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) |
|
\[ {}y^{\prime } = a f \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.254 |
|
|
\[ {}y^{\prime } = x +\sin \left (x \right )+y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.786 |
|
|
\[ {}y^{\prime } = x^{2}+3 \cosh \left (x \right )+2 y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
2.769 |
|
|
\[ {}y^{\prime } = a +b x +c y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.579 |
|
|
\[ {}y^{\prime } = a \cos \left (b x +c \right )+k y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
2.238 |
|
|
\[ {}y^{\prime } = a \sin \left (b x +c \right )+k y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
2.006 |
|
|
\[ {}y^{\prime } = a +b \,{\mathrm e}^{k x}+c y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.674 |
|
|
\[ {}y^{\prime } = x \left (x^{2}-y\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.328 |
|
|
\[ {}y^{\prime } = x \left ({\mathrm e}^{-x^{2}}+a y\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.71 |
|
|
\[ {}y^{\prime } = x^{2} \left (a \,x^{3}+b y\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.734 |
|
|
\[ {}y^{\prime } = a \,x^{n} y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.195 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \cos \left (x \right )+y \cos \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.138 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{\sin \left (x \right )}+y \cos \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.684 |
|
|
\[ {}y^{\prime } = y \cot \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.142 |
|
|
\[ {}y^{\prime } = 1-y \cot \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.454 |
|
|
\[ {}y^{\prime } = x \csc \left (x \right )-y \cot \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.486 |
|
|
\[ {}y^{\prime } = \left (2 \csc \left (2 x \right )+\cot \left (x \right )\right ) y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.082 |
|
|
\[ {}y^{\prime } = \sec \left (x \right )-y \cot \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.72 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x} \sin \left (x \right )+y \cot \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.099 |
|
|
\[ {}y^{\prime }+\csc \left (x \right )+2 y \cot \left (x \right ) = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.667 |
|
|
\[ {}y^{\prime } = 4 \csc \left (x \right ) x \sec \left (x \right )^{2}-2 y \cot \left (2 x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
4.425 |
|
|
\[ {}y^{\prime } = 2 \cot \left (x \right )^{2} \cos \left (2 x \right )-2 y \csc \left (2 x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.731 |
|
|
\[ {}y^{\prime } = 4 \csc \left (x \right ) x \left (\sin \left (x \right )^{3}+y\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
8.142 |
|
|
\[ {}y^{\prime } = 4 \csc \left (x \right ) x \left (1-\tan \left (x \right )^{2}+y\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
76.021 |
|
|
\[ {}y^{\prime } = y \sec \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.773 |
|
|
\[ {}y^{\prime }+\tan \left (x \right ) = \left (1-y\right ) \sec \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.483 |
|
|
\[ {}y^{\prime } = y \tan \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.871 |
|
|
\[ {}y^{\prime } = \cos \left (x \right )+y \tan \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.931 |
|
|
\[ {}y^{\prime } = \cos \left (x \right )-y \tan \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.838 |
|
|
\[ {}y^{\prime } = \sec \left (x \right )-y \tan \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.866 |
|
|
\[ {}y^{\prime } = \sin \left (2 x \right )+y \tan \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.98 |
|
|
\[ {}y^{\prime } = \sin \left (2 x \right )-y \tan \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.915 |
|
|
\[ {}y^{\prime } = \sin \left (x \right )+2 y \tan \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.968 |
|
|
\[ {}y^{\prime } = 2+2 \sec \left (2 x \right )+2 y \tan \left (2 x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.793 |
|
|
\[ {}y^{\prime } = \csc \left (x \right )+3 y \tan \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.534 |
|
|
\[ {}y^{\prime } = \left (a +\cos \left (\ln \left (x \right )\right )+\sin \left (\ln \left (x \right )\right )\right ) y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.095 |
|
|
\[ {}y^{\prime } = 6 \,{\mathrm e}^{2 x}-y \tanh \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.088 |
|
|
\[ {}y^{\prime } = f \left (x \right ) f^{\prime }\left (x \right )+f^{\prime }\left (x \right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.803 |
|
|
\[ {}y^{\prime } = f \left (x \right )+g \left (x \right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.711 |
|
|
\[ {}y^{\prime } = x^{2}-y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.545 |
|
|
\[ {}y^{\prime }+f \left (x \right )^{2} = f^{\prime }\left (x \right )+y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
0.612 |
|
|
\[ {}y^{\prime }+1-x = y \left (x +y\right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.854 |
|
|
\[ {}y^{\prime } = \left (x +y\right )^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.727 |
|
|
\[ {}y^{\prime } = \left (x -y\right )^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
1.586 |
|
|
\[ {}y^{\prime } = 3-3 x +3 y+\left (x -y\right )^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
1.682 |
|
|
\[ {}y^{\prime } = 2 x -\left (x^{2}+1\right ) y+y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.368 |
|
|
\[ {}y^{\prime } = x \left (x^{3}+2\right )-\left (2 x^{2}-y\right ) y \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
0.895 |
|
|
\[ {}y^{\prime } = 1+x \left (-x^{3}+2\right )+\left (2 x^{2}-y\right ) y \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
1.597 |
|
|
\[ {}y^{\prime } = \cos \left (x \right )-\left (\sin \left (x \right )-y\right ) y \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.085 |
|
|
\[ {}y^{\prime } = \cos \left (2 x \right )+\left (\sin \left (2 x \right )+y\right ) y \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.227 |
|
|
\[ {}y^{\prime } = f \left (x \right )+x f \left (x \right ) y+y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.501 |
|
|
\[ {}y^{\prime } = \left (3+x -4 y\right )^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.947 |
|
|
\[ {}y^{\prime } = \left (1+4 x +9 y\right )^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.922 |
|
|
\[ {}y^{\prime } = 3 a +3 b x +3 b y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.067 |
|
|
\[ {}y^{\prime } = a +b y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.26 |
|
|
\[ {}y^{\prime } = x a +b y^{2} \] |
1 |
1 |
1 |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.648 |
|
|
\[ {}y^{\prime } = a +b x +c y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.926 |
|
|
\[ {}y^{\prime } = a \,x^{n -1}+b \,x^{2 n}+c y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
64.083 |
|
|
\[ {}y^{\prime } = x^{2} a +b y^{2} \] |
1 |
1 |
1 |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.826 |
|
|
\[ {}y^{\prime } = \operatorname {a0} +\operatorname {a1} y+\operatorname {a2} y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.44 |
|
|
\[ {}y^{\prime } = f \left (x \right )+a y+b y^{2} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
0.623 |
|
|
\[ {}y^{\prime } = 1+a \left (x -y\right ) y \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.647 |
|
|
\[ {}y^{\prime } = f \left (x \right )+g \left (x \right ) y+a y^{2} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
0.804 |
|
|
\[ {}y^{\prime } = x y \left (3+y\right ) \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.513 |
|
|
\[ {}y^{\prime } = 1-x -x^{3}+\left (2 x^{2}+1\right ) y-x y^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
2.721 |
|
|
\[ {}y^{\prime } = x \left (2+x^{2} y-y^{2}\right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.99 |
|
|
\[ {}y^{\prime } = x +\left (1-2 x \right ) y-\left (1-x \right ) y^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
2.438 |
|
|
\[ {}y^{\prime } = a x y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime } = x^{n} \left (a +b y^{2}\right ) \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.323 |
|
|
\[ {}y^{\prime } = a \,x^{m}+b \,x^{n} y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.122 |
|
|
\[ {}y^{\prime } = \left (a +b y \cos \left (k x \right )\right ) y \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.233 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \left (2 \sec \left (x \right )^{2}-y\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.546 |
|
|
\[ {}y^{\prime }+4 \csc \left (x \right ) = \left (3-\cot \left (x \right )\right ) y+y^{2} \sin \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
9.078 |
|
|
\[ {}y^{\prime } = y \sec \left (x \right )+\left (\sin \left (x \right )-1\right )^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.793 |
|
|
\[ {}y^{\prime }+\tan \left (x \right ) \left (1-y^{2}\right ) = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.99 |
|
|
\[ {}y^{\prime } = f \left (x \right )+g \left (x \right ) y+h \left (x \right ) y^{2} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
1.212 |
|
|
\[ {}y^{\prime } = \left (a +b y+c y^{2}\right ) f \left (x \right ) \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.484 |
|
|
\[ {}y^{\prime }+\left (x a +y\right ) y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
