-x^{2}+4}+\sqrt {x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.967 |
|
|
\[ {}y^{\prime } = \frac {y}{-x^{2}+4}+\sqrt {x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
7.509 |
|
|
\[ {}y^{\prime } = \cot \left (x \right ) y+\csc \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.372 |
|
|
\[ {}y^{\prime } = -x \sqrt {1-y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.98 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.729 |
|
|
\[ {}y^{\prime } = 1+3 x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.296 |
|
|
\[ {}y^{\prime } = x +\frac {1}{x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.382 |
|
|
\[ {}y^{\prime } = 2 \sin \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.422 |
|
|
\[ {}y^{\prime } = x \sin \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.49 |
|
|
\[ {}y^{\prime } = \frac {1}{-1+x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.378 |
|
|
\[ {}y^{\prime } = \frac {1}{-1+x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.293 |
|
|
\[ {}y^{\prime } = \frac {1}{x^{2}-1} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.386 |
|
|
\[ {}y^{\prime } = \frac {1}{x^{2}-1} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.332 |
|
|
\[ {}y^{\prime } = \tan \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.403 |
|
|
\[ {}y^{\prime } = \tan \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.325 |
|
|
\[ {}y^{\prime } = 3 y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.435 |
|
|
\[ {}y^{\prime } = 1-y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.327 |
|
|
\[ {}y^{\prime } = 1-y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.336 |
|
|
\[ {}y^{\prime } = x \,{\mathrm e}^{-x^{2}+y} \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.371 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.887 |
|
|
\[ {}y^{\prime } = \frac {2 x}{y} \] |
1 |
1 |
1 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.724 |
|
|
\[ {}y^{\prime } = -2 y+y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.783 |
|
|
\[ {}y^{\prime } = x y+x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.167 |
|
|
\[ {}x \,{\mathrm e}^{y}+y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.982 |
|
|
\[ {}y-x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.328 |
|
|
\[ {}2 y y^{\prime } = 1 \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.308 |
|
|
\[ {}2 x y y^{\prime }+y^{2} = -1 \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.467 |
|
|
\[ {}y^{\prime } = \frac {1-x y}{x^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.72 |
|
|
\[ {}y^{\prime } = -\frac {y \left (y+2 x \right )}{x \left (2 y+x \right )} \] |
1 |
1 |
2 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.775 |
|
|
\[ {}y^{\prime } = \frac {y^{2}}{1-x y} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.053 |
|
|
\[ {}y^{\prime } = 4 y+1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.484 |
|
|
\[ {}y^{\prime } = x y+2 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.075 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.926 |
|
|
\[ {}y^{\prime } = \frac {y}{-1+x}+x^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.954 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\sin \left (x^{2}\right ) \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.747 |
|
|
\[ {}y^{\prime } = \frac {2 y}{x}+{\mathrm e}^{x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
13.245 |
|
|
\[ {}y^{\prime } = \cot \left (x \right ) y+\sin \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.187 |
|
|
\[ {}x -y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.227 |
|
|
\[ {}y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.654 |
|
|
\[ {}x^{2}-y+x y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.721 |
|
|
\[ {}x y \left (1-y\right )-2 y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.379 |
|
|
\[ {}x \left (1-y^{3}\right )-3 y^{2} y^{\prime } = 0 \] |
1 |
1 |
3 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.635 |
|
|
\[ {}\left (2 x -1\right ) y+x \left (1+x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.972 |
|
|
\[ {}y^{\prime } = \frac {1}{-1+x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.238 |
|
|
\[ {}y^{\prime } = x +y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.783 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.887 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.849 |
|
|
\[ {}y^{\prime } = \frac {y}{-x^{2}+1}+\sqrt {x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.382 |
