|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{1} \\ y_{2}^{\prime }=y_{1}+y_{2} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.253 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{2} \\ y_{2}^{\prime }=6 y_{1}+y_{2} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.39 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=y_{1}+y_{2} \\ y_{2}^{\prime }=y_{1}+y_{2}+{\mathrm e}^{3 x} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.546 |
|
|
\[ {}\left [\begin {array}{c} y_{1}^{\prime }=3 y_{1}+x y_{3} \\ y_{2}^{\prime }=y_{2}+x^{3} y_{3} \\ y_{3}^{\prime }=2 x y_{2}-y_{2}+{\mathrm e}^{x} y_{3} \end {array}\right ] \] |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.033 |
|
|
\[ {}y^{\prime } = 2 x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.141 |
|
|
\[ {}x y^{\prime } = 2 y \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.975 |
|
|
\[ {}y y^{\prime } = {\mathrm e}^{2 x} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.77 |
|
|
\[ {}y^{\prime } = k y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.658 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.396 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.219 |
|
|
\[ {}x y^{\prime }+y = y^{\prime } \sqrt {1-x^{2} y} \] |
unknown |
[_rational] |
❇ |
N/A |
16.67 |
|
|
\[ {}x y^{\prime } = y+x^{2}+y^{2} \] |
riccati, exactByInspection, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.287 |
|
|
\[ {}y^{\prime } = \frac {x y}{x^{2}+y^{2}} \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.279 |
|
|
\[ {}2 x y y^{\prime } = x^{2}+y^{2} \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.457 |
|
|
\[ {}x y^{\prime }+y = x^{4} {y^{\prime }}^{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
7.297 |
|
|
\[ {}y^{\prime } = \frac {y^{2}}{x y-x^{2}} \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.466 |
|
|
\[ {}\left (y \cos \left (y\right )-\sin \left (y\right )+x \right ) y^{\prime } = y \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.437 |
|
|
\[ {}1+y^{2}+y^{2} y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.329 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{3 x}-x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.183 |
|
|
\[ {}y^{\prime } = x \,{\mathrm e}^{x^{2}} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.168 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.198 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.191 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = \arctan \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.419 |
|
|
\[ {}x y^{\prime } = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.128 |
|
|
\[ {}y^{\prime } = \arcsin \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.172 |
|
|
\[ {}y^{\prime } \sin \left (x \right ) = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.266 |
|
|
\[ {}\left (x^{3}+1\right ) y^{\prime } = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.243 |
|
|
\[ {}\left (x^{2}-3 x +2\right ) y^{\prime } = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.362 |
|
|
\[ {}y^{\prime } = x \,{\mathrm e}^{x} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.275 |
|
|
\[ {}y^{\prime } = 2 \sin \left (x \right ) \cos \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.489 |
|
|
\[ {}y^{\prime } = \ln \left (x \right ) \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.3 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime } = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.385 |
|
|
\[ {}x \left (x^{2}-4\right ) y^{\prime } = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.425 |
|
|
\[ {}\left (1+x \right ) \left (x^{2}+1\right ) y^{\prime } = 2 x^{2}+x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.49 |
|
|
\[ {}y^{\prime } = 2 x y+1 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.773 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+4 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.26 |
