|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime } = 3 y^{\frac {2}{3}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.273 |
|
|
\[ {}x \ln \left (x \right ) y^{\prime }-\left (1+\ln \left (x \right )\right ) y = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.98 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.462 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.498 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.596 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.645 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 x y^{\prime }-6 y = 0 \] |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.439 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
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 |
|
|
\[ {}y^{\prime } = 1-x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.184 |
|
|
\[ {}y^{\prime } = -1+x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.171 |
|
|
\[ {}y^{\prime } = 1-y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.281 |
|
|
\[ {}y^{\prime } = y+1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.254 |
|
|
\[ {}y^{\prime } = y^{2}-4 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.426 |
|
|
\[ {}y^{\prime } = 4-y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.404 |
|
|
\[ {}y^{\prime } = x y \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.642 |
|
|
\[ {}y^{\prime } = -x y \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.712 |
|
|
\[ {}y^{\prime } = x^{2}-y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
0.895 |
|
|
\[ {}y^{\prime } = -x^{2}+y^{2} \] |
riccati |
[_Riccati] |
✓ |
✓ |
0.799 |
|
|
\[ {}y^{\prime } = x +y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.621 |
|
|
\[ {}y^{\prime } = x y \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.557 |
|
|
\[ {}y^{\prime } = \frac {x}{y} \] |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.197 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.602 |
|
|
\[ {}y^{\prime } = 1+y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.211 |
|
|
\[ {}y^{\prime } = y^{2}-3 y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.434 |
|
|
\[ {}y^{\prime } = x^{3}+y^{3} \] |
abelFirstKind |
[_Abel] |
❇ |
N/A |
0.439 |
|
|
\[ {}y^{\prime } = {| y|} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x -y} \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.626 |
|
|
\[ {}y^{\prime } = \ln \left (x +y\right ) \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.801 |
|
|
\[ {}y^{\prime } = \frac {2 x -y}{x +3 y} \] |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.579 |
|
|
\[ {}y^{\prime } = \frac {1}{\sqrt {15-x^{2}-y^{2}}} \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
0.855 |
|
|
\[ {}y^{\prime } = \frac {3 y}{\left (x -5\right ) \left (x +3\right )}+{\mathrm e}^{-x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.183 |
|
|
\[ {}y^{\prime } = \frac {x y}{x^{2}+y^{2}} \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.0 |
|
|
\[ {}y^{\prime } = \frac {1}{x y} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.677 |
|
|
\[ {}y^{\prime } = \ln \left (y-1\right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.23 |
|
|
\[ {}y^{\prime } = \sqrt {\left (y+2\right ) \left (y-1\right )} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.705 |
|
|
\[ {}y^{\prime } = \frac {y}{y-x} \] |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.806 |
|
|
\[ {}y^{\prime } = \frac {x}{y^{2}} \] |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
5.332 |
|
|
\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.188 |
|
|
\[ {}y^{\prime } = \frac {x y}{1-y} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.788 |
|
|
\[ {}y^{\prime } = \left (x y\right )^{\frac {1}{3}} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
96.204 |
|
|
\[ {}y^{\prime } = \sqrt {\frac {y-4}{x}} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
3.463 |
|
|
\[ {}y^{\prime } = -\frac {y}{x}+y^{\frac {1}{4}} \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
2.401 |
|
|
\[ {}y^{\prime } = 4 y-5 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.545 |
|
|
\[ {}y^{\prime }+3 y = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.487 |
|
|
\[ {}y^{\prime } = a y+b \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.597 |
|
|
\[ {}y^{\prime } = x^{2}+{\mathrm e}^{x}-\sin \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.977 |
|
|
\[ {}y^{\prime } = x y+\frac {1}{x^{2}+1} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.97 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\cos \left (x \right ) \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.202 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\tan \left (x \right ) \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
5.545 |
|
|
\[ {}y^{\prime } = \frac {y}{-x^{2}+4}+\sqrt {x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.967 |
|
|
\[ {}y^{\prime } = \frac {y}{-x^{2}+4}+\sqrt {x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
7.509 |
|
|
\[ {}y^{\prime } = y \cot \left (x \right )+\csc \left (x \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.372 |
|
|
\[ {}y^{\prime } = -x \sqrt {1-y^{2}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.98 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.729 |
|
|
\[ {}y^{\prime } = 1+3 x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.296 |
|
|
\[ {}y^{\prime } = x +\frac {1}{x} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.382 |
|
|
\[ {}y^{\prime } = 2 \sin \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.422 |
|
|
\[ {}y^{\prime } = x \sin \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.49 |
|
|
\[ {}y^{\prime } = \frac {1}{-1+x} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.378 |
|
|
\[ {}y^{\prime } = \frac {1}{-1+x} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.293 |
|
|
\[ {}y^{\prime } = \frac {1}{x^{2}-1} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.386 |
|
|
\[ {}y^{\prime } = \frac {1}{x^{2}-1} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.332 |
|
|
\[ {}y^{\prime } = \tan \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.403 |
|
|
\[ {}y^{\prime } = \tan \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.325 |
|
|
\[ {}y^{\prime } = 3 y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.435 |
|
|
\[ {}y^{\prime } = 1-y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.327 |
|
|
\[ {}y^{\prime } = 1-y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.336 |
|
|
\[ {}y^{\prime } = x \,{\mathrm e}^{-x^{2}+y} \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.371 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.887 |
|
|
\[ {}y^{\prime } = \frac {2 x}{y} \] |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.724 |
|
|
\[ {}y^{\prime } = -2 y+y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.783 |
|
|
\[ {}y^{\prime } = x y+x \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.167 |
|
|
\[ {}x \,{\mathrm e}^{y}+y^{\prime } = 0 \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.982 |
|
|
\[ {}y-x^{2} y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.328 |
|
|
\[ {}2 y y^{\prime } = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.308 |
|
|
\[ {}2 x y y^{\prime }+y^{2} = -1 \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.467 |
|
|
\[ {}y^{\prime } = \frac {1-x y}{x^{2}} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.72 |
|
|
\[ {}y^{\prime } = -\frac {y \left (y+2 x \right )}{x \left (2 y+x \right )} \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.775 |
|
|
\[ {}y^{\prime } = \frac {y^{2}}{1-x y} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.053 |
|
|
\[ {}y^{\prime } = 4 y+1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.484 |
|
|
\[ {}y^{\prime } = x y+2 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.075 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.926 |
|
|
\[ {}y^{\prime } = \frac {y}{-1+x}+x^{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.954 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\sin \left (x^{2}\right ) \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.747 |
|
|
\[ {}y^{\prime } = \frac {2 y}{x}+{\mathrm e}^{x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
13.245 |
|
|
\[ {}y^{\prime } = y \cot \left (x \right )+\sin \left (x \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.187 |
|
|
\[ {}x -y y^{\prime } = 0 \] |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.227 |
|
|
\[ {}y-x y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.654 |
|
|
\[ {}x^{2}-y+x y^{\prime } = 0 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.721 |
|
|
\[ {}x y \left (1-y\right )-2 y^{\prime } = 0 \] |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.379 |
|
|
\[ {}x \left (1-y^{3}\right )-3 y^{2} y^{\prime } = 0 \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.635 |
|
|
\[ {}y \left (2 x -1\right )+x \left (1+x \right ) y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.972 |
|
|
\[ {}y^{\prime } = \frac {1}{-1+x} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.238 |
|
|
\[ {}y^{\prime } = x +y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.783 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.887 |
|
|
|
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