|
|
\[ {}y^{\prime } = \frac {y}{-x^{2}+1}+\sqrt {x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.951 |
|
|
\[ {}y^{\prime } = \frac {y}{-x^{2}+1}+\sqrt {x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.293 |
|
|
\[ {}y^{\prime } = y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.273 |
|
|
\[ {}y^{\prime } = y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.278 |
|
|
\[ {}y^{\prime } = y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.289 |
|
|
\[ {}y^{\prime } = y^{3} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.335 |
|
|
\[ {}y^{\prime } = y^{3} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.255 |
|
|
\[ {}y^{\prime } = y^{3} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.28 |
|
|
\[ {}y^{\prime } = -\frac {3 x^{2}}{2 y} \] |
1 |
1 |
1 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.566 |
|
|
\[ {}y^{\prime } = -\frac {3 x^{2}}{2 y} \] |
1 |
1 |
1 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.958 |
|
|
\[ {}y^{\prime } = -\frac {3 x^{2}}{2 y} \] |
1 |
1 |
2 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.991 |
|
|
\[ {}y^{\prime } = -\frac {3 x^{2}}{2 y} \] |
1 |
1 |
1 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.318 |
|
|
\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.206 |
|
|
\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.059 |
|
|
\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.874 |
|
|
\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.662 |
|
|
\[ {}y^{\prime } = 3 x y^{\frac {1}{3}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.885 |
|
|
\[ {}y^{\prime } = 3 x y^{\frac {1}{3}} \] |
1 |
0 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.247 |
|
|
\[ {}y^{\prime } = 3 x y^{\frac {1}{3}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.645 |
|
|
\[ {}y^{\prime } = 3 x y^{\frac {1}{3}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.622 |
|
|
\[ {}y^{\prime } = 3 x y^{\frac {1}{3}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.546 |
|
|
\[ {}y^{\prime } = \sqrt {\left (y+2\right ) \left (y-1\right )} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.613 |
|
|
\[ {}y^{\prime } = \sqrt {\left (y+2\right ) \left (y-1\right )} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.289 |
|
|
\[ {}y^{\prime } = \sqrt {\left (y+2\right ) \left (y-1\right )} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.829 |
|
|
\[ {}y^{\prime } = \frac {y}{y-x} \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.643 |
|
|
\[ {}y^{\prime } = \frac {y}{y-x} \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.719 |
|
|
\[ {}y^{\prime } = \frac {y}{y-x} \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.141 |
|
|
\[ {}y^{\prime } = \frac {y}{y-x} \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.289 |
|
|
\[ {}y^{\prime } = \frac {x y}{x^{2}+y^{2}} \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.948 |
|
|
\[ {}y^{\prime } = \frac {x y}{x^{2}+y^{2}} \] |
1 |
0 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.625 |
|
|
\[ {}y^{\prime } = \frac {x y}{x^{2}+y^{2}} \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.802 |
|
|
\[ {}y^{\prime } = x \sqrt {1-y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.947 |
|
|
\[ {}y^{\prime } = x \sqrt {1-y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
9.96 |
|
|
\[ {}y^{\prime } = x \sqrt {1-y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
12.649 |
|
|
\[ {}y^{\prime } = x \sqrt {1-y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.989 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.59 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.036 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.143 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \] |
1 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.897 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.395 |
|
|
\[ {}3 y^{\prime \prime }-2 y^{\prime }+4 y = x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.704 |
|
|
\[ {}x y^{\prime \prime \prime }+x y^{\prime } = 4 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
15.169 |
|
|
\[ {}x \left (x -3\right ) y^{\prime \prime }+3 y^{\prime } = x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.86 |
|
|
\[ {}x \left (x -3\right ) y^{\prime \prime }+3 y^{\prime } = x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.83 |
|
|
\[ {}\sqrt {1-x}\, y^{\prime \prime }-4 y = \sin \left (x \right ) \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