1.5 |
|
|
\[ {}y^{\prime } = \left (a \,{\mathrm e}^{x}+y\right ) y^{2} \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
2.904 |
|
|
\[ {}y^{\prime }+3 a \left (y+2 x \right ) y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
1.433 |
|
|
\[ {}y^{\prime } = y \left (a +b y^{2}\right ) \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.902 |
|
|
\[ {}y^{\prime } = \operatorname {a0} +\operatorname {a1} y+\operatorname {a2} y^{2}+\operatorname {a3} y^{3} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.193 |
|
|
\[ {}y^{\prime } = x y^{3} \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.73 |
|
|
\[ {}y^{\prime }+y \left (1-x y^{2}\right ) = 0 \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.682 |
|
|
\[ {}y^{\prime } = \left (a +b x y\right ) y^{2} \] |
1 |
1 |
1 |
abelFirstKind, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
✓ |
8.756 |
|
|
\[ {}y^{\prime }+2 x y \left (1+a x y^{2}\right ) = 0 \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.995 |
|
|
\[ {}y^{\prime }+\left (\tan \left (x \right )+y^{2} \sec \left (x \right )\right ) y = 0 \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.552 |
|
|
\[ {}y^{\prime }+y^{3} \sec \left (x \right ) \tan \left (x \right ) = 0 \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.954 |
|
|
\[ {}y^{\prime } = \operatorname {f0} \left (x \right )+\operatorname {f1} \left (x \right ) y+\operatorname {f2} \left (x \right ) y^{2}+\operatorname {f3} \left (x \right ) y^{3} \] |
1 |
0 |
0 |
abelFirstKind |
[_Abel] |
❇ |
N/A |
5.928 |
|
|
\[ {}y^{\prime } = a \,x^{\frac {n}{-n +1}}+b y^{n} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
✓ |
1.043 |
|
|
\[ {}y^{\prime } = f \left (x \right ) y+g \left (x \right ) y^{k} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.525 |
|
|
\[ {}y^{\prime } = f \left (x \right )+g \left (x \right ) y+h \left (x \right ) y^{n} \] |
1 |
0 |
0 |
unknown |
[_Chini] |
❇ |
N/A |
0.779 |
|
|
\[ {}y^{\prime } = \sqrt {{| y|}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.488 |
|
|
\[ {}y^{\prime } = a +b y+\sqrt {\operatorname {A0} +\operatorname {B0} y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.852 |
|
|
\[ {}y^{\prime } = x a +b \sqrt {y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
✓ |
3.365 |
|
|
\[ {}y^{\prime }+x^{3} = x \sqrt {x^{4}+4 y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.598 |
|
|
\[ {}y^{\prime }+2 y \left (1-x \sqrt {y}\right ) = 0 \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.003 |
|
|
\[ {}y^{\prime } = \sqrt {a +b y^{2}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.509 |
|
|
\[ {}y^{\prime } = y \sqrt {a +b y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.464 |
|
|
\[ {}y^{\prime }+\left (f \left (x \right )-y\right ) g \left (x \right ) \sqrt {\left (y-a \right ) \left (y-b \right )} = 0 \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.296 |
|
|
\[ {}y^{\prime } = \sqrt {X Y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.316 |
|
|
\[ {}y^{\prime } = \cos \left (x \right )^{2} \cos \left (y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.817 |
|
|
\[ {}y^{\prime } = \sec \left (x \right )^{2} \cot \left (y\right ) \cos \left (y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.894 |
|
|
\[ {}y^{\prime } = a +b \cos \left (A x +B y\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
90.74 |
|
|
\[ {}y^{\prime }+f \left (x \right )+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) \cos \left (a y\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
2.158 |
|
|
\[ {}y^{\prime } = a +b \cos \left (y\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.411 |
|
|
\[ {}y^{\prime }+x \left (\sin \left (2 y\right )-x^{2} \cos \left (y\right )^{2}\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.875 |
|
|
\[ {}y^{\prime }+\tan \left (x \right ) \sec \left (x \right ) \cos \left (y\right )^{2} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.779 |
|
|
\[ {}y^{\prime } = \cot \left (x \right ) \cot \left (y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.116 |
|
|
\[ {}y^{\prime }+\cot \left (x \right ) \cot \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.964 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \left (\csc \left (y\right )-\cot \left (y\right )\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.105 |
|
|
\[ {}y^{\prime } = \tan \left (x \right ) \cot \left (y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.789 |
|
|
\[ {}y^{\prime }+\tan \left (x \right ) \cot \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime }+\sin \left (2 x \right ) \csc \left (2 y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.735 |
|
|
\[ {}y^{\prime } = \tan \left (x \right ) \left (\tan \left (y\right )+\sec \left (x \right ) \sec \left (y\right )\right ) \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.905 |
|
|
\[ {}y^{\prime } = \cos \left (x \right ) \sec \left (y\right )^{2} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.069 |
|
|
\[ {}y^{\prime } = \sec \left (x \right )^{2} \sec \left (y\right )^{3} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.51 |
|
|
\[ {}y^{\prime } = a +b \sin \left (y\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.404 |
|
|
\[ {}y^{\prime } = \left (1+\cos \left (x \right ) \sin \left (y\right )\right ) \tan \left (y\right ) \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
2.441 |
|
|
\[ {}y^{\prime }+\csc \left (2 x \right ) \sin \left (2 y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.963 |
|
|
\[ {}y^{\prime }+f \left (x \right )+g \left (x \right ) \tan \left (y\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.06 |
|
|
\[ {}y^{\prime } = \sqrt {a +b \cos \left (y\right )} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.765 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{y}+x \] |
1 |
1 |
1 |
first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
0.646 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x +y} \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.585 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x} \left (a +b \,{\mathrm e}^{-y}\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.281 |
|
|
\[ {}y^{\prime }+y \ln \left (x \right ) \ln \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.967 |
|
|
\[ {}y^{\prime } = x^{m -1} y^{-n +1} f \left (a \,x^{m}+b y^{n}\right ) \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.194 |
|
|
\[ {}y^{\prime } = a f \left (y\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.277 |
|
|
\[ {}y^{\prime } = f \left (a +b x +c y\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.694 |
|
|
\[ {}y^{\prime } = f \left (x \right ) g \left (y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.615 |
|
|
\[ {}y^{\prime } = \sec \left (x \right )^{2}+y \sec \left (x \right ) \operatorname {Csx} \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.252 |
|
|
\[ {}2 y^{\prime } = 2 \sin \left (y\right )^{2} \tan \left (y\right )-x \sin \left (2 y\right ) \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
22.05 |
|
|
\[ {}2 y^{\prime }+x a = \sqrt {a^{2} x^{2}-4 b \,x^{2}-4 c y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.02 |
|
|
\[ {}3 y^{\prime } = x +\sqrt {x^{2}-3 y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.079 |
|
|
\[ {}x y^{\prime } = \sqrt {a^{2}-x^{2}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.331 |
|
|
\[ {}x y^{\prime }+x +y = 0 \] |
1 |
1 |
1 |
exact, linear, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.751 |
|
|
\[ {}x y^{\prime }+x^{2}-y = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.603 |
|
|
\[ {}x y^{\prime } = x^{3}-y \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.597 |
|
|
\[ {}x y^{\prime } = 1+x^{3}+y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.631 |
|
|
\[ {}x y^{\prime } = x^{m}+y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.914 |
|
|
\[ {}x y^{\prime } = x \sin \left (x \right )-y \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.633 |
|
|
\[ {}x y^{\prime } = x^{2} \sin \left (x \right )+y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.682 |
|
|
\[ {}x y^{\prime } = x^{n} \ln \left (x \right )-y \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.89 |
|
|
\[ {}x y^{\prime } = \sin \left (x \right )-2 y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.685 |
|
|
\[ {}x y^{\prime } = a y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.015 |
|
|
\[ {}x y^{\prime } = 1+x +a y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.866 |
|
|
\[ {}x y^{\prime } = x a +b y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.046 |
|
|
\[ {}x y^{\prime } = x^{2} a +b y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.842 |
|
|
\[ {}x y^{\prime } = a +b \,x^{n}+c y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.005 |
|
|
\[ {}x y^{\prime }+2+\left (-x +3\right ) y = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.68 |
|
|
\[ {}x y^{\prime }+x +\left (x a +2\right ) y = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.884 |
|
|
\[ {}x y^{\prime }+\left (b x +a \right ) y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.05 |
|
|
\[ {}x y^{\prime } = x^{3}+\left (-2 x^{2}+1\right ) y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.874 |
|
|
\[ {}x y^{\prime } = x a -\left (-b \,x^{2}+1\right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.952 |
|
|
\[ {}x y^{\prime }+x +\left (-x^{2} a +2\right ) y = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.012 |