|
|
\[ {}y^{\prime } = \frac {2 x y^{2}}{1-x^{2} y} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.313 |
|
|
\[ {}2 y^{\prime \prime \prime }+y^{\prime \prime }-5 y^{\prime }+2 y = 0 \] |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.25 |
|
|
\[ {}x^{5} y^{\prime }+y^{5} = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.279 |
|
|
\[ {}y^{\prime } = 4 x y \] |
separable |
[_separable] |
✓ |
✓ |
0.278 |
|
|
\[ {}y^{\prime }+\tan \left (x \right ) y = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.499 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+1+y^{2} = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.289 |
|
|
\[ {}y \ln \left (y\right )-x y^{\prime } = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.781 |
|
|
\[ {}x y^{\prime } = \left (-4 x^{2}+1\right ) \tan \left (y\right ) \] |
separable |
[_separable] |
✓ |
✓ |
0.354 |
|
|
\[ {}y^{\prime } \sin \left (y\right ) = x^{2} \] |
separable |
[_separable] |
✓ |
✓ |
0.329 |
|
|
\[ {}y^{\prime }-\tan \left (x \right ) y = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.445 |
|
|
\[ {}x y y^{\prime } = y-1 \] |
separable |
[_separable] |
✓ |
✓ |
0.381 |
|
|
\[ {}x y^{2}-x^{2} y^{\prime } = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.161 |
|
|
\[ {}y y^{\prime } = 1+x \] |
separable |
[_separable] |
✓ |
✓ |
0.855 |
|
|
\[ {}x^{2} y^{\prime } = y \] |
separable |
[_separable] |
✓ |
✓ |
0.589 |
|
|
\[ {}\frac {y^{\prime }}{x^{2}+1} = \frac {x}{y} \] |
separable |
[_separable] |
✓ |
✓ |
0.887 |
|
|
\[ {}y^{2} y^{\prime } = 2+x \] |
separable |
[_separable] |
✓ |
✓ |
1.717 |
|
|
\[ {}y^{\prime } = x^{2} y^{2} \] |
separable |
[_separable] |
✓ |
✓ |
0.424 |
|
|
\[ {}\left (y+1\right ) y^{\prime } = -x^{2}+1 \] |
separable |
[_separable] |
✓ |
✓ |
1.011 |
|
|
\[ {}\frac {y^{\prime \prime }}{y^{\prime }} = x^{2} \] |
second_order_integrable_as_is, second_order_ode_missing_y, exact nonlinear second order ode |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.948 |
|
|
\[ {}y^{\prime } y^{\prime \prime } = \left (1+x \right ) x \] |
second_order_integrable_as_is, second_order_ode_missing_y, exact nonlinear second order ode |
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
3.369 |
|
|
\[ {}y^{\prime }-x y = 0 \] |
linear |
[_separable] |
✓ |
✓ |
0.277 |
|
|
\[ {}y^{\prime }+x y = x \] |
linear |
[_separable] |
✓ |
✓ |
0.284 |
|
|
\[ {}y^{\prime }+y = \frac {1}{{\mathrm e}^{2 x}+1} \] |
linear |
[_linear] |
✓ |
✓ |
0.287 |
|
|
\[ {}y^{\prime }+y = 2 x \,{\mathrm e}^{-x}+x^{2} \] |
linear |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.249 |
|
|
\[ {}2 y-x^{3} = x y^{\prime } \] |
linear |
[_linear] |
✓ |
✓ |
0.21 |
|
|
\[ {}y^{\prime }+2 x y = 0 \] |
linear |
[_separable] |
✓ |
✓ |
0.267 |
|
|
\[ {}x y^{\prime }-3 y = x^{4} \] |
linear |
[_linear] |
✓ |
✓ |
0.2 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+2 x y = \cot \left (x \right ) \] |
linear |
[_linear] |
✓ |
✓ |
0.23 |
|
|
\[ {}y^{\prime }+y \cot \left (x \right ) = 2 x \csc \left (x \right ) \] |
linear |
[_linear] |
✓ |
✓ |
0.308 |
|
|
\[ {}y-x +x y \cot \left (x \right )+x y^{\prime } = 0 \] |
linear |
[_linear] |
✓ |
✓ |
0.405 |
|
|
\[ {}y^{\prime }-x y = 0 \] |
linear |
[_separable] |
✓ |
✓ |
0.468 |
|
|
\[ {}y^{\prime }-2 x y = 6 x \,{\mathrm e}^{x^{2}} \] |
linear |
[_linear] |
✓ |
✓ |
0.582 |
|
|
\[ {}x \ln \left (x \right ) y^{\prime }+y = 3 x^{3} \] |
linear |
[_linear] |
❇ |
N/A |
0.852 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = x^{2} \] |
linear |
[_linear] |
✓ |
✓ |
0.53 |
|
|
\[ {}y^{\prime }+4 y = {\mathrm e}^{-x} \] |
linear |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.481 |
|
|
\[ {}x^{2} y^{\prime }+x y = 2 x \] |
linear |
[_separable] |
✓ |
✓ |
0.595 |
|
|
\[ {}x y^{\prime }+y = x^{4} y^{3} \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.231 |
|
|
\[ {}x y^{2} y^{\prime }+y^{3} = x \cos \left (x \right ) \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