18.188 |
|
|
\[ {}\left (x^{2}-4\right ) y^{\prime \prime }+y \ln \left (x \right ) = x \,{\mathrm e}^{x} \] |
1 |
0 |
0 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
0.964 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.774 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.88 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.391 |
|
|
\[ {}2 y y^{\prime \prime }-{y^{\prime }}^{2} = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.208 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.7 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.589 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
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]] |
✓ |
✓ |
2.388 |
|
|
\[ {}y^{\prime \prime }-4 y = 31 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.849 |
|
|
\[ {}y^{\prime \prime }+9 y = 27 x +18 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = -3 x -\frac {3}{x} \] |
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]] |
✓ |
✓ |
2.217 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.296 |
|
|
\[ {}y^{\prime \prime \prime }-4 y^{\prime \prime }+6 y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.373 |
|
|
\[ {}y^{\prime \prime \prime \prime }-16 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.372 |
|
|
\[ {}y^{\prime \prime \prime \prime }+16 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.874 |
|
|
\[ {}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+8 y^{\prime \prime }-8 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.187 |
|
|
\[ {}y^{\prime \prime \prime \prime }-8 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.595 |
|
|
\[ {}36 y^{\prime \prime \prime \prime }-12 y^{\prime \prime \prime }-11 y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.45 |
|
|
\[ {}y^{\left (5\right )}-3 y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }-3 y^{\prime \prime }+2 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.406 |
|
|
\[ {}y^{\left (5\right )}-y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }+35 y^{\prime \prime }+16 y^{\prime }-52 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.63 |
|
|
\[ {}y^{\left (8\right )}+8 y^{\prime \prime \prime \prime }+16 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.238 |
|
|
\[ {}y^{\prime \prime }+\alpha y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.414 |
|
|
\[ {}y^{\prime \prime \prime }+\left (-3-4 i\right ) y^{\prime \prime }+\left (-4+12 i\right ) y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.221 |
|
|
\[ {}y^{\prime \prime \prime \prime }+\left (-3-i\right ) y^{\prime \prime \prime }+\left (4+3 i\right ) y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.204 |
|
|
\[ {}y^{\prime }-i y = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.472 |
|
|
\[ {}y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+13 y^{\prime \prime }-12 y^{\prime }+4 y = 2 \,{\mathrm e}^{x}-4 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.326 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime } = 24 x^{2}-6 x +14+32 \cos \left (2 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.846 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y = 3+\cos \left (2 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.311 |
|
|
\[ {}y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+3 y^{\prime \prime }-y^{\prime } = 6 x -20-120 x^{2} {\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.323 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+21 y^{\prime }-26 y = 36 \,{\mathrm e}^{2 x} \sin \left (3 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.947 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }-y^{\prime }-y = \left (2 x^{2}+4 x +8\right ) \cos \left (x \right )+\left (6 x^{2}+8 x +12\right ) \sin \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.184 |
|
|
\[ {}y^{\left (6\right )}-12 y^{\left (5\right )}+63 y^{\prime \prime \prime \prime }-18 y^{\prime \prime \prime }+315 y^{\prime \prime }-300 y^{\prime }+125 y = {\mathrm e}^{x} \left (48 \cos \left (x \right )+96 \sin \left (x \right )\right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
66.164 |
|
|
\[ {}y^{\prime \prime \prime }-3 y^{\prime \prime }-4 y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.447 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+2 y^{\prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.349 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = 2 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.343 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y = 3 x +4 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.609 |
