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) |
|
\[ {}x y^{\prime } = 2 y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.257 |
|
|
\[ {}y y^{\prime }+x = 0 \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.714 |
|
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{4} \] |
4 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.519 |
|
|
\[ {}2 x^{3} y^{\prime } = y \left (y^{2}+3 x^{2}\right ) \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.546 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.271 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.739 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.613 |
|
|
\[ {}y^{\prime \prime }-y = 4-x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.348 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.238 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 2 \,{\mathrm e}^{x} \left (1-x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.421 |
|
|
\[ {}4 y+x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.306 |
|
|
\[ {}1+2 y+\left (-x^{2}+4\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.143 |
|
|
\[ {}y^{2}-x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.185 |
|
|
\[ {}1+y-\left (1+x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.408 |
|
|
\[ {}x y^{2}+y+\left (x^{2} y-x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.543 |
|
|
\[ {}x \sin \left (\frac {y}{x}\right )-y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.992 |
|
|
\[ {}y^{2} \left (x^{2}+2\right )+\left (x^{3}+y^{3}\right ) \left (y-x y^{\prime }\right ) = 0 \] |
1 |
1 |
3 |
homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational] |
✓ |
✓ |
1.699 |
|
|
\[ {}y \sqrt {x^{2}+y^{2}}-x \left (x +\sqrt {x^{2}+y^{2}}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
✓ |
4.592 |
|
|
\[ {}x +y+1+\left (2 x +2 y+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.803 |
|
|
\[ {}1+2 y-\left (4-x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.612 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.198 |
|
|
\[ {}2 x y+\left (2 x +3 y\right ) y^{\prime } = 0 \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.399 |
|
|
\[ {}2 x y^{\prime }-2 y = \sqrt {x^{2}+4 y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.997 |
|
|
\[ {}3 y-7 x +7+\left (7 y-3 x +3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.744 |
|
|
\[ {}x y y^{\prime } = \left (y+1\right ) \left (1-x \right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.465 |
|
|
\[ {}y^{2}-x^{2}+x y y^{\prime } = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.685 |
|
|
\[ {}y \left (2 x y+1\right )+x \left (1-x y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.641 |
|
|
\[ {}1+\left (-x^{2}+1\right ) \cot \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.492 |
|
|
\[ {}x^{3}+y^{3}+3 x y^{2} y^{\prime } = 0 \] |
1 |
1 |
3 |
exact, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
2.03 |
|
|
\[ {}3 x +2 y+1-\left (3 x +2 y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.863 |
|
|
\[ {}x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.536 |
|
|
\[ {}x^{2}+y^{2}+x y y^{\prime } = 0 \] |
1 |
1 |
1 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
2.356 |
|
|
\[ {}\cos \left (y\right )+\left (1+{\mathrm e}^{-x}\right ) \sin \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.595 |
|
|
\[ {}y^{2}+x y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
28.431 |
|
|
\[ {}y^{\prime } = -2 \left (2 x +3 y\right )^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.867 |
|
|
\[ {}x -2 \sin \left (y\right )+3+\left (2 x -4 \sin \left (y\right )-3\right ) \cos \left (y\right ) y^{\prime } = 0 \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
3.741 |
|
|
\[ {}x^{2}-y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_linear] |
✓ |
✓ |
0.237 |
|
|
\[ {}x^{2}+y^{2}+2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.259 |
|
|
\[ {}x +y \cos \left (x \right )+\sin \left (x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_linear] |
✓ |
✓ |
0.286 |
|
|
\[ {}2 x +3 y+4+\left (3 x +4 y+5\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.289 |
|
|
\[ {}4 x^{3} y^{3}+\frac {1}{x}+\left (3 x^{4} y^{2}-\frac {1}{y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
✓ |
0.796 |
|
|
\[ {}2 u^{2}+2 u v+\left (u^{2}+v^{2}\right ) v^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
0.323 |
|
|
\[ {}x \sqrt {x^{2}+y^{2}}-y+\left (y \sqrt {x^{2}+y^{2}}-x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.389 |
|
|
\[ {}x +y+1-\left (y-x +3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.283 |
|
|
\[ {}y^{2}-\frac {y}{x \left (x +y\right )}+2+\left (\frac {1}{x +y}+2 \left (1+x \right ) y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, _rational] |
✓ |
✓ |
0.455 |
|
|