|
|
\[ {}x y^{\prime }+x^{2}+y^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
0.814 |
|
|
\[ {}x y^{\prime } = x^{2}+y \left (y+1\right ) \] |
1 |
1 |
1 |
riccati, exactByInspection, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.05 |
|
|
\[ {}x y^{\prime }-y+y^{2} = x^{\frac {2}{3}} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
6.918 |
|
|
\[ {}x y^{\prime } = a +b y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.995 |
|
|
\[ {}x y^{\prime } = x^{2} a +y+b y^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.128 |
|
|
\[ {}x y^{\prime } = a \,x^{2 n}+\left (n +b y\right ) y \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.394 |
|
|
\[ {}x y^{\prime } = a \,x^{n}+b y+c y^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.53 |
|
|
\[ {}x y^{\prime } = k +a \,x^{n}+b y+c y^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.668 |
|
|
\[ {}x y^{\prime }+a +x y^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
0.944 |
|
|
\[ {}x y^{\prime }+\left (1-x y\right ) y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.727 |
|
|
\[ {}x y^{\prime } = \left (1-x y\right ) y \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.707 |
|
|
\[ {}x y^{\prime } = \left (1+x y\right ) y \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.716 |
|
|
\[ {}x y^{\prime } = a \,x^{3} \left (1-x y\right ) y \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.191 |
|
|
\[ {}x y^{\prime } = x^{3}+\left (2 x^{2}+1\right ) y+x y^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.095 |
|
|
\[ {}x y^{\prime } = y \left (2 x y+1\right ) \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.711 |
|
|
\[ {}x y^{\prime }+b x +\left (2+a x y\right ) y = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.043 |
|
|
\[ {}x y^{\prime }+\operatorname {a0} +\operatorname {a1} x +\left (\operatorname {a2} +\operatorname {a3} x y\right ) y = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.273 |
|
|
\[ {}x y^{\prime }+a \,x^{2} y^{2}+2 y = b \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.247 |
|
|
\[ {}x y^{\prime }+x^{m}+\frac {\left (n -m \right ) y}{2}+x^{n} y^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.462 |
|
|
\[ {}x y^{\prime }+\left (a +b \,x^{n} y\right ) y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.899 |
|
|
\[ {}x y^{\prime } = a \,x^{m}-b y-c \,x^{n} y^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.053 |
|
|
\[ {}x y^{\prime } = 2 x -y+a \,x^{n} \left (x -y\right )^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
1.286 |
|
|
\[ {}x y^{\prime }+\left (1-a y \ln \left (x \right )\right ) y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.99 |
|
|
\[ {}x y^{\prime } = y+\left (x^{2}-y^{2}\right ) f \left (x \right ) \] |
1 |
1 |
1 |
riccati, exactByInspection, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
1.607 |
|
|
\[ {}x y^{\prime } = y \left (1+y^{2}\right ) \] |
1 |
1 |
2 |
exact, bernoulli, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.431 |
|
|
\[ {}x y^{\prime }+y \left (1-x y^{2}\right ) = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.803 |
|
|
\[ {}x y^{\prime }+y = a \left (x^{2}+1\right ) y^{3} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.915 |
|
|
\[ {}x y^{\prime } = a y+b \left (x^{2}+1\right ) y^{3} \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.003 |
|
|
\[ {}x y^{\prime }+2 y = a \,x^{2 k} y^{k} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.387 |
|
|
\[ {}x y^{\prime } = 4 y-4 \sqrt {y} \] |
1 |
1 |
1 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.38 |
|
|
\[ {}x y^{\prime }+2 y = \sqrt {1+y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.276 |
|
|
\[ {}x y^{\prime } = y+\sqrt {x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.869 |
|
|
\[ {}x y^{\prime } = y+\sqrt {x^{2}-y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.917 |
|
|
\[ {}x y^{\prime } = y+x \sqrt {x^{2}+y^{2}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.05 |
|
|
\[ {}x y^{\prime } = y-x \left (x -y\right ) \sqrt {x^{2}+y^{2}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.285 |
|
|
\[ {}x y^{\prime } = y+a \sqrt {y^{2}+b^{2} x^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.506 |
|
|
\[ {}x y^{\prime }+\left (\sin \left (y\right )-3 x^{2} \cos \left (y\right )\right ) \cos \left (y\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.823 |
|
|
\[ {}x y^{\prime }+x -y+x \cos \left (\frac {y}{x}\right ) = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.011 |
|
|
\[ {}x y^{\prime } = y-x \cos \left (\frac {y}{x}\right )^{2} \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.992 |
|
|
\[ {}x y^{\prime } = \left (-2 x^{2}+1\right ) \cot \left (y\right )^{2} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.883 |
|
|
\[ {}x y^{\prime } = y-\cot \left (y\right )^{2} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.818 |
|
|
\[ {}x y^{\prime }+y+2 x \sec \left (x y\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
3.558 |
|
|
\[ {}x y^{\prime }-y+x \sec \left (\frac {y}{x}\right ) = 0 \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.012 |
|
|
\[ {}x y^{\prime } = y+x \sec \left (\frac {y}{x}\right )^{2} \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.026 |
|
|
\[ {}x y^{\prime } = \sin \left (x -y\right ) \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.537 |
|
|
\[ {}x y^{\prime } = y+x \sin \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.843 |
|
|
\[ {}x y^{\prime }+\tan \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.08 |
|
|
\[ {}x y^{\prime }+x +\tan \left (x +y\right ) = 0 \] |
1 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.714 |
|
|
\[ {}x y^{\prime } = y-x \tan \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.0 |
|
|
\[ {}x y^{\prime } = \left (1+y^{2}\right ) \left (x^{2}+\arctan \left (y\right )\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.962 |
|
|
\[ {}x y^{\prime } = y+x \,{\mathrm e}^{\frac {y}{x}} \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.691 |
|
|
\[ {}x y^{\prime } = x +y+x \,{\mathrm e}^{\frac {y}{x}} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.001 |
|
|
\[ {}x y^{\prime } = y \ln \left (y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.782 |
|
|
\[ {}x y^{\prime } = \left (1+\ln \left (x \right )-\ln \left (y\right )\right ) y \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.326 |
|
|
\[ {}x y^{\prime }+\left (1-\ln \left (x \right )-\ln \left (y\right )\right ) y = 0 \] |
1 |
1 |
1 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.304 |
|
|
\[ {}x y^{\prime } = y-2 x \tanh \left (\frac {y}{x}\right ) \] |
1 |
1 |
2 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.108 |
|
|
\[ {}x y^{\prime }+n y = f \left (x \right ) g \left (x^{n} y\right ) \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.086 |
|
|
\[ {}x y^{\prime } = y f \left (x^{m} y^{n}\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
0.991 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = x^{3} \left (3 x +4\right )+y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.649 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = \left (1+x \right )^{4}+2 y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.658 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = {\mathrm e}^{x} \left (1+x \right )^{n +1}+n y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.98 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = a y+b x y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.895 |
|
|
\[ {}\left (1+x \right ) y^{\prime }+y+\left (1+x \right )^{4} y^{3} = 0 \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_1st_order, _with_linear_symmetries], _rational, _Bernoulli] |
✓ |
✓ |
0.829 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = \left (1-x y^{3}\right ) y \] |
1 |
3 |
3 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.029 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = 1+y+\left (1+x \right ) \sqrt {y+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.514 |
|
|
\[ {}\left (x +a \right ) y^{\prime } = b x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.276 |
|
|
\[ {}\left (x +a \right ) y^{\prime } = b x +y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.889 |
|
|
\[ {}\left (x +a \right ) y^{\prime }+b \,x^{2}+y = 0 \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.796 |
|
|
\[ {}\left (x +a \right ) y^{\prime } = 2 \left (x +a \right )^{5}+3 y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.908 |
|
|
\[ {}\left (x +a \right ) y^{\prime } = b +c y \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.058 |
|
|
\[ {}\left (x +a \right ) y^{\prime } = b x +c y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.96 |
|
|
\[ {}\left (x +a \right ) y^{\prime } = y \left (1-a y\right ) \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.45 |
|
|
\[ {}\left (a -x \right ) y^{\prime } = y+\left (c x +b \right ) y^{3} \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.899 |
|
|
\[ {}2 x y^{\prime } = 2 x^{3}-y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.668 |
|
|
\[ {}2 x y^{\prime }+1 = 4 i x y+y^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.587 |
|
|
\[ {}2 x y^{\prime } = y \left (1+y^{2}\right ) \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.427 |
|
|
\[ {}2 x y^{\prime }+y \left (1+y^{2}\right ) = 0 \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.734 |
|
|
\[ {}2 x y^{\prime } = \left (1+x -6 y^{2}\right ) y \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.672 |