|
|
\[ {}y^{\prime }-y = 0 \] |
1 |
1 |
1 |
first_order_laplace |
[_quadrature] |
✓ |
✓ |
0.23 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.265 |
|
|
\[ {}y^{\prime }+2 y = 4 \] |
1 |
1 |
1 |
first_order_laplace |
[_quadrature] |
✓ |
✓ |
0.267 |
|
|
\[ {}y^{\prime \prime }-9 y = 2 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.509 |
|
|
\[ {}y^{\prime \prime }+9 y = 2 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.52 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = x \,{\mathrm e}^{x}-3 x^{2} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.407 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+y^{\prime \prime } = x \,{\mathrm e}^{x}-3 x^{2} \] |
1 |
1 |
1 |
higher_order_laplace |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.428 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x} \] |
1 |
1 |
1 |
first_order_laplace |
[_quadrature] |
✓ |
✓ |
0.257 |
|
|
\[ {}y^{\prime }-y = 2 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.475 |
|
|
\[ {}y^{\prime \prime }-9 y = 2+x \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.457 |
|
|
\[ {}y^{\prime \prime }+9 y = 2+x \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.603 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+6 y = -2 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.189 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = -x^{2}+1 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.54 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }+2 y^{\prime } = x +\cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_laplace |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.836 |
|
|
\[ {}y^{\prime }-2 y = 6 \] |
1 |
1 |
1 |
first_order_laplace |
[_quadrature] |
✓ |
✓ |
0.413 |
|
|
\[ {}y^{\prime }+y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.473 |
|
|
\[ {}y^{\prime \prime }+9 y = 1 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.447 |
|
|
\[ {}y^{\prime \prime }+9 y = 18 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.52 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.37 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = x^{2} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.44 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 \sin \left (x \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.49 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+4 y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
higher_order_laplace |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.504 |
|
|
\[ {}y^{\prime }+2 y = \left \{\begin {array}{cc} 2 & 0\le x <1 \\ 1 & 1\le x \end {array}\right . \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.227 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = \left \{\begin {array}{cc} 1 & 2\le x <4 \\ 0 & \operatorname {otherwise} \end {array}\right . \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.85 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = \left \{\begin {array}{cc} 0 & 0\le x <1 \\ \left (-1+x \right )^{2} & 1\le x \end {array}\right . \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.462 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \left \{\begin {array}{cc} 0 & 0\le x <1 \\ x^{2}-2 x +3 & 1\le x \end {array}\right . \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.473 |
|
|
\[ {}y^{\prime \prime }+4 y = \left \{\begin {array}{cc} 0 & 0\le x <\pi \\ -\sin \left (3 x \right ) & \pi \le x \end {array}\right . \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.03 |
|
|
\[ {}y^{\prime \prime }-4 y = \left \{\begin {array}{cc} x & 0\le x <1 \\ 1 & 1\le x \end {array}\right . \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.082 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = \left \{\begin {array}{cc} x & 0\le x <1 \\ 1 & 1\le x \end {array}\right . \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.207 |
|
|
\[ {}y^{\prime }+3 y = \delta \left (-2+x \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.819 |
|
|
\[ {}y^{\prime }-3 y = \delta \left (-1+x \right )+2 \operatorname {Heaviside}\left (-2+x \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.398 |
|
|
