|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}x y y^{\prime }+x^{2}+y^{2} = 0 \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
2.448 |
|
|
\[ {}y^{\prime } = \frac {y^{2}-3 x y-5 x^{2}}{x^{2}} \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
3.981 |
|
|
\[ {}x^{2} y^{\prime } = 2 x^{2}+y^{2}+4 x y \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
2.976 |
|
|
\[ {}x y y^{\prime } = 3 x^{2}+4 y^{2} \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
2.984 |
|
|
\[ {}y^{\prime } = \frac {x +y}{x -y} \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.582 |
|
|
\[ {}\left (-y+x y^{\prime }\right ) \left (\ln \left (y\right )-\ln \left (x \right )\right ) = x \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘]] |
✓ |
✓ |
4.368 |
|
|
\[ {}y^{\prime } = \frac {y^{3}+2 x y^{2}+x^{2} y+x^{3}}{x \left (x +y\right )^{2}} \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.686 |
|
|
\[ {}y^{\prime } = \frac {2 y+x}{y+2 x} \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.706 |
|
|
\[ {}y^{\prime } = \frac {y}{-2 x +y} \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.826 |
|
|
\[ {}y^{\prime } = \frac {x y^{2}+2 y^{3}}{x^{3}+x^{2} y+x y^{2}} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
87.84 |
|
|
\[ {}y^{\prime } = \frac {x^{3}+x^{2} y+3 y^{3}}{x^{3}+3 x y^{2}} \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.707 |
|
|
\[ {}x^{2} y^{\prime } = y^{2}+x y-4 x^{2} \] |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
3.735 |
|
|
\[ {}x y y^{\prime } = x^{2}-x y+y^{2} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.616 |
|
|
\[ {}y^{\prime } = \frac {2 y^{2}-x y+2 x^{2}}{x y+2 x^{2}} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
5.976 |
|
|
\[ {}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x y} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.73 |
|
|
\[ {}y^{\prime } = \frac {-6 x +y-3}{2 x -y-1} \] |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.35 |
|
|
\[ {}y^{\prime } = \frac {2 x +y+1}{x +2 y-4} \] |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.258 |
|
|
\[ {}y^{\prime } = \frac {-x +3 y-14}{x +y-2} \] |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.929 |
|
|
\[ {}3 x y^{2} y^{\prime } = y^{3}+x \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.648 |
|
|
\[ {}x y y^{\prime } = 3 x^{6}+6 y^{2} \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.386 |
|
|
\[ {}x^{3} y^{\prime } = 2 y^{2}+2 x^{2} y-2 x^{4} \] |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
3.026 |
|
|
\[ {}y^{\prime } = y^{2} {\mathrm e}^{-x}+4 y+2 \,{\mathrm e}^{x} \] |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
1.691 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+y \tan \left (x \right )+\tan \left (x \right )^{2}}{\sin \left (x \right )^{2}} \] |
riccati |
[_Riccati] |
✓ |
✓ |
55.009 |
|
|
\[ {}x \ln \left (x \right )^{2} y^{\prime } = -4 \ln \left (x \right )^{2}+y \ln \left (x \right )+y^{2} \] |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Riccati] |
✓ |
✓ |
2.782 |
|
|
\[ {}2 x \left (y+2 \sqrt {x}\right ) y^{\prime } = \left (y+\sqrt {x}\right )^{2} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.043 |
|
|
\[ {}\left (y+{\mathrm e}^{x^{2}}\right ) y^{\prime } = 2 x \left (y^{2}+y \,{\mathrm e}^{x^{2}}+{\mathrm e}^{2 x^{2}}\right ) \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.753 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x} = \frac {3 x^{2} y^{2}+6 x y+2}{x^{2} \left (2 x y+3\right )} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
16.555 |
|
|
\[ {}y^{\prime }+\frac {3 y}{x} = \frac {3 x^{4} y^{2}+10 x^{2} y+6}{x^{3} \left (2 x^{2} y+5\right )} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
12.26 |
|
|
\[ {}y^{\prime } = 1+x -\left (2 x +1\right ) y+x y^{2} \] |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
3.421 |
|
|
\[ {}6 x^{2} y^{2}+4 x^{3} y y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.474 |
|
|
\[ {}3 y \cos \left (x \right )+4 x \,{\mathrm e}^{x}+2 x^{3} y+\left (3 \sin \left (x \right )+3\right ) y^{\prime } = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
153.251 |
|
|
\[ {}14 x^{2} y^{3}+21 x^{2} y^{2} y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.444 |
|
|
\[ {}2 x -2 y^{2}+\left (12 y^{2}-4 x y\right ) y^{\prime } = 0 \] |
exact |
[_exact, _rational] |
✓ |
✓ |
2.083 |
|
|
\[ {}\left (x +y\right )^{2}+\left (x +y\right )^{2} y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.16 |
|
|