|
|
\[ {}2 x y^{\prime }+4 y+a +\sqrt {a^{2}-4 b -4 c y} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
18.974 |
|
|
\[ {}\left (1-2 x \right ) y^{\prime } = 16+32 x -6 y \] |
1 |
1 |
1 |
linear, homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.954 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime } = 4 \,{\mathrm e}^{-y}-2 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.921 |
|
|
\[ {}2 \left (1-x \right ) y^{\prime } = 4 x \sqrt {1-x}+y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.608 |
|
|
\[ {}2 \left (1+x \right ) y^{\prime }+2 y+\left (1+x \right )^{4} y^{3} = 0 \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_1st_order, _with_linear_symmetries], _rational, _Bernoulli] |
✓ |
✓ |
0.63 |
|
|
\[ {}3 x y^{\prime } = 3 x^{\frac {2}{3}}+\left (-3 y+1\right ) y \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.006 |
|
|
\[ {}3 x y^{\prime } = \left (2+x y^{3}\right ) y \] |
1 |
3 |
3 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.784 |
|
|
\[ {}3 x y^{\prime } = \left (1+3 x y^{3} \ln \left (x \right )\right ) y \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.441 |
|
|
\[ {}x^{2} y^{\prime } = -y+a \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.778 |
|
|
\[ {}x^{2} y^{\prime } = a +b x +c \,x^{2}+x y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.619 |
|
|
\[ {}x^{2} y^{\prime } = a +b x +c \,x^{2}-x y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.617 |
|
|
\[ {}x^{2} y^{\prime }+\left (1-2 x \right ) y = x^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.543 |
|
|
\[ {}x^{2} y^{\prime } = a +b x y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.555 |
|
|
\[ {}x^{2} y^{\prime } = \left (b x +a \right ) y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.684 |
|
|
\[ {}x^{2} y^{\prime }+x \left (2+x \right ) y = x \left (1-{\mathrm e}^{-2 x}\right )-2 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.626 |
|
|
\[ {}x^{2} y^{\prime }+2 x \left (1-x \right ) y = {\mathrm e}^{x} \left (2 \,{\mathrm e}^{x}-1\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.617 |
|
|
\[ {}x^{2} y^{\prime }+x^{2}+x y+y^{2} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.688 |
|
|
\[ {}x^{2} y^{\prime } = \left (1+2 x -y\right )^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
✓ |
1.639 |
|
|
\[ {}x^{2} y^{\prime } = a +b y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.772 |
|
|
\[ {}x^{2} y^{\prime } = \left (a y+x \right ) y \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.651 |
|
|
\[ {}x^{2} y^{\prime } = \left (x a +b y\right ) y \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.786 |
|
|
\[ {}x^{2} y^{\prime }+x^{2} a +b x y+c y^{2} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.764 |
|
|
\[ {}x^{2} y^{\prime } = a +b \,x^{n}+x^{2} y^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.918 |
|
|
\[ {}x^{2} y^{\prime }+2+x y \left (4+x y\right ) = 0 \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
1.003 |
|
|
\[ {}x^{2} y^{\prime }+2+a x \left (1-x y\right )-x^{2} y^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.275 |
|
|
\[ {}x^{2} y^{\prime } = a +b \,x^{2} y^{2} \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
1.735 |
|
|
\[ {}x^{2} y^{\prime } = a +b \,x^{n}+c \,x^{2} y^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.972 |
|
|
\[ {}x^{2} y^{\prime } = a +b x y+c \,x^{2} y^{2} \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
2.024 |
|
|
\[ {}x^{2} y^{\prime } = a +b x y+c \,x^{4} y^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.576 |
|
|
\[ {}x^{2} y^{\prime }+\left (x^{2}+y^{2}-x \right ) y = 0 \] |
1 |
2 |
2 |
bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.843 |
|
|
\[ {}x^{2} y^{\prime } = 2 y \left (x -y^{2}\right ) \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.528 |
|
|
\[ {}x^{2} y^{\prime } = a \,x^{2} y^{2}-a y^{3} \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
1.148 |
|
|
\[ {}x^{2} y^{\prime }+a y^{2}+b \,x^{2} y^{3} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
1.58 |
|
|
\[ {}x^{2} y^{\prime } = \left (x a +b y^{3}\right ) y \] |
1 |
3 |
3 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.39 |
|
|
\[ {}x^{2} y^{\prime }+x y+\sqrt {y} = 0 \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.869 |
|
|
\[ {}x^{2} y^{\prime } = \sec \left (y\right )+3 x \tan \left (y\right ) \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.119 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime } = -x^{2}+y+1 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.68 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+1 = x y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.638 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime } = 5-x y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.663 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+a +x y = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.552 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+a -x y = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.687 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+a -x y = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.573 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }-x +x y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.63 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }-x^{2}+x y = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.615 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+x^{2}+x y = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.617 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = \left (x^{2}+1\right ) x -x y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.511 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = x \left (3 x^{2}-y\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.514 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+2 x y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.552 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = 2 x \left (x -y\right ) \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.527 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = 2 x \left (x^{2}+1\right )^{2}+2 x y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.504 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+\cos \left (x \right ) = 2 x y \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.158 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = \tan \left (x \right )-2 x y \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.573 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime } = a +4 x y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.566 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = \left (2 b x +a \right ) y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.69 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = 1+y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.555 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime } = 1-y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.515 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime } = 1-\left (2 x -y\right ) y \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.006 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime } = n \left (1-2 x y+y^{2}\right ) \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.596 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+x y \left (1-y\right ) = 0 \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.104 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime } = x y \left (1+a y\right ) \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.698 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = 1+y^{2}-2 x y \left (1+y^{2}\right ) \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
65.437 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right ) = x \left (x^{2}+1\right ) \cos \left (y\right )^{2} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.553 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime } = 1+x^{2}-y \,\operatorname {arccot}\left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.236 |
|
|
\[ {}\left (-x^{2}+4\right ) y^{\prime }+4 y = \left (2+x \right ) y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.651 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) y^{\prime } = b +x y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.928 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) y^{\prime } = \left (b +y\right ) \left (x +\sqrt {a^{2}+x^{2}}\right ) \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.938 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) y^{\prime }+\left (x -y\right ) y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.622 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) y^{\prime } = a^{2}+3 x y-2 y^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, _Riccati] |
✓ |
✓ |
3.102 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) y^{\prime }+x y+b x y^{2} = 0 \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.346 |
|
|
\[ {}x \left (1-x \right ) y^{\prime } = a +\left (1+x \right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.7 |
|
|
\[ {}x \left (1-x \right ) y^{\prime } = 2 x y+2 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.61 |
|
|
\[ {}x \left (1-x \right ) y^{\prime } = 2 x y-2 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.554 |
|
|
\[ {}x \left (1+x \right ) y^{\prime } = \left (1-2 x \right ) y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.655 |
|
|
