|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime \prime } = y \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.48 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.475 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.25 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.815 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y = 0 \] |
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_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.131 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.043 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.088 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.326 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.998 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.569 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.391 |
|
|
\[ {}y^{\prime }-\sin \left (x +y\right ) = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.845 |
|
|
\[ {}y^{\prime } = 4 y^{2}-3 y+1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.418 |
|
|
\[ {}s^{\prime } = t \ln \left (s^{2 t}\right )+8 t^{2} \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
0.575 |
|
|
\[ {}y^{\prime } = \frac {y \,{\mathrm e}^{x +y}}{x^{2}+2} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.726 |
|
|
\[ {}\left (x y^{2}+3 y^{2}\right ) y^{\prime }-2 x = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.747 |
|
|
\[ {}s^{2}+s^{\prime } = \frac {s+1}{s t} \] |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
❇ |
N/A |
0.42 |
|
|
\[ {}x y^{\prime } = \frac {1}{y^{3}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.439 |
|
|
\[ {}x^{\prime } = 3 x t^{2} \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.547 |
|
|
\[ {}x^{\prime } = \frac {t \,{\mathrm e}^{-t -2 x}}{x} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.743 |
|
|
\[ {}y^{\prime } = \frac {x}{y^{2} \sqrt {1+x}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.605 |
|
|
\[ {}x v^{\prime } = \frac {1-4 v^{2}}{3 v} \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.539 |
|
|
\[ {}y^{\prime } = \frac {\sec \left (y\right )^{2}}{x^{2}+1} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
82.737 |
|
|
\[ {}y^{\prime } = 3 x^{2} \left (1+y^{2}\right )^{\frac {3}{2}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.89 |
|
|
\[ {}x^{\prime }-x^{3} = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.964 |
|
|
\[ {}x +x y^{2}+{\mathrm e}^{x^{2}} y y^{\prime } = 0 \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.51 |
|
|
\[ {}\frac {y^{\prime }}{y}+y \,{\mathrm e}^{\cos \left (x \right )} \sin \left (x \right ) = 0 \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.726 |
|
|
\[ {}y^{\prime } = \left (1+y^{2}\right ) \tan \left (x \right ) \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.17 |
|
|
\[ {}y^{\prime } = x^{3} \left (1-y\right ) \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.085 |
|
|
\[ {}\frac {y^{\prime }}{2} = \sqrt {y+1}\, \cos \left (x \right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.295 |
|
|
\[ {}x^{2} y^{\prime } = \frac {4 x^{2}-x -2}{\left (1+x \right ) \left (y+1\right )} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
10.971 |
|
|
\[ {}\frac {y^{\prime }}{\theta } = \frac {y \sin \left (\theta \right )}{y^{2}+1} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.456 |
|
|
\[ {}x^{2}+2 y y^{\prime } = 0 \] |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.619 |
|
|
\[ {}y^{\prime } = 2 t \cos \left (y\right )^{2} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.257 |
|
|
\[ {}y^{\prime } = 8 x^{3} {\mathrm e}^{-2 y} \] |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.196 |
|
|
\[ {}y^{\prime } = x^{2} \left (y+1\right ) \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.071 |
|
|
\[ {}\sqrt {y}+\left (1+x \right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.5 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x^{2}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.31 |
|
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{x^{2}}}{y^{2}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.085 |
|
|
\[ {}y^{\prime } = \sqrt {\sin \left (x \right )+1}\, \left (1+y^{2}\right ) \] |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
76.019 |
|
|
\[ {}y^{\prime } = 2 y-2 t y \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.283 |
|
|
\[ {}y^{\prime } = y^{\frac {1}{3}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.207 |
|
|
\[ {}y^{\prime } = y^{\frac {1}{3}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.249 |
|
|
\[ {}y^{\prime } = \left (x -3\right ) \left (y+1\right )^{\frac {2}{3}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.638 |
|
|
\[ {}y^{\prime } = x y^{3} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.57 |
|
|
\[ {}y^{\prime } = x y^{3} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.247 |
|
|
\[ {}y^{\prime } = x y^{3} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.102 |
|
|
\[ {}y^{\prime } = x y^{3} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.092 |
|
|
\[ {}y^{\prime } = y^{2}-3 y+2 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.556 |
|
|
\[ {}x^{2} y^{\prime }+\sin \left (x \right )-y = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.517 |
|
|
\[ {}x^{\prime }+x t = {\mathrm e}^{x} \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
0.414 |
|
|
\[ {}\left (t^{2}+1\right ) y^{\prime } = t y-y \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.053 |
|
|
\[ {}3 t = {\mathrm e}^{t} y^{\prime }+y \ln \left (t \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
4.544 |
|
|
\[ {}x x^{\prime }+x t^{2} = \sin \left (t \right ) \] |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.865 |
|
|