\[ {}4 x +7 y+\left (3 x +4 y\right ) y^{\prime } = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.97 |
|
|
\[ {}-2 \sin \left (x \right ) y^{2}+3 y^{3}-2 x +\left (4 y \cos \left (x \right )+9 x y^{2}\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
35.72 |
|
|
\[ {}2 x +y+\left (2 y+2 x \right ) y^{\prime } = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.662 |
|
|
\[ {}3 x^{2}+2 x y+4 y^{2}+\left (x^{2}+8 x y+18 y\right ) y^{\prime } = 0 \] |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.122 |
|
|
\[ {}2 x^{2}+8 x y+y^{2}+\left (2 x^{2}+\frac {x y^{3}}{3}\right ) y^{\prime } = 0 \] |
unknown |
[_rational] |
❇ |
N/A |
1.431 |
|
|
\[ {}\frac {1}{x}+2 x +\left (\frac {1}{y}+2 y\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.542 |
|
|
\[ {}y \sin \left (x y\right )+x y^{2} \cos \left (x y\right )+\left (x \sin \left (x y\right )+x y^{2} \cos \left (x y\right )\right ) y^{\prime } = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
53.93 |
|
|
\[ {}\frac {x}{\left (x^{2}+y^{2}\right )^{\frac {3}{2}}}+\frac {y y^{\prime }}{\left (x^{2}+y^{2}\right )^{\frac {3}{2}}} = 0 \] |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.012 |
|
|
\[ {}{\mathrm e}^{x} \left (x^{2} y^{2}+2 x y^{2}\right )+6 x +\left (2 x^{2} y \,{\mathrm e}^{x}+2\right ) y^{\prime } = 0 \] |
exact |
[_exact, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.899 |
|
|
\[ {}x^{2} {\mathrm e}^{y+x^{2}} \left (2 x^{2}+3\right )+4 x +\left (x^{3} {\mathrm e}^{y+x^{2}}-12 y^{2}\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
3.645 |
|
|
\[ {}{\mathrm e}^{x y} \left (y x^{4}+4 x^{3}\right )+3 y+\left (x^{5} {\mathrm e}^{x y}+3 x \right ) y^{\prime } = 0 \] |
exact, first_order_ode_lie_symmetry_calculated |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
16.681 |
|
|
\[ {}3 x^{2} \cos \left (x \right ) y-x^{3} y^{2} \sin \left (x \right )+4 x +\left (8 y-x^{4} \sin \left (x \right ) y\right ) y^{\prime } = 0 \] |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
52.588 |
|
|
\[ {}4 x^{3} y^{2}-6 x^{2} y-2 x -3+\left (2 y x^{4}-2 x^{3}\right ) y^{\prime } = 0 \] |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.983 |
|
|
\[ {}-4 y \cos \left (x \right )+4 \sin \left (x \right ) \cos \left (x \right )+\sec \left (x \right )^{2}+\left (4 y-4 \sin \left (x \right )\right ) y^{\prime } = 0 \] |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
8.937 |
|
|
\[ {}\left (y^{3}-1\right ) {\mathrm e}^{x}+3 y^{2} \left (1+{\mathrm e}^{x}\right ) y^{\prime } = 0 \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
5.071 |
|
|
\[ {}\sin \left (x \right )-\sin \left (x \right ) y-2 \cos \left (x \right )+\cos \left (x \right ) y^{\prime } = 0 \] |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.128 |
|
|
\[ {}\left (2 x -1\right ) \left (y-1\right )+\left (2+x \right ) \left (x -3\right ) y^{\prime } = 0 \] |
exact, linear, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.746 |
|
|
\[ {}7 x +4 y+\left (4 x +3 y\right ) y^{\prime } = 0 \] |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.935 |
|
|
\[ {}{\mathrm e}^{x} \left (x^{4} y^{2}+4 x^{3} y^{2}+1\right )+\left (2 x^{4} y \,{\mathrm e}^{x}+2 y\right ) y^{\prime } = 0 \] |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[_exact, _Bernoulli] |
✓ |
✓ |
2.301 |
|
|
\[ {}x^{3} y^{4}+x +\left (x^{4} y^{3}+y\right ) y^{\prime } = 0 \] |
exact |
[_exact, _rational] |
✓ |
✓ |
2.538 |
|
|
\[ {}3 x^{2}+2 y+\left (2 y+2 x \right ) y^{\prime } = 0 \] |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.579 |
|
|
\[ {}x^{3} y^{4}+2 x +\left (x^{4} y^{3}+3 y\right ) y^{\prime } = 0 \] |
exact |
[_exact, _rational] |
✓ |
✓ |
2.586 |
|
|
\[ {}x^{2}+y^{2}+2 x y y^{\prime } = 0 \] |
exact, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
2.431 |
|
|
\[ {}y^{\prime }+\frac {2 y}{x} = -\frac {2 x y}{x^{2}+2 x^{2} y+1} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
5.012 |
|
|
\[ {}y^{\prime }-\frac {3 y}{x} = \frac {2 x^{4} \left (4 x^{3}-3 y\right )}{3 x^{5}+3 x^{3}+2 y} \] |
exactWithIntegrationFactor |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.682 |
|
|
\[ {}y^{\prime }+2 x y = -\frac {{\mathrm e}^{-x^{2}} \left (3 x +2 y \,{\mathrm e}^{x^{2}}\right )}{2 x +3 y \,{\mathrm e}^{x^{2}}} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
44.569 |
|
|
\[ {}y+\left (2 x +\frac {1}{y}\right ) y^{\prime } = 0 \] |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.641 |
|
|
\[ {}-y^{2}+x^{2} y^{\prime } = 0 \] |