\[ {}x \left (1-x \right ) y^{\prime }+\left (2 x +1\right ) y = a \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.641 |
|
|
\[ {}x \left (1-x \right ) y^{\prime } = a +2 \left (2-x \right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.671 |
|
|
\[ {}x \left (1-x \right ) y^{\prime }+2-3 x y+y = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.667 |
|
|
\[ {}x \left (1+x \right ) y^{\prime } = \left (1+x \right ) \left (x^{2}-1\right )+\left (x^{2}+x -1\right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.694 |
|
|
\[ {}\left (-2+x \right ) \left (x -3\right ) y^{\prime }+x^{2}-8 y+3 x y = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.669 |
|
|
\[ {}x \left (x +a \right ) y^{\prime } = \left (b +c y\right ) y \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.375 |
|
|
\[ {}\left (x +a \right )^{2} y^{\prime } = 2 \left (x +a \right ) \left (b +y\right ) \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.874 |
|
|
\[ {}\left (x -a \right )^{2} y^{\prime }+k \left (x +y-a \right )^{2}+y^{2} = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
✓ |
1.631 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }+k y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.49 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime } = \left (x -a \right ) \left (-b +x \right )+\left (2 x -a -b \right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.894 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime } = c y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.829 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }+k \left (y-a \right ) \left (y-b \right ) = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.154 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }+k \left (x +y-a \right ) \left (x +y-b \right )+y^{2} = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
2.243 |
|
|
\[ {}2 x^{2} y^{\prime } = y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.647 |
|
|
\[ {}2 x^{2} y^{\prime }+x \cot \left (x \right )-1+2 x^{2} y \cot \left (x \right ) = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.017 |
|
|
\[ {}2 x^{2} y^{\prime }+1+2 x y-x^{2} y^{2} = 0 \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
0.979 |
|
|
\[ {}2 x^{2} y^{\prime } = 2 x y+\left (1-x \cot \left (x \right )\right ) \left (x^{2}-y^{2}\right ) \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
1.714 |
|
|
\[ {}2 \left (-x^{2}+1\right ) y^{\prime } = \sqrt {-x^{2}+1}+\left (1+x \right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.693 |
|
|
\[ {}x \left (1-2 x \right ) y^{\prime }+1+\left (1-4 x \right ) y = 0 \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.632 |
|
|
\[ {}x \left (1-2 x \right ) y^{\prime } = 4 x -\left (1+4 x \right ) y+y^{2} \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.909 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime }+x +\left (1-2 x \right ) y = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.797 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime }+x +\left (1-x \right ) y^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.364 |
|
|
\[ {}2 \left (x^{2}+x +1\right ) y^{\prime } = 1+8 x^{2}-\left (2 x +1\right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.979 |
|
|
\[ {}4 \left (x^{2}+1\right ) y^{\prime }-4 x y-x^{2} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.651 |
|
|
\[ {}a \,x^{2} y^{\prime } = x^{2}+a x y+y^{2} b^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.956 |
|
|
\[ {}\left (b \,x^{2}+a \right ) y^{\prime } = A +B y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.681 |
|
|
\[ {}\left (b \,x^{2}+a \right ) y^{\prime } = c x y \ln \left (y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.227 |
|
|
\[ {}x \left (x a +1\right ) y^{\prime }+a -y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.753 |
|
|
\[ {}\left (b x +a \right )^{2} y^{\prime }+c y^{2}+\left (b x +a \right ) y^{3} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
4.153 |
|
|
\[ {}x^{3} y^{\prime } = a +b \,x^{2} y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.656 |
|
|
\[ {}x^{3} y^{\prime } = 3-x^{2}+x^{2} y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.605 |
|
|
\[ {}x^{3} y^{\prime } = x^{4}+y^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
0.602 |
|
|
\[ {}x^{3} y^{\prime } = y \left (y+x^{2}\right ) \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.691 |
|
|
\[ {}x^{3} y^{\prime } = x^{2} \left (y-1\right )+y^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.47 |
|
|
\[ {}x^{3} y^{\prime } = \left (1+x \right ) y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.538 |
|
|
\[ {}x^{3} y^{\prime }+20+x^{2} y \left (1-x^{2} y\right ) = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
1.165 |
|
|
\[ {}x^{3} y^{\prime }+3+\left (3-2 x \right ) x^{2} y-x^{6} y^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.024 |
|
|
\[ {}x^{3} y^{\prime } = \left (2 x^{2}+y^{2}\right ) y \] |
1 |
2 |
2 |
bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.748 |
|
|
\[ {}x^{3} y^{\prime } = \cos \left (y\right ) \left (\cos \left (y\right )-2 x^{2} \sin \left (y\right )\right ) \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.839 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime } = x^{2} a +y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.743 |
|
|
\[ {}x \left (-x^{2}+1\right ) y^{\prime } = x^{2} a +y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.806 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime } = a \,x^{3}+y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.788 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime } = a -x^{2} y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.709 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime } = \left (-x^{2}+1\right ) y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.647 |
|
|
\[ {}x \left (-x^{2}+1\right ) y^{\prime } = \left (x^{2}-x +1\right ) y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.71 |
|
|
\[ {}x \left (-x^{2}+1\right ) y^{\prime } = a \,x^{3}+\left (-2 x^{2}+1\right ) y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.964 |
|
|
\[ {}x \left (-x^{2}+1\right ) y^{\prime } = x^{3} \left (-x^{2}+1\right )+\left (-2 x^{2}+1\right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.645 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime } = 2-4 x^{2} y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.67 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime } = x -\left (5 x^{2}+3\right ) y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.613 |
|
|
\[ {}x \left (-x^{2}+1\right ) y^{\prime }+x^{2}+\left (-x^{2}+1\right ) y^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.477 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime } = \left (2-x \right ) x y-y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.813 |
|
|
\[ {}2 x^{3} y^{\prime } = \left (x^{2}-y^{2}\right ) y \] |
1 |
2 |
2 |
bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.756 |
|
|
\[ {}2 x^{3} y^{\prime } = \left (3 x^{2}+a y^{2}\right ) y \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.921 |
|
|
\[ {}6 x^{3} y^{\prime } = 4 x^{2} y+\left (1-3 x \right ) y^{4} \] |
1 |
3 |
3 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.968 |
|
|
\[ {}x \left (c \,x^{2}+b x +a \right ) y^{\prime }+x^{2}-\left (c \,x^{2}+b x +a \right ) y = y^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.576 |
|
|
\[ {}x^{4} y^{\prime } = \left (x^{3}+y\right ) y \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.574 |
|
|
\[ {}x^{4} y^{\prime }+a^{2}+x^{4} y^{2} = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
1.111 |
|
|
\[ {}x^{4} y^{\prime }+x^{3} y+\csc \left (x y\right ) = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
5.848 |
|
|
\[ {}\left (-x^{4}+1\right ) y^{\prime } = 2 x \left (1-y^{2}\right ) \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.381 |
|
|
\[ {}x \left (-x^{3}+1\right ) y^{\prime } = 2 x -\left (-4 x^{3}+1\right ) y \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.726 |
|
|
\[ {}x \left (-x^{3}+1\right ) y^{\prime } = x^{2}+\left (1-2 x y\right ) y \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.112 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime } = \left (x -3 x^{3} y\right ) y \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.868 |
|
|
\[ {}x \left (-2 x^{3}+1\right ) y^{\prime } = 2 \left (-x^{3}+1\right ) y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.826 |
|
|
\[ {}\left (c \,x^{2}+b x +a \right )^{2} \left (y^{\prime }+y^{2}\right )+A = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, _Riccati] |
✓ |
✓ |
3.77 |
|
|
\[ {}x^{5} y^{\prime } = 1-3 x^{4} y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.605 |
|
|
\[ {}x \left (-x^{4}+1\right ) y^{\prime } = 2 x \left (x^{2}-y^{2}\right )+\left (-x^{4}+1\right ) y \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.354 |
|
|
\[ {}x^{7} y^{\prime }+5 x^{3} y^{2}+2 \left (x^{2}+1\right ) y^{3} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
61.092 |
|
|
\[ {}x^{n} y^{\prime } = a +b \,x^{n -1} y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.692 |
|
|
\[ {}x^{n} y^{\prime } = x^{2 n -1}-y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.288 |
|
|
\[ {}x^{n} y^{\prime }+x^{2 n -2}+y^{2}+\left (-n +1\right ) x^{n -1} = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
12.794 |
|
|
\[ {}x^{n} y^{\prime } = a^{2} x^{2 n -2}+y^{2} b^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
3.025 |