\[ {}3 r = r^{\prime }-\theta ^{3} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.584 |
|
|
\[ {}y^{\prime }-y-{\mathrm e}^{3 x} = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.553 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+2 x +1 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.583 |
|
|
\[ {}r^{\prime }+r \tan \left (\theta \right ) = \sec \left (\theta \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.714 |
|
|
\[ {}x y^{\prime }+2 y = \frac {1}{x^{3}} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.576 |
|
|
\[ {}t +y+1-y^{\prime } = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.528 |
|
|
\[ {}y^{\prime } = x^{2} {\mathrm e}^{-4 x}-4 y \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.579 |
|
|
\[ {}y y^{\prime }+2 x = 5 y^{3} \] |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
❇ |
N/A |
0.389 |
|
|
\[ {}x y^{\prime }+3 y+3 x^{2} = \frac {\sin \left (x \right )}{x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.75 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+x y-x = 0 \] |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.967 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }-x^{2} y = \left (1+x \right ) \sqrt {-x^{2}+1} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.342 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = x \,{\mathrm e}^{x} \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.122 |
|
|
\[ {}y^{\prime }+4 y-{\mathrm e}^{-x} = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.069 |
|
|
\[ {}t^{2} x^{\prime }+3 x t = t^{4} \ln \left (t \right )+1 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.249 |
|
|
\[ {}y^{\prime }+\frac {3 y}{x}+2 = 3 x \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.112 |
|
|
\[ {}\cos \left (x \right ) y^{\prime }+y \sin \left (x \right ) = 2 x \cos \left (x \right )^{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.098 |
|
|
\[ {}y^{\prime } \sin \left (x \right )+y \cos \left (x \right ) = x \sin \left (x \right ) \] |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.739 |
|
|
\[ {}y^{\prime }+y \sqrt {1+\sin \left (x \right )^{2}} = x \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
19.125 |
|
|
\[ {}\left ({\mathrm e}^{4 y}+2 x \right ) y^{\prime }-1 = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
✓ |
4.855 |
|
|
\[ {}y^{\prime }+2 y = \frac {x}{y^{2}} \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.266 |
|
|
\[ {}y^{\prime }+\frac {3 y}{x} = x^{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.552 |
|
|
\[ {}x^{\prime } = \alpha -\beta \cos \left (\frac {\pi t}{12}\right )-k x \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.755 |
|
|
\[ {}u^{\prime } = \alpha \left (1-u\right )-\beta u \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.375 |
|
|
\[ {}x^{2} y+x^{4} \cos \left (x \right )-x^{3} y^{\prime } = 0 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.118 |
|
|
\[ {}x^{\frac {10}{3}}-2 y+x y^{\prime } = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.59 |
|
|
\[ {}\sqrt {-2 y-y^{2}}+\left (-x^{2}+2 x +3\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.96 |
|
|
\[ {}y \,{\mathrm e}^{x y}+2 x +\left (x \,{\mathrm e}^{x y}-2 y\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
0.312 |
|
|
\[ {}y^{\prime }+x y = 0 \] |
exact |
[_separable] |
✓ |
✓ |
0.279 |
|
|
\[ {}y^{2}+\left (2 x y+\cos \left (y\right )\right ) y^{\prime } = 0 \] |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
0.283 |
|
|
\[ {}2 x +y \cos \left (x y\right )+\left (x \cos \left (x y\right )-2 y\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
0.399 |
|
|
\[ {}\theta r^{\prime }+3 r-\theta -1 = 0 \] |
exact |
[_linear] |
✓ |
✓ |
0.266 |
|
|
\[ {}2 x y+3+\left (x^{2}-1\right ) y^{\prime } = 0 \] |
exact |
[_linear] |
✓ |
✓ |
0.26 |
|
|
\[ {}2 x +y+\left (x -2 y\right ) y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.24 |
|
|
\[ {}{\mathrm e}^{x} \sin \left (y\right )-3 x^{2}+\left ({\mathrm e}^{x} \cos \left (y\right )+\frac {1}{3 y^{\frac {2}{3}}}\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
0.556 |
|
|
\[ {}\cos \left (x \right ) \cos \left (y\right )+2 x -\left (\sin \left (x \right ) \sin \left (y\right )+2 y\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
7.503 |
|
|
\[ {}{\mathrm e}^{t} \left (-t +y\right )+\left (1+{\mathrm e}^{t}\right ) y^{\prime } = 0 \] |
exact |
[_linear] |
✓ |
✓ |
0.265 |
|
|
\[ {}\frac {t y^{\prime }}{y}+1+\ln \left (y\right ) = 0 \] |
exact |
[_separable] |
✓ |
✓ |
0.477 |
|
|
\[ {}\cos \left (\theta \right ) r^{\prime }-r \sin \left (\theta \right )+{\mathrm e}^{\theta } = 0 \] |
exact |
[_linear] |
✓ |
✓ |
0.293 |
|
|
\[ {}y \,{\mathrm e}^{x y}-\frac {1}{y}+\left (x \,{\mathrm e}^{x y}+\frac {x}{y^{2}}\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
0.319 |
|
|
\[ {}\frac {1}{y}-\left (3 y-\frac {x}{y^{2}}\right ) y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.204 |
|
|
\[ {}2 x +y^{2}-\cos \left (x +y\right )-\left (2 x y-\cos \left (x +y\right )-{\mathrm e}^{y}\right ) y^{\prime } = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
57.155 |
|
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{x +y}}{y-1} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.007 |
|
|
\[ {}y^{\prime }-4 y = 32 x^{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.918 |
|
|
\[ {}\left (x^{2}-\frac {2}{y^{3}}\right ) y^{\prime }+2 x y-3 x^{2} = 0 \] |
exact |
[_exact, _rational] |
✓ |
✓ |
1.542 |
|
|
\[ {}y^{\prime }+\frac {3 y}{x} = x^{2}-4 x +3 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.97 |
|
|
\[ {}2 x y^{3}-\left (-x^{2}+1\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.431 |
|
|
|
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