\[ {}y^{\prime \prime }+9 y = \delta \left (x -\pi \right )+\delta \left (x -3 \pi \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.817 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 \delta \left (-1+x \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.796 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = \cos \left (x \right )+2 \delta \left (x -\pi \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.654 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (x \right ) \delta \left (x -\pi \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.614 |
|
|
\[ {}y^{\prime \prime }+a^{2} y = \delta \left (x -\pi \right ) f \left (x \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.027 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=2 y_{1}-3 y_{2} \\ y_{2}^{\prime }=y_{1}-2 y_{2} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.328 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{1}-2 y_{2} \\ y_{2}^{\prime }=y_{1}+3 y_{2} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.47 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{1}+2 y_{2}+x -1 \\ y_{2}^{\prime }=3 y_{1}+2 y_{2}-5 x -2 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.6 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=\frac {2 y_{1}}{x}-\frac {y_{2}}{x^{2}}-3+\frac {1}{x}-\frac {1}{x^{2}} \\ y_{2}^{\prime }=2 y_{1}+1-6 x \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.026 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=\frac {5 y_{1}}{x}+\frac {4 y_{2}}{x}-2 x \\ y_{2}^{\prime }=-\frac {6 y_{1}}{x}-\frac {5 y_{2}}{x}+5 x \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.029 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=3 y_{1}-2 y_{2} \\ y_{2}^{\prime }=-y_{1}+y_{2} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.642 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=\sin \left (x \right ) y_{1}+\sqrt {x}\, y_{2}+\ln \left (x \right ) \\ y_{2}^{\prime }=\tan \left (x \right ) y_{1}-{\mathrm e}^{x} y_{2}+1 \end {array}\right ] \] |
1 |
0 |
0 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.028 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=\sin \left (x \right ) y_{1}+\sqrt {x}\, y_{2}+\ln \left (x \right ) \\ y_{2}^{\prime }=\tan \left (x \right ) y_{1}-{\mathrm e}^{x} y_{2}+1 \end {array}\right ] \] |
1 |
0 |
0 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.026 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }={\mathrm e}^{-x} y_{1}-\sqrt {1+x}\, y_{2}+x^{2} \\ y_{2}^{\prime }=\frac {y_{1}}{\left (-2+x \right )^{2}} \end {array}\right ] \] |
1 |
0 |
0 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.027 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }={\mathrm e}^{-x} y_{1}-\sqrt {1+x}\, y_{2}+x^{2} \\ y_{2}^{\prime }=\frac {y_{1}}{\left (-2+x \right )^{2}} \end {array}\right ] \] |
1 |
0 |
0 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.03 |
|
|
\(\left [\begin {array}{cc} -2 & -4 \\ 1 & 3 \end {array}\right ]\) |
|
|
|
Eigenvectors |
N/A |
✓ |
N/A |
0.125 |
|
|
\(\left [\begin {array}{cc} -3 & -1 \\ 2 & -1 \end {array}\right ]\) |
|
|
|
Eigenvectors |
N/A |
✓ |
N/A |
0.175 |
|
|
\(\left [\begin {array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -1 \\ -2 & 0 & -1 \end {array}\right ]\) |
|
|
|
Eigenvectors |
N/A |
✓ |
N/A |
0.257 |
|
|
\(\left [\begin {array}{ccc} 3 & 1 & -1 \\ 1 & 3 & -1 \\ 3 & 3 & -1 \end {array}\right ]\) |
|
|
|
Eigenvectors |
N/A |
✓ |
N/A |
0.166 |
|
|
\(\left [\begin {array}{ccc} 7 & -1 & 6 \\ -10 & 4 & -12 \\ -2 & 1 & -1 \end {array}\right ]\) |
|
|
|
Eigenvectors |
N/A |
✓ |
N/A |
0.223 |
|
|
\(\left [\begin {array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end {array}\right ]\) |
|
|
|
Eigenvectors |
N/A |
✓ |
N/A |
0.199 |
|
|
\(\left [\begin {array}{cccc} 1 & 3 & 5 & 7 \\ 2 & 6 & 10 & 14 \\ 3 & 9 & 15 & 21 \\ 6 & 18 & 30 & 42 \end {array}\right ]\) |
|
|
|
Eigenvectors |
N/A |
✓ |
N/A |
0.243 |
|
|
\(\left [\begin {array}{ccccc} 1 & 3 & 5 & 2 & 4 \\ 5 & 2 & 4 & 1 & 3 \\ 4 & 1 & 3 & 5 & 2 \\ 3 & 5 & 2 & 4 & 1 \\ 2 & 4 & 1 & 3 & 5 \end {array}\right ]\) |
|
|
|
Eigenvectors |
N/A |
✓ |
N/A |
6.331 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=2 y_{1}-3 y_{2}+5 \,{\mathrm e}^{x} \\ y_{2}^{\prime }=y_{1}+4 y_{2}-2 \,{\mathrm e}^{-x} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
2.425 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{2}-2 y_{1}+\sin \left (2 x \right ) \\ y_{2}^{\prime }=-3 y_{1}+y_{2}-2 \cos \left (3 x \right ) \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