|
|
\[ {}x^{n} y^{\prime } = x^{n -1} \left (a \,x^{2 n}+n y-b y^{2}\right ) \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.429 |
|
|
\[ {}x^{k} y^{\prime } = a \,x^{m}+b y^{n} \] |
1 |
0 |
0 |
unknown |
[_Chini] |
❇ |
N/A |
0.528 |
|
|
\[ {}y^{\prime } \sqrt {x^{2}+1} = 2 x -y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.746 |
|
|
\[ {}y^{\prime } \sqrt {-x^{2}+1} = 1+y^{2} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.656 |
|
|
\[ {}\left (x -\sqrt {x^{2}+1}\right ) y^{\prime } = y+\sqrt {1+y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.009 |
|
|
\[ {}y^{\prime } \sqrt {a^{2}+x^{2}}+x +y = \sqrt {a^{2}+x^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.826 |
|
|
\[ {}y^{\prime } \sqrt {b^{2}+x^{2}} = \sqrt {y^{2}+a^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
11.581 |
|
|
\[ {}y^{\prime } \sqrt {b^{2}-x^{2}} = \sqrt {a^{2}-y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.438 |
|
|
\[ {}x y^{\prime } \sqrt {a^{2}+x^{2}} = y \sqrt {b^{2}+y^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
9.665 |
|
|
\[ {}x y^{\prime } \sqrt {-a^{2}+x^{2}} = y \sqrt {y^{2}-b^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.847 |
|
|
\[ {}y^{\prime } \sqrt {X}+\sqrt {Y} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.12 |
|
|
\[ {}y^{\prime } \sqrt {X} = \sqrt {Y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.105 |
|
|
\[ {}x^{\frac {3}{2}} y^{\prime } = a +b \,x^{\frac {3}{2}} y^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
1.547 |
|
|
\[ {}y^{\prime } \sqrt {x^{3}+1} = \sqrt {y^{3}+1} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
77.213 |
|
|
\[ {}y^{\prime } \sqrt {x \left (1-x \right ) \left (-x a +1\right )} = \sqrt {y \left (1-y\right ) \left (1-a y\right )} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
9.26 |
|
|
\[ {}y^{\prime } \sqrt {-x^{4}+1} = \sqrt {1-y^{4}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.47 |
|
|
\[ {}y^{\prime } \sqrt {x^{4}+x^{2}+1} = \sqrt {1+y^{2}+y^{4}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.326 |
|
|
\[ {}y^{\prime } \sqrt {X} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.059 |
|
|
\[ {}y^{\prime } \sqrt {X}+\sqrt {Y} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.099 |
|
|
\[ {}y^{\prime } \sqrt {X} = \sqrt {Y} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.085 |
|
|
\[ {}y^{\prime } \left (x^{3}+1\right )^{\frac {2}{3}}+\left (y^{3}+1\right )^{\frac {2}{3}} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
6.859 |
|
|
\[ {}y^{\prime } \left (4 x^{3}+\operatorname {a1} x +\operatorname {a0} \right )^{\frac {2}{3}}+\left (\operatorname {a0} +\operatorname {a1} y+4 y^{3}\right )^{\frac {2}{3}} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.793 |
|
|
|
|||||||||
|
\[ {}X^{\frac {2}{3}} y^{\prime } = Y^{\frac {2}{3}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.128 |
|
|
\[ {}y^{\prime } \left (a +\cos \left (\frac {x}{2}\right )^{2}\right ) = y \tan \left (\frac {x}{2}\right ) \left (1+a +\cos \left (\frac {x}{2}\right )^{2}-y\right ) \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
10.263 |
|
|
\[ {}\left (1-4 \cos \left (x \right )^{2}\right ) y^{\prime } = \tan \left (x \right ) \left (1+4 \cos \left (x \right )^{2}\right ) y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.974 |
|
|
\[ {}\left (1-\sin \left (x \right )\right ) y^{\prime }+y \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.358 |
|
|
\[ {}\left (\cos \left (x \right )-\sin \left (x \right )\right ) y^{\prime }+y \left (\cos \left (x \right )+\sin \left (x \right )\right ) = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.652 |
|
|
\[ {}\left (\operatorname {a0} +\operatorname {a1} \sin \left (x \right )^{2}\right ) y^{\prime }+\operatorname {a2} x \left (\operatorname {a3} +\operatorname {a1} \sin \left (x \right )^{2}\right )+\operatorname {a1} y \sin \left (2 x \right ) = 0 \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.44 |
|
|
\[ {}\left (x -{\mathrm e}^{x}\right ) y^{\prime }+x \,{\mathrm e}^{x}+\left (-{\mathrm e}^{x}+1\right ) y = 0 \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.624 |
|
|
\[ {}y^{\prime } x \ln \left (x \right ) = a x \left (1+\ln \left (x \right )\right )-y \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.815 |
|
|
\[ {}y y^{\prime }+x = 0 \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.987 |
|
|
\[ {}y y^{\prime }+x \,{\mathrm e}^{x^{2}} = 0 \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.475 |
|
|
\[ {}y y^{\prime }+x^{3}+y = 0 \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.399 |
|
|
\[ {}y y^{\prime }+x a +b y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
11.408 |
|
|
\[ {}y y^{\prime }+x \,{\mathrm e}^{-x} \left (y+1\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.954 |
|
|
\[ {}y y^{\prime }+f \left (x \right ) = g \left (x \right ) y \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.362 |
|
|
\[ {}y y^{\prime }+4 \left (1+x \right ) x +y^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.223 |
|
|
\[ {}y y^{\prime } = x a +b y^{2} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.854 |
|
|
\[ {}y y^{\prime } = b \cos \left (x +c \right )+a y^{2} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.502 |
|
|
\[ {}y y^{\prime } = \operatorname {a0} +\operatorname {a1} y+\operatorname {a2} y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.535 |
|
|
\[ {}y y^{\prime } = x a +b x y^{2} \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.19 |
|
|
\[ {}y y^{\prime } = \csc \left (x \right )^{2}-y^{2} \cot \left (x \right ) \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
21.66 |
|
|
\[ {}y y^{\prime } = \sqrt {y^{2}+a^{2}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.248 |
|
|
\[ {}y y^{\prime } = \sqrt {y^{2}-a^{2}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.211 |
|
|
\[ {}y y^{\prime }+x +f \left (x^{2}+y^{2}\right ) g \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
1.201 |
|
|
\[ {}\left (y+1\right ) y^{\prime } = x +y \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
6.916 |
|
|
\[ {}\left (y+1\right ) y^{\prime } = x^{2} \left (1-y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.658 |
|
|
\[ {}\left (x +y\right ) y^{\prime }+y = 0 \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.407 |
|
|
\[ {}\left (x -y\right ) y^{\prime } = y \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.863 |
|
|
\[ {}\left (x +y\right ) y^{\prime }+x -y = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.908 |
|
|
\[ {}\left (x +y\right ) y^{\prime } = x -y \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.351 |
|
|
\[ {}1-y^{\prime } = x +y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.459 |
|
|
\[ {}\left (x -y\right ) y^{\prime } = y \left (2 x y+1\right ) \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor |
[[_homogeneous, ‘class D‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.059 |
|
|
\[ {}\left (x +y\right ) y^{\prime }+\tan \left (y\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.849 |
|
|
\[ {}\left (x -y\right ) y^{\prime } = \left ({\mathrm e}^{-\frac {x}{y}}+1\right ) y \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.217 |
|
|
\[ {}\left (1+x +y\right ) y^{\prime }+1+4 x +3 y = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.18 |
|
|
\[ {}\left (x +y+2\right ) y^{\prime } = 1-x -y \] |
1 |
1 |
2 |
exact, differentialType, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.888 |
|
|
\[ {}\left (3-x -y\right ) y^{\prime } = 1+x -3 y \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.21 |
|
|
\[ {}\left (3-x +y\right ) y^{\prime } = 11-4 x +3 y \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.207 |
|
|
\[ {}\left (y+2 x \right ) y^{\prime }+x -2 y = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.872 |
|
|
\[ {}\left (2 x -y+2\right ) y^{\prime }+3+6 x -3 y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.756 |
|
|
\[ {}\left (3+2 x -y\right ) y^{\prime }+2 = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.776 |
|
|
\[ {}\left (4+2 x -y\right ) y^{\prime }+5+x -2 y = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.676 |
|
|
\[ {}\left (5-2 x -y\right ) y^{\prime }+4-x -2 y = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.998 |
|
|
\[ {}\left (1-3 x +y\right ) y^{\prime } = 2 x -2 y \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.745 |
|
|
\[ {}\left (2-3 x +y\right ) y^{\prime }+5-2 x -3 y = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.981 |
|
|
\[ {}\left (4 x -y\right ) y^{\prime }+2 x -5 y = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.383 |
|
|
\[ {}\left (6-4 x -y\right ) y^{\prime } = 2 x -y \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.574 |
|
|
\[ {}\left (1+5 x -y\right ) y^{\prime }+5+x -5 y = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.514 |
|
|
\[ {}\left (a +b x +y\right ) y^{\prime }+a -b x -y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.61 |
|
|
\[ {}\left (x^{2}-y\right ) y^{\prime }+x = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
0.72 |
|
|
\[ {}\left (x^{2}-y\right ) y^{\prime } = 4 x y \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.272 |
|
|
\[ {}\left (y-\cot \left (x \right ) \csc \left (x \right )\right ) y^{\prime }+\csc \left (x \right ) \left (1+y \cos \left (x \right )\right ) y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
41.14 |
|
|
\[ {}2 y y^{\prime }+2 x +x^{2}+y^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.248 |