\[ {}2 x y \,{\mathrm e}^{x^{2} y}+y^{2} {\mathrm e}^{x y^{2}}+1+\left (x^{2} {\mathrm e}^{x^{2} y}+2 x y \,{\mathrm e}^{x y^{2}}-2 y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.46 |
|
|
\[ {}y \left (x -2 y\right )-x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.345 |
|
|
\[ {}x^{2}+y^{2}+x y y^{\prime } = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.536 |
|
|
\[ {}x^{2}+y^{2}+2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
1.734 |
|
|
\[ {}1-\sqrt {a^{2}-x^{2}}\, y^{\prime } = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.491 |
|
|
\[ {}x +y+1-\left (x -y-3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.921 |
|
|
\[ {}x -x^{2}-y^{2}+y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.274 |
|
|
\[ {}2 y-3 x +x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_linear] |
✓ |
✓ |
0.249 |
|
|
\[ {}x -y^{2}+2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.256 |
|
|
\[ {}-y-3 x^{2} \left (x^{2}+y^{2}\right )+x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
0.334 |
|
|
\[ {}y-\ln \left (x \right )-x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_linear] |
✓ |
✓ |
0.296 |
|
|
\[ {}3 x^{2}+y^{2}-2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.231 |
|
|
\[ {}x y-2 y^{2}-\left (x^{2}-3 x y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.441 |
|
|
\[ {}x +y-\left (x -y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.286 |
|
|
\[ {}2 y-3 x y^{2}-x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
10.922 |
|
|
\[ {}y+x \left (x^{2} y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
9 |
exact |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.292 |
|
|
\[ {}y+x^{3} y+2 x^{2}+\left (x +4 y^{4} x +8 y^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_rational] |
✓ |
✓ |
0.365 |
|
|
\[ {}-y-x^{2} {\mathrm e}^{x}+x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_linear] |
✓ |
✓ |
0.295 |
|
|
\[ {}1+y^{2} = \left (x^{2}+x \right ) y^{\prime } \] |
1 |
1 |
1 |
exact |
[_separable] |
✓ |
✓ |
0.252 |
|
|
\[ {}2 y-x^{3}+x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_linear] |
✓ |
✓ |
0.26 |
|
|
\[ {}y+\left (y^{2}-x \right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.217 |
|
|
\[ {}3 y^{3}-x y-\left (x^{2}+6 x y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.429 |
|
|
\[ {}3 x^{2} y^{2}+4 \left (x^{3} y-3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.261 |
|
|
\[ {}y \left (x +y\right )-x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.237 |
|
|
\[ {}2 y+3 x y^{2}+\left (x +2 x^{2} y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.253 |
|
|
\[ {}y \left (y^{2}-2 x^{2}\right )+x \left (2 y^{2}-x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.372 |
|
|
\[ {}-y+x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_separable] |
✓ |
✓ |
0.264 |
|
|
\[ {}y^{\prime }+y = 2 x +2 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.349 |
|
|
\[ {}y^{\prime }-y = x y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.299 |
|
|
\[ {}-3 y-\left (-2+x \right ) {\mathrm e}^{x}+x y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.332 |
|
|
\[ {}i^{\prime }-6 i = 10 \sin \left (2 t \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.397 |
|
|
\[ {}y^{\prime }+y = y^{2} {\mathrm e}^{x} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_1st_order, _with_linear_symmetries], _Bernoulli] |
✓ |
✓ |
0.582 |
|
|
\[ {}y+\left (x y+x -3 y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.443 |
|
|
\[ {}\left (2 s-{\mathrm e}^{2 t}\right ) s^{\prime } = 2 s \,{\mathrm e}^{2 t}-2 \cos \left (2 t \right ) \] |
1 |
1 |
2 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
5.523 |
|
|
\[ {}x y^{\prime }+y-x^{3} y^{6} = 0 \] |
1 |
1 |
5 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
2.846 |
|
|
\[ {}r^{\prime }+2 r \cos \left (\theta \right )+\sin \left (2 \theta \right ) = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.566 |
|
|
\[ {}y \left (1+y^{2}\right ) = 2 \left (1-2 x y^{2}\right ) y^{\prime } \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.914 |
|
|
\[ {}y y^{\prime }-x y^{2}+x = 0 \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.288 |
|
|
\[ {}\left (x -x \sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y = 0 \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.86 |
|
|
\[ {}2 x^{\prime }-\frac {x}{y}+x^{3} \cos \left (y \right ) = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
4.125 |
|
|
\[ {}x y^{\prime } = y \left (1-x \tan \left (x \right )\right )+x^{2} \cos \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
15.706 |
|
|
\[ {}2+y^{2}-\left (x y+2 y+y^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
6.585 |
|
|
\[ {}1+y^{2} = \left (\arctan \left (y\right )-x \right ) y^{\prime } \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
10.002 |
|
|
\[ {}2 x y^{5}-y+2 x y^{\prime } = 0 \] |
1 |
4 |