6.501 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=2 y_{2} \\ y_{2}^{\prime }=3 y_{1} \\ y_{3}^{\prime }=2 y_{3}-y_{1} \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.733 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=2 x y_{1}-x^{2} y_{2}+4 x \\ y_{2}^{\prime }={\mathrm e}^{x} y_{1}+3 \,{\mathrm e}^{-x} y_{2}-\cos \left (3 x \right ) \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.03 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=2 y_{1}-3 y_{2} \\ y_{2}^{\prime }=y_{1}-2 y_{2} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.307 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=2 y_{1}-3 y_{2}+4 x -2 \\ y_{2}^{\prime }=y_{1}-2 y_{2}+3 x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.758 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=\frac {5 y_{1}}{x}+\frac {4 y_{2}}{x} \\ y_{2}^{\prime }=-\frac {6 y_{1}}{x}-\frac {5 y_{2}}{x} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.025 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=\frac {5 y_{1}}{x}+\frac {4 y_{2}}{x}-2 x \\ y_{2}^{\prime }=-\frac {6 y_{1}}{x}-\frac {5 y_{2}}{x}+5 x \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.026 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=2 y_{1}+y_{2}-2 y_{3} \\ y_{2}^{\prime }=3 y_{2}-2 y_{3} \\ y_{3}^{\prime }=3 y_{1}+y_{2}-3 y_{3} \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.497 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=5 y_{1}-5 y_{2}-5 y_{3} \\ y_{2}^{\prime }=-y_{1}+4 y_{2}+2 y_{3} \\ y_{3}^{\prime }=3 y_{1}-5 y_{2}-3 y_{3} \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.776 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=4 y_{1}+6 y_{2}+6 y_{3} \\ y_{2}^{\prime }=y_{1}+3 y_{2}+2 y_{3} \\ y_{3}^{\prime }=-y_{1}-4 y_{2}-3 y_{3} \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.554 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{1}+2 y_{2}-3 y_{3} \\ y_{2}^{\prime }=-3 y_{1}+4 y_{2}-2 y_{3} \\ y_{3}^{\prime }=2 y_{1}+y_{3} \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.921 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=-2 y_{1}-y_{2}+y_{3} \\ y_{2}^{\prime }=-y_{1}-2 y_{2}-y_{3} \\ y_{3}^{\prime }=y_{1}-y_{2}-2 y_{3} \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.466 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{1}+y_{2}+2 y_{3} \\ y_{2}^{\prime }=y_{1}+y_{2}+2 y_{3} \\ y_{3}^{\prime }=2 y_{1}+2 y_{2}+4 y_{3} \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.419 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=2 y_{1}+y_{2} \\ y_{2}^{\prime }=-y_{1}+2 y_{2} \\ y_{3}^{\prime }=3 y_{3}-4 y_{4} \\ y_{4}^{\prime }=4 y_{3}+3 y_{4} \end {array}\right ] \] |
1 |
1 |
4 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.944 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{2} \\ y_{2}^{\prime }=-3 y_{1}+2 y_{3} \\ y_{3}^{\prime }=y_{4} \\ y_{4}^{\prime }=2 y_{1}-5 y_{3} \end {array}\right ] \] |
1 |
1 |
4 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
4.61 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=3 y_{1}+2 y_{2} \\ y_{2}^{\prime }=-2 y_{1}+3 y_{2} \\ y_{3}^{\prime }=y_{3} \\ y_{4}^{\prime }=2 y_{4} \end {array}\right ] \] |
1 |
1 |
4 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.71 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{2}+y_{4} \\ y_{2}^{\prime }=y_{1}-y_{3} \\ y_{3}^{\prime }=y_{4} \\ y_{4}^{\prime }=y_{3} \end {array}\right ] \] |
1 |
1 |
4 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.611 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-2 x+3 y \\ y^{\prime }=-x+2 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.319 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-x+2 y \\ y^{\prime }=-2 x+3 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.344 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-x-2 y \\ y^{\prime }=2 x-3 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.873 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-x-2 y \\ y^{\prime }=5 x+y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.493 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-x+2 y \\ y^{\prime }=-2 x-y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.454 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x-2 y \\ y^{\prime }=2 x+y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.426 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-5 x-y+2 \\ y^{\prime }=3 x-y-3 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.754 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=3 x-2 y-6 \\ y^{\prime }=4 x-y+2 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.625 |
|
|
|
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