|
|
\[ {}2 y y^{\prime } = x y^{2}+x^{3} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.678 |
|
|
\[ {}\left (x -2 y\right ) y^{\prime } = y \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.911 |
|
|
\[ {}\left (2 y+x \right ) y^{\prime }+2 x -y = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.891 |
|
|
\[ {}\left (x -2 y\right ) y^{\prime }+2 x +y = 0 \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.19 |
|
|
\[ {}\left (1+x -2 y\right ) y^{\prime } = 1+2 x -y \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.033 |
|
|
\[ {}\left (1+x +2 y\right ) y^{\prime }+1-x -2 y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.711 |
|
|
\[ {}\left (1+x +2 y\right ) y^{\prime }+7+x -4 y = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.697 |
|
|
\[ {}2 \left (x +y\right ) y^{\prime }+x^{2}+2 y = 0 \] |
1 |
1 |
2 |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.186 |
|
|
\[ {}\left (3+2 x -2 y\right ) y^{\prime } = 1+6 x -2 y \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.346 |
|
|
\[ {}\left (1-4 x -2 y\right ) y^{\prime }+2 x +y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.0 |
|
|
\[ {}\left (6 x -2 y\right ) y^{\prime } = 2+3 x -y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.848 |
|
|
\[ {}\left (19+9 x +2 y\right ) y^{\prime }+18-2 x -6 y = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.625 |
|
|
\[ {}\left (x^{3}+2 y\right ) y^{\prime } = 3 x \left (2-x y\right ) \] |
1 |
1 |
2 |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.348 |
|
|
\[ {}\left (\tan \left (x \right ) \sec \left (x \right )-2 y\right ) y^{\prime }+\sec \left (x \right ) \left (1+2 y \sin \left (x \right )\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
6.388 |
|
|
\[ {}\left (x \,{\mathrm e}^{-x}-2 y\right ) y^{\prime } = 2 x \,{\mathrm e}^{-2 x}-\left ({\mathrm e}^{-x}+x \,{\mathrm e}^{-x}-2 y\right ) y \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.798 |
|
|
\[ {}3 y y^{\prime }+5 \cot \left (x \right ) \cot \left (y\right ) \cos \left (y\right )^{2} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
5.5 |
|
|
\[ {}3 \left (2-y\right ) y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.803 |
|
|
\[ {}\left (x -3 y\right ) y^{\prime }+4+3 x -y = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.618 |
|
|
\[ {}\left (4-x -3 y\right ) y^{\prime }+3-x -3 y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.747 |
|
|
\[ {}\left (2+2 x +3 y\right ) y^{\prime } = 1-2 x -3 y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.739 |
|
|
\[ {}\left (5-2 x -3 y\right ) y^{\prime }+1-2 x -3 y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.742 |
|
|
\[ {}\left (1+9 x -3 y\right ) y^{\prime }+2+3 x -y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.846 |
|
|
\[ {}\left (x +4 y\right ) y^{\prime }+4 x -y = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.891 |
|
|
\[ {}\left (3+2 x +4 y\right ) y^{\prime } = 1+x +2 y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.718 |
|
|
\[ {}\left (5+2 x -4 y\right ) y^{\prime } = 3+x -2 y \] |
1 |
1 |
2 |
exact, differentialType, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.953 |
|
|
\[ {}\left (5+3 x -4 y\right ) y^{\prime } = 2+7 x -3 y \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.467 |
|
|
\[ {}4 \left (1-x -y\right ) y^{\prime }+2-x = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.184 |
|
|
\[ {}\left (11-11 x -4 y\right ) y^{\prime } = 62-8 x -25 y \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.689 |
|
|
\[ {}\left (6+3 x +5 y\right ) y^{\prime } = 2+x +7 y \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.254 |
|
|
\[ {}\left (7 x +5 y\right ) y^{\prime }+10 x +8 y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.203 |
|
|
\[ {}\left (x +4 x^{3}+5 y\right ) y^{\prime }+7 x^{3}+3 x^{2} y+4 y = 0 \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.106 |
|
|
\[ {}\left (5-x +6 y\right ) y^{\prime } = 3-x +4 y \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.732 |
|
|
\[ {}3 \left (2 y+x \right ) y^{\prime } = 1-x -2 y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.709 |
|
|
\[ {}\left (3-3 x +7 y\right ) y^{\prime }+7-7 x +3 y = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.565 |
|
|
\[ {}\left (1+x +9 y\right ) y^{\prime }+1+x +5 y = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.249 |
|
|
\[ {}\left (8+5 x -12 y\right ) y^{\prime } = 3+2 x -5 y \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.115 |
|
|
\[ {}\left (140+7 x -16 y\right ) y^{\prime }+25+8 x +y = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.699 |
|
|
\[ {}\left (3+9 x +21 y\right ) y^{\prime } = 45+7 x -5 y \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.666 |
|
|
\[ {}\left (x a +b y\right ) y^{\prime }+x = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
17.747 |
|
|
\[ {}\left (x a +b y\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.525 |
|
|
\[ {}\left (x a +b y\right ) y^{\prime }+b x +a y = 0 \] |
1 |
1 |
2 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.478 |
|
|
\[ {}\left (x a +b y\right ) y^{\prime } = b x +a y \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.796 |
|
|
\[ {}x y y^{\prime }+1+y^{2} = 0 \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.368 |
|
|
\[ {}x y y^{\prime } = x +y^{2} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.72 |
|
|
\[ {}x y y^{\prime }+x^{2}+y^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.955 |
|
|
\[ {}x y y^{\prime }+x^{4}-y^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.75 |
|
|
\[ {}x y y^{\prime } = a \,x^{3} \cos \left (x \right )+y^{2} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _Bernoulli] |
✓ |
✓ |
0.971 |
|
|
\[ {}x y y^{\prime } = x^{2}-x y+y^{2} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.839 |
|
|
\[ {}x y y^{\prime }+2 x^{2}-2 x y-y^{2} = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.984 |
|
|
\[ {}x y y^{\prime } = a +b y^{2} \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.295 |
|
|
\[ {}x y y^{\prime } = a \,x^{n}+b y^{2} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.063 |
|
|
\[ {}x y y^{\prime } = \left (x^{2}+1\right ) \left (1-y^{2}\right ) \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
12.514 |
|
|
\[ {}x y y^{\prime }+x^{2} \operatorname {arccot}\left (\frac {y}{x}\right )-y^{2} = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.49 |
|
|
\[ {}x y y^{\prime }+x^{2} {\mathrm e}^{-\frac {2 y}{x}}-y^{2} = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.916 |
|
|
\[ {}\left (1+x y\right ) y^{\prime }+y^{2} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.757 |
|
|
\[ {}x \left (y+1\right ) y^{\prime }-\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.731 |
|
|
\[ {}x \left (1-y\right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.76 |
|
|
\[ {}x \left (1-y\right ) y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.755 |
|
|
\[ {}x \left (y+2\right ) y^{\prime }+x a = 0 \] |
1 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.319 |
|
|
\[ {}\left (2+3 x -x y\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.839 |
|
|
\[ {}x \left (4+y\right ) y^{\prime } = 2 x +2 y+y^{2} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
5.195 |
|
|
\[ {}x \left (a +y\right ) y^{\prime }+b x +c y = 0 \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.438 |
|
|
\[ {}x \left (a +y\right ) y^{\prime } = y \left (B x +A \right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.269 |
|
|
\[ {}x \left (x +y\right ) y^{\prime }+y^{2} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.396 |
|
|
\[ {}x \left (x -y\right ) y^{\prime }+y^{2} = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.925 |
|
|
\[ {}x \left (x +y\right ) y^{\prime } = x^{2}+y^{2} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.95 |
|
|
\[ {}x \left (x -y\right ) y^{\prime }+2 x^{2}+3 x y-y^{2} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.451 |
|
|
\[ {}x \left (x +y\right ) y^{\prime }-y \left (x +y\right )+x \sqrt {x^{2}-y^{2}} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.695 |
|
|
\[ {}\left (a +x \left (x +y\right )\right ) y^{\prime } = b \left (x +y\right ) y \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.633 |
|
|
\[ {}x \left (y+2 x \right ) y^{\prime } = x^{2}+x y-y^{2} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.181 |
|
|
\[ {}x \left (4 x -y\right ) y^{\prime }+4 x^{2}-6 x y-y^{2} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
5.661 |
|
|
\[ {}x \left (x^{3}+y\right ) y^{\prime } = \left (x^{3}-y\right ) y \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.257 |
|
|
\[ {}x \left (2 x^{3}+y\right ) y^{\prime } = \left (2 x^{3}-y\right ) y \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.253 |
|
|
\[ {}x \left (2 x^{3}+y\right ) y^{\prime } = 6 y^{2} \] |
1 |
1 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.349 |
|
|
\[ {}y \left (1-x \right ) y^{\prime }+x \left (1-y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.775 |
|
|
\[ {}\left (x +a \right ) \left (x +b \right ) y^{\prime } = x y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.069 |
|
|
\[ {}2 x y y^{\prime }+1-2 x^{3}-y^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.704 |
|
|
\[ {}2 x y y^{\prime }+a +y^{2} = 0 \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.216 |
|
|
\[ {}2 x y y^{\prime } = x a +y^{2} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.686 |
|
|
\[ {}2 x y y^{\prime }+x^{2}+y^{2} = 0 \] |
1 |
1 |
2 |