4 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.774 |
|
|
\[ {}1+\sin \left (y\right ) = \left (2 y \cos \left (y\right )-x \left (\sec \left (y\right )+\tan \left (y\right )\right )\right ) y^{\prime } \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
31.581 |
|
|
\[ {}x y^{\prime } = 2 y+x^{3} {\mathrm e}^{x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.503 |
|
|
\[ {}L i^{\prime }+R i = E \sin \left (2 t \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.95 |
|
|
\[ {}x^{2} \cos \left (y\right ) y^{\prime } = 2 x \sin \left (y\right )-1 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
2.265 |
|
|
\[ {}4 x^{2} y y^{\prime } = 3 x \left (3 y^{2}+2\right )+2 \left (3 y^{2}+2\right )^{3} \] |
1 |
0 |
4 |
unknown |
[_rational] |
✗ |
N/A |
1.793 |
|
|
\[ {}x y^{3}-y^{3}-x^{2} {\mathrm e}^{x}+3 x y^{2} y^{\prime } = 0 \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.971 |
|
|
\[ {}y^{\prime }+x \left (x +y\right ) = x^{3} \left (x +y\right )^{3}-1 \] |
1 |
1 |
2 |
abelFirstKind |
[_Abel] |
✓ |
✓ |
3.294 |
|
|
\[ {}y+{\mathrm e}^{y}-{\mathrm e}^{-x}+\left (1+{\mathrm e}^{y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
2.033 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+x y y^{\prime }-6 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.877 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (-x^{2}+y-1\right ) y^{\prime }-x \left (y-1\right ) = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.586 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+4 x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.341 |
|
|
\[ {}3 x^{4} {y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.763 |
|
|
\[ {}8 y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
5 |
5 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.699 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
5.545 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.297 |
|
|
\[ {}16 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
7.541 |
|
|
\[ {}x {y^{\prime }}^{5}-y {y^{\prime }}^{4}+\left (x^{2}+1\right ) {y^{\prime }}^{3}-2 x y {y^{\prime }}^{2}+\left (x +y^{2}\right ) y^{\prime }-y = 0 \] |
5 |
3 |
6 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.808 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.482 |
|
|
\[ {}y = 2 x y^{\prime }+y^{2} {y^{\prime }}^{3} \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
94.018 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.438 |
|
|
\[ {}y = \left (1+y^{\prime }\right ) x +{y^{\prime }}^{2} \] |
2 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.404 |
|
|
\[ {}y = 2 y^{\prime }+\sqrt {1+{y^{\prime }}^{2}} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
16.452 |
|
|
\[ {}y {y^{\prime }}^{2}-x y^{\prime }+3 y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.709 |
|
|
\[ {}y = x y^{\prime }-2 {y^{\prime }}^{2} \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.317 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
4.674 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+4 x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.368 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.423 |
|
|
\[ {}\left (3 y-1\right )^{2} {y^{\prime }}^{2} = 4 y \] |
2 |
2 |
7 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.476 |
|
|
\[ {}y = -x y^{\prime }+x^{4} {y^{\prime }}^{2} \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.43 |
|
|
\[ {}2 y = {y^{\prime }}^{2}+4 x y^{\prime } \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.429 |
|
|
\[ {}y \left (3-4 y\right )^{2} {y^{\prime }}^{2} = 4-4 y \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.737 |
|
|
\[ {}{y^{\prime }}^{3}-4 x^{4} y^{\prime }+8 x^{3} y = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
117.097 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \left (x -y\right )^{2} = \left (y y^{\prime }+x \right )^{2} \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
4.073 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.237 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.125 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = {\mathrm e}^{5 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.368 |
|
|
\[ {}y^{\prime \prime }+9 y = x \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.829 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.108 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+x y^{\prime }-y = 3 x^{4} \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.309 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.246 |
|
|
\[ {}y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.531 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 2 \] |
1 |
1 |
2 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.636 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{3} = 0 \] |
1 |