exact, riccati, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.15 |
|
|
\[ {}y-x y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.003 |
|
|
\[ {}3 x^{2} y+2 x^{3} y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.039 |
|
|
\[ {}2 y^{3}+3 y^{2} y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.388 |
|
|
\[ {}5 x y+2 y+5+2 x y^{\prime } = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.224 |
|
|
\[ {}x y+x +2 y+1+\left (1+x \right ) y^{\prime } = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.129 |
|
|
\[ {}27 x y^{2}+8 y^{3}+\left (18 x^{2} y+12 x y^{2}\right ) y^{\prime } = 0 \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
4.234 |
|
|
\[ {}6 x y^{2}+2 y+\left (12 x^{2} y+6 x +3\right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.299 |
|
|
\[ {}y^{2}+\left (x y^{2}+6 x y+\frac {1}{y}\right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.101 |
|
|
\[ {}12 x^{3} y+24 x^{2} y^{2}+\left (9 x^{4}+32 x^{3} y+4 y\right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.471 |
|
|
\[ {}x^{2} y+4 x y+2 y+\left (x^{2}+x \right ) y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.771 |
|
|
\[ {}-y+\left (x^{4}-x \right ) y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.483 |
|
|
\[ {}\cos \left (x \right ) \cos \left (y\right )+\left (\sin \left (x \right ) \cos \left (y\right )-\sin \left (x \right ) \sin \left (y\right )+y\right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
42.992 |
|
|
\[ {}2 x y+y^{2}+\left (2 x y+x^{2}-2 x y^{2}-2 x y^{3}\right ) y^{\prime } = 0 \] |
unknown |
[_rational] |
❇ |
N/A |
33.105 |
|
|
\[ {}y \sin \left (y\right )+x \left (\sin \left (y\right )-y \cos \left (y\right )\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.327 |
|
|
\[ {}a y+b x y+\left (c x +d x y\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.798 |
|
|
\[ {}3 x^{2} y^{3}-y^{2}+y+\left (-x y+2 x \right ) y^{\prime } = 0 \] |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
2.23 |
|
|
\[ {}2 y+3 \left (x^{2}+x^{2} y^{3}\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.284 |
|
|
\[ {}a \cos \left (x \right ) y-\sin \left (x \right ) y^{2}+\left (b \cos \left (x \right ) y-x \sin \left (x \right ) y\right ) y^{\prime } = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
33.192 |
|
|
\[ {}x^{4} y^{4}+x^{5} y^{3} y^{\prime } = 0 \] |
exact, linear, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.446 |
|
|
\[ {}y \left (x \cos \left (x \right )+2 \sin \left (x \right )\right )+x \left (y+1\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.591 |
|
|
\[ {}x^{4} y^{3}+y+\left (x^{5} y^{2}-x \right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.156 |
|
|
\[ {}3 x y+2 y^{2}+y+\left (x^{2}+2 x y+x +2 y\right ) y^{\prime } = 0 \] |
homogeneousTypeD2, exactWithIntegrationFactor |
[[_homogeneous, ‘class D‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.622 |
|
|
\[ {}12 x y+6 y^{3}+\left (9 x^{2}+10 x y^{2}\right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.186 |
|
|
\[ {}3 x^{2} y^{2}+2 y+2 x y^{\prime } = 0 \] |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.291 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+10 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.808 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.943 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.715 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.873 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.709 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+4 x y^{\prime }+2 y = 0 \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.006 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 0 \] |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.345 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.436 |
|
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+a^{2} y = 0 \] |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.447 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.412 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
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]] |
✓ |
✓ |
2.379 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 a -1\right ) x y^{\prime }+a^{2} y = 0 \] |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.911 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.308 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.304 |
|
|
|
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|
|
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