8.563 |
|
|
\[ {}x y^{\prime }+y = x y^{2} \] |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.415 |
|
|
\[ {}y^{\prime }+x y = y^{4} x \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
7.695 |
|
|
\[ {}\left ({\mathrm e}^{y}-2 x y\right ) y^{\prime } = y^{2} \] |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.072 |
|
|
\[ {}y-x y^{\prime } = y^{\prime } y^{2} {\mathrm e}^{y} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.62 |
|
|
\[ {}x y^{\prime }+2 = x^{3} \left (y-1\right ) y^{\prime } \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
5.012 |
|
|
\[ {}x y^{\prime } = 2 x^{2} y+y \ln \left (x \right ) \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.501 |
|
|
\[ {}y^{\prime } \sin \left (2 x \right ) = 2 y+2 \cos \left (x \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
4.145 |
|
|
\[ {}\left (x +\frac {2}{y}\right ) y^{\prime }+y = 0 \] |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.368 |
|
|
\[ {}\sin \left (x \right ) \tan \left (y\right )+1+\cos \left (x \right ) \sec \left (x \right )^{2} y y^{\prime } = 0 \] |
unknown |
[‘x=_G(y,y’)‘] |
❇ |
N/A |
45.073 |
|
|
\[ {}y-x^{3}+\left (x +y^{3}\right ) y^{\prime } = 0 \] |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
2.069 |
|
|
\[ {}2 y^{2}-4 x +5 = \left (4-2 y+4 x y\right ) y^{\prime } \] |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.807 |
|
|
\[ {}y+y \cos \left (x y\right )+\left (x +x \cos \left (x y\right )\right ) y^{\prime } = 0 \] |
exact, linear, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.158 |
|
|
\[ {}\cos \left (x \right ) \cos \left (y\right )^{2}+2 \sin \left (x \right ) \sin \left (y\right ) \cos \left (y\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
8.104 |
|
|
\[ {}\left (\sin \left (x \right ) \sin \left (y\right )-x \,{\mathrm e}^{y}\right ) y^{\prime } = {\mathrm e}^{y}+\cos \left (x \right ) \cos \left (y\right ) \] |
exact |
[_exact] |
✓ |
✓ |
15.332 |
|
|
\[ {}-\frac {\sin \left (\frac {x}{y}\right )}{y}+\frac {x \sin \left (\frac {x}{y}\right ) y^{\prime }}{y^{2}} = 0 \] |
exact |
[_separable] |
✓ |
✓ |
0.669 |
|
|
\[ {}1+y+\left (1-x \right ) y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.761 |
|
|
\[ {}2 x y^{3}+y \cos \left (x \right )+\left (3 x^{2} y^{2}+\sin \left (x \right )\right ) y^{\prime } = 0 \] |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
54.746 |
|
|
\[ {}\frac {y}{1-x^{2} y^{2}}+\frac {x y^{\prime }}{1-x^{2} y^{2}} = 1 \] |
exact, riccati |
[_exact, _rational, _Riccati] |
✓ |
✓ |
2.265 |
|
|
\[ {}2 y^{4} x +\sin \left (y\right )+\left (4 x^{2} y^{3}+x \cos \left (y\right )\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
4.274 |
|
|
\[ {}\frac {x y^{\prime }+y}{1-x^{2} y^{2}}+x = 0 \] |
exact, riccati |
[_exact, _rational, _Riccati] |
✓ |
✓ |
2.368 |
|
|
\[ {}2 x \left (1+\sqrt {x^{2}-y}\right ) = \sqrt {x^{2}-y}\, y^{\prime } \] |
exact, first_order_ode_lie_symmetry_calculated |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
7.158 |
|
|
\[ {}x \ln \left (y\right )+x y+\left (y \ln \left (x \right )+x y\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.919 |
|
|
\[ {}{\mathrm e}^{y^{2}}-\csc \left (y\right ) \csc \left (x \right )^{2}+\left (2 x y \,{\mathrm e}^{y^{2}}-\csc \left (y\right ) \cot \left (y\right ) \cot \left (x \right )\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
84.118 |
|
|
\[ {}1+y^{2} \sin \left (2 x \right )-2 y \cos \left (x \right )^{2} y^{\prime } = 0 \] |
exact |
[_exact, _Bernoulli] |
✓ |
✓ |
0.707 |
|
|
\[ {}\frac {x}{\left (x^{2}+y^{2}\right )^{\frac {3}{2}}}+\frac {y y^{\prime }}{\left (x^{2}+y^{2}\right )^{\frac {3}{2}}} = 0 \] |
exact |
[_separable] |
✓ |
✓ |
0.289 |
|
|
\[ {}3 x^{2} \left (1+\ln \left (y\right )\right )+\left (\frac {x^{3}}{y}-2 y\right ) y^{\prime } = 0 \] |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
3.157 |
|
|
|
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