1 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-15 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.266 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }-2 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.217 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.273 |
|
|
\[ {}y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+12 y^{\prime \prime }-8 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.14 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.436 |
|
|
\[ {}y^{\prime \prime }+25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.001 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+9 y^{\prime }-9 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.302 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.297 |
|
|
\[ {}y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+13 y^{\prime \prime }-12 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.326 |
|
|
\[ {}y^{\left (6\right )}+9 y^{\prime \prime \prime \prime }+24 y^{\prime \prime }+16 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.181 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 1 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.358 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime } = 5 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.078 |
|
|
\[ {}y^{\prime \prime \prime }-4 y^{\prime \prime } = 5 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.161 |
|
|
\[ {}y^{\left (5\right )}-4 y^{\prime \prime \prime } = 5 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.197 |
|
|
\[ {}y^{\prime \prime \prime }-4 y^{\prime } = x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.474 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.455 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = -2 x^{2}+2 x +2 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.382 |
|
|
\[ {}y^{\prime \prime }-y = 4 x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.445 |
|
|
\[ {}y^{\prime \prime }-y = \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.66 |
|
|
\[ {}y^{\prime \prime }-y = \frac {1}{\left (1+{\mathrm e}^{-x}\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.531 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.524 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \sin \left ({\mathrm e}^{-x}\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.575 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }+4 y = 4 \sec \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.807 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = \frac {1}{1+{\mathrm e}^{-x}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.536 |
|
|
\[ {}y^{\prime \prime }-y = {\mathrm e}^{-x} \sin \left ({\mathrm e}^{-x}\right )+\cos \left ({\mathrm e}^{-x}\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.073 |
|
|
\[ {}y^{\prime \prime }-y = \frac {1}{\left (1+{\mathrm e}^{-x}\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.463 |
|
|
\[ {}y^{\prime \prime }+2 y = 2+{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.614 |
|
|
\[ {}y^{\prime \prime }-y = {\mathrm e}^{x} \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.612 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = x^{2}+\sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.691 |
|
|
\[ {}y^{\prime \prime }-9 y = x +{\mathrm e}^{2 x}-\sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.812 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }+2 y^{\prime } = x^{2}+4 x +8 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.454 |
|
|
\[ {}y^{\prime \prime }+y = -2 \sin \left (x \right )+4 x \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.802 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }-4 y^{\prime }+4 y = 2 x^{2}-4 x -1+2 \,{\mathrm e}^{2 x} x^{2}+5 \,{\mathrm e}^{2 x} x +{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.691 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = {\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{-2 x}+5 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.391 |
|
|
\[ {}y^{\prime \prime }-y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.391 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = {\mathrm e}^{x}+{\mathrm e}^{2 x} x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.552 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y = \sin \left (2 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.01 |
|
|
\[ {}y^{\prime \prime \prime }+y = \cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.415 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.611 |
|
|
\[ {}y^{\prime \prime }+5 y = \cos \left (\sqrt {5}\, x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.73 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y = {\mathrm e}^{x}+{\mathrm e}^{-x}+\sin \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.429 |
|
|
\[ {}y^{\prime \prime }-y = x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.378 |
|
|
\[ {}y^{\prime \prime }+2 y = x^{3}+x^{2}+{\mathrm e}^{-2 x}+\cos \left (3 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.331 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-y = {\mathrm e}^{x} \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.578 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \frac {{\mathrm e}^{2 x}}{x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.534 |
|
|
\[ {}y^{\prime \prime }-y = x \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.404 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = {\mathrm e}^{-2 x} \sec \left (x \right )^{2} \left (1+2 \tan \left (x \right )\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.136 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = x +\ln \left (x \right ) x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.908 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = \ln \left (x \right )^{2}-\ln \left (x^{2}\right ) \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
11.81 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime } = x +\sin \left (\ln \left (x \right )\right ) \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.463 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+x y^{\prime }-y = 3 x^{4} \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.336 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }-y = \ln \left (1+x \right )^{2}+x -1 \] |
1 |
1 |
1 |
kovacic, 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_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.375 |
|
|
\[ {}\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }-12 y = 6 x \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.28 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[_Laguerre] |
✓ |
✓ |
1.691 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 2 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.546 |
|
|
\[ {}\left (x^{2}+4\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 8 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.884 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+\left (2+x \right ) y = \left (x^{2}+2 x +1\right ) {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.653 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-10 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.606 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2 x +3\right ) y^{\prime }+\left (x^{2}+3 x +3\right ) y = \left (-x^{2}+6\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.617 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right )^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.719 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-4 x^{2}+x \right ) y^{\prime }+\left (4 x^{2}-2 x +1\right ) y = \left (x^{2}-x +1\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.564 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.25 |
|
|
\[ {}x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+y = \frac {1+x}{x} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.923 |
|
|
\[ {}x^{8} y^{\prime \prime }+4 x^{7} y^{\prime }+y = \frac {1}{x^{3}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.064 |
|
|
\[ {}\left (x \sin \left (x \right )+\cos \left (x \right )\right ) y^{\prime \prime }-x \cos \left (x \right ) y^{\prime }+y \cos \left (x \right ) = x \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.556 |
|
|
\[ {}x y^{\prime \prime }-3 y^{\prime }+\frac {3 y}{x} = 2+x \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.818 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (3 x +4\right ) y^{\prime }+3 y = \left (3 x +2\right ) {\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.077 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (9 x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 x y = 4 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.108 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = \frac {-x^{2}+1}{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.576 |
|
|
\[ {}y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.191 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime } = \frac {2}{x^{3}} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.133 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = -\frac {2}{x}-\ln \left (x \right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.816 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime } = x^{2} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.174 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{3} = 0 \] |
1 |
1 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.63 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 0 \] |
1 |
2 |
3 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.932 |
|
|
\[ {}y y^{\prime \prime } = {y^{\prime }}^{2} \left (1-\cos \left (y\right ) y^{\prime }+y y^{\prime } \sin \left (y\right )\right ) \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
3.074 |
|
|
\[ {}\left (2 x -3\right ) y^{\prime \prime \prime }-\left (6 x -7\right ) y^{\prime \prime }+4 x y^{\prime }-4 y = 8 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.174 |
|
|
\[ {}\left (2 x^{3}-1\right ) y^{\prime \prime \prime }-6 x^{2} y^{\prime \prime }+6 x y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
3.004 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{2} \ln \left (y\right ) \] |
1 |
2 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.854 |
|
|
\[ {}\left (2 y+x \right ) y^{\prime \prime }+2 {y^{\prime }}^{2}+2 y^{\prime } = 2 \] |
1 |
1 |
2 |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.543 |
|
|
\[ {}\left (1+2 y+3 y^{2}\right ) y^{\prime \prime \prime }+6 y^{\prime } \left (y^{\prime \prime }+{y^{\prime }}^{2}+3 y y^{\prime \prime }\right ) = x \] |
1 |
1 |
3 |
unknown |
[[_3rd_order, _exact, _nonlinear]] |
✗ |
N/A |
0.0 |
|
|
\[ {}3 x \left (y^{2} y^{\prime \prime \prime }+6 y y^{\prime } y^{\prime \prime }+2 {y^{\prime }}^{3}\right )-3 y \left (y y^{\prime \prime }+2 {y^{\prime }}^{2}\right ) = -\frac {2}{x} \] |
1 |
1 |
3 |
unknown |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y y^{\prime \prime \prime }+3 y^{\prime } y^{\prime \prime }-2 y y^{\prime \prime }-2 {y^{\prime }}^{2}+y y^{\prime } = {\mathrm e}^{2 x} \] |
1 |
1 |
2 |
unknown |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}2 \left (y+1\right ) y^{\prime \prime }+2 {y^{\prime }}^{2}+y^{2}+2 y = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.474 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-y^{\prime }+y=-{\mathrm e}^{t} \\ x+y^{\prime }-y={\mathrm e}^{2 t} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.679 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+2 x+y^{\prime }+y=t \\ 5 x+y^{\prime }+3 y=t^{2} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.152 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+x+2 y^{\prime }+7 y={\mathrm e}^{t}+2 \\ -2 x+y^{\prime }+3 y={\mathrm e}^{t}-1 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.399 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-x+y^{\prime }+3 y={\mathrm e}^{-t}-1 \\ x^{\prime }+2 x+y^{\prime }+3 y={\mathrm e}^{2 t}+1 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.407 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-x+y^{\prime }+2 y=1+{\mathrm e}^{t} \\ y^{\prime }+2 y+z^{\prime }+z={\mathrm e}^{t}+2 \\ x^{\prime }-x+z^{\prime }+z=3+{\mathrm e}^{t} \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.734 |
|
|
\[ {}\left (1-x \right ) y^{\prime } = x^{2}-y \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
0.369 |
|
|
\[ {}x y^{\prime } = 1-x +2 y \] |
1 |
1 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
0.432 |
|
|
\[ {}x y^{\prime } = 1-x +2 y \] |
1 |
1 |
1 |
linear, homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.968 |
|
|
\[ {}y^{\prime } = 2 x^{2}+3 y \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.52 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = x^{2}-2 x +y \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
0.385 |
|
|
\[ {}y^{\prime \prime }+x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.401 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.416 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+x^{2} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.938 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+p \left (p +1\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[_Gegenbauer] |
✓ |
✓ |
0.976 |
|
|
\[ {}y^{\prime \prime }+x^{2} y = x^{2}+x +1 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.563 |
|
|
\[ {}2 \left (x^{3}+x^{2}\right ) y^{\prime \prime }-\left (-3 x^{2}+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.044 |
|
|
\[ {}4 x y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.099 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.872 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[_Lienard] |
✓ |
✓ |
0.77 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.845 |
|
|
\[ {}x y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.696 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[_Lienard] |
✓ |
✓ |
0.826 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.016 |
|
|
\[ {}2 x y^{\prime \prime }+y^{\prime }-y = 1+x \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.929 |
|
|
\[ {}2 x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.925 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Repeated root |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.901 |
|
|
\[ {}z^{\prime \prime }+t z^{\prime }+\left (t^{2}-\frac {1}{9}\right ) z = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.991 |
|
|
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