exact, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
1.293 |
|
|
\[ {}2 x y y^{\prime } = x^{2}+y^{2} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.924 |
|
|
\[ {}2 x y y^{\prime } = 4 x^{2} \left (2 x +1\right )+y^{2} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.618 |
|
|
\[ {}2 x y y^{\prime }+x^{2} \left (a \,x^{3}+1\right ) = 6 y^{2} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.755 |
|
|
\[ {}\left (3-x +2 x y\right ) y^{\prime }+3 x^{2}-y+y^{2} = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.229 |
|
|
\[ {}x \left (x -2 y\right ) y^{\prime }+y^{2} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.421 |
|
|
\[ {}x \left (2 y+x \right ) y^{\prime }+\left (2 x -y\right ) y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.434 |
|
|
\[ {}x \left (x -2 y\right ) y^{\prime }+\left (2 x -y\right ) y = 0 \] |
1 |
1 |
2 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.526 |
|
|
\[ {}x \left (1+x -2 y\right ) y^{\prime }+\left (1-2 x +y\right ) y = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.194 |
|
|
\[ {}x \left (1-x -2 y\right ) y^{\prime }+\left (2 x +y+1\right ) y = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.174 |
|
|
\[ {}2 x \left (2 x^{2}+y\right ) y^{\prime }+\left (12 x^{2}+y\right ) y = 0 \] |
1 |
1 |
2 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.415 |
|
|
\[ {}2 \left (1+x \right ) y y^{\prime }+2 x -3 x^{2}+y^{2} = 0 \] |
1 |
1 |
2 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[_exact, _rational, _Bernoulli] |
✓ |
✓ |
0.92 |
|
|
\[ {}x \left (2 x +3 y\right ) y^{\prime } = y^{2} \] |
1 |
1 |
3 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.437 |
|
|
\[ {}x \left (2 x +3 y\right ) y^{\prime }+3 \left (x +y\right )^{2} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.763 |
|
|
\[ {}\left (3+6 x y+x^{2}\right ) y^{\prime }+2 x +2 x y+3 y^{2} = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.997 |
|
|
\[ {}3 x \left (2 y+x \right ) y^{\prime }+x^{3}+3 y \left (y+2 x \right ) = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.068 |
|
|
\[ {}a x y y^{\prime } = x^{2}+y^{2} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.403 |
|
|
\[ {}a x y y^{\prime }+x^{2}-y^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.078 |
|
|
\[ {}x \left (a +b y\right ) y^{\prime } = c y \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.72 |
|
|
\[ {}x \left (x -a y\right ) y^{\prime } = y \left (-x a +y\right ) \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.023 |
|
|
\[ {}x \left (x^{n}+a y\right ) y^{\prime }+\left (b +c y\right ) y^{2} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
6.516 |
|
|
\[ {}\left (1-x^{2} y\right ) y^{\prime }+1-x y^{2} = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.925 |
|
|
\[ {}\left (1-x^{2} y\right ) y^{\prime }-1+x y^{2} = 0 \] |
1 |
0 |
3 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.747 |
|
|
\[ {}x \left (1-x y\right ) y^{\prime }+\left (1+x y\right ) y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.041 |
|
|
\[ {}x \left (x y+2\right ) y^{\prime } = 3+2 x^{3}-2 y-x y^{2} \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.267 |
|
|
\[ {}x \left (2-x y\right ) y^{\prime }+2 y-x y^{2} \left (1+x y\right ) = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
0.848 |
|
|
\[ {}x \left (3-x y\right ) y^{\prime } = y \left (x y-1\right ) \] |
1 |
1 |
3 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.184 |
|
|
\[ {}x^{2} \left (1-y\right ) y^{\prime }+\left (1-x \right ) y = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.03 |
|
|
\[ {}x^{2} \left (1-y\right ) y^{\prime }+\left (1+x \right ) y^{2} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.244 |
|
|
\[ {}\left (x^{2}+1\right ) y y^{\prime }+x \left (1-y^{2}\right ) = 0 \] |
1 |
1 |
2 |
exact, bernoulli, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.071 |
|
|
\[ {}\left (-x^{2}+1\right ) y y^{\prime }+2 x^{2}+x y^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.832 |
|
|
\[ {}2 x^{2} y y^{\prime } = x^{2} \left (2 x +1\right )-y^{2} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.65 |
|
|
\[ {}x \left (1-2 x y\right ) y^{\prime }+y \left (2 x y+1\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.144 |
|
|
\[ {}x \left (2 x y+1\right ) y^{\prime }+\left (2+3 x y\right ) y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.707 |
|
|
\[ {}x \left (2 x y+1\right ) y^{\prime }+\left (1+2 x y-x^{2} y^{2}\right ) y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
0.883 |
|
|
\[ {}x^{2} \left (x -2 y\right ) y^{\prime } = 2 x^{3}-4 x y^{2}+y^{3} \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.968 |
|
|
\[ {}2 \left (1+x \right ) x y y^{\prime } = 1+y^{2} \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.658 |
|
|
\[ {}3 x^{2} y y^{\prime }+1+2 x y^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.836 |
|
|
\[ {}x^{2} \left (4 x -3 y\right ) y^{\prime } = \left (6 x^{2}-3 x y+2 y^{2}\right ) y \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.125 |
|
|
\[ {}\left (1-x^{3} y\right ) y^{\prime } = x^{2} y^{2} \] |
1 |
1 |
9 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.39 |
|
|
\[ {}2 x^{3} y y^{\prime }+a +3 x^{2} y^{2} = 0 \] |
1 |
1 |
2 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.829 |
|
|
\[ {}x \left (3-2 x^{2} y\right ) y^{\prime } = 4 x -3 y+3 x^{2} y^{2} \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.14 |
|
|
\[ {}x \left (3+2 x^{2} y\right ) y^{\prime }+\left (4+3 x^{2} y\right ) y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
6.575 |
|
|
\[ {}8 x^{3} y y^{\prime }+3 x^{4}-6 x^{2} y^{2}-y^{4} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.09 |
|
|
\[ {}x y \left (b \,x^{2}+a \right ) y^{\prime } = A +B y^{2} \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.143 |
|
|
\[ {}3 x^{4} y y^{\prime } = 1-2 x^{3} y^{2} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.97 |
|
|
\[ {}x^{7} y y^{\prime } = 2 x^{2}+2+5 x^{3} y \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.408 |
|
|
\[ {}y y^{\prime } \sqrt {x^{2}+1}+x \sqrt {1+y^{2}} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.895 |
|
|
\[ {}\left (y+1\right ) y^{\prime } \sqrt {x^{2}+1} = y^{3} \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.23 |
|
|
\[ {}\left (\operatorname {g0} \left (x \right )+y \operatorname {g1} \left (x \right )\right ) y^{\prime } = \operatorname {f0} \left (x \right )+\operatorname {f1} \left (x \right ) y+\operatorname {f2} \left (x \right ) y^{2}+\operatorname {f3} \left (x \right ) y^{3} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
❇ |
N/A |
1.832 |
|
|
\[ {}y^{2} y^{\prime }+x \left (2-y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{2} y^{\prime } = x \left (1+y^{2}\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.704 |
|
|
\[ {}\left (x +y^{2}\right ) y^{\prime }+y = b x +a \] |
1 |
1 |
3 |
exact |
[_exact, _rational] |
✓ |
✓ |
1.427 |
|
|
\[ {}\left (x -y^{2}\right ) y^{\prime } = x^{2}-y \] |
1 |
1 |
3 |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
9.654 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
1 |
1 |
4 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.749 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime } = x y \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.924 |
|
|
\[ {}\left (x^{2}-y^{2}\right ) y^{\prime } = 2 x y \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.559 |
|
|
\[ {}\left (x^{2}-y^{2}\right ) y^{\prime }+x \left (2 y+x \right ) = 0 \] |
1 |
1 |
3 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
7.746 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (y+2 x \right ) = 0 \] |
1 |
1 |
3 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
9.385 |
|
|
\[ {}\left (1-x^{2}+y^{2}\right ) y^{\prime } = 1+x^{2}-y^{2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
0.654 |
|
|
\[ {}\left (a^{2}+x^{2}+y^{2}\right ) y^{\prime }+2 x y = 0 \] |
1 |
1 |
3 |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
8.592 |
|
|
\[ {}\left (a^{2}+x^{2}+y^{2}\right ) y^{\prime }+b^{2}+x^{2}+2 x y = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational] |
✓ |
✓ |
1.213 |
|
|
\[ {}\left (x +x^{2}+y^{2}\right ) y^{\prime } = y \] |
1 |
1 |
1 |
exactByInspection |
[_rational] |
✓ |
✓ |
0.873 |
|
|
\[ {}\left (3 x^{2}-y^{2}\right ) y^{\prime } = 2 x y \] |
1 |
1 |
3 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.833 |
|
|
\[ {}\left (x^{4}+y^{2}\right ) y^{\prime } = 4 x^{3} y \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.571 |
|
|
\[ {}y \left (y+1\right ) y^{\prime } = \left (1+x \right ) x \] |
1 |
1 |
3 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
165.359 |
|
|
\[ {}\left (x +2 y+y^{2}\right ) y^{\prime }+y \left (y+1\right )+\left (x +y\right )^{2} y^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.139 |
|
|
\[ {}\left (x^{2}+2 y+y^{2}\right ) y^{\prime }+2 x = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
0.951 |
|
|
\[ {}\left (x^{3}+2 y-y^{2}\right ) y^{\prime }+3 x^{2} y = 0 \] |
1 |
1 |
3 |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
10.477 |
|
|
|
|||||||||
|
\[ {}\left (1+y+x y+y^{2}\right ) y^{\prime }+1+y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
2.296 |
|
|
\[ {}\left (x +y\right )^{2} y^{\prime } = a^{2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.724 |
|
|
\[ {}\left (x -y\right )^{2} y^{\prime } = a^{2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.306 |
|
|
\[ {}\left (x^{2}+2 x y-y^{2}\right ) y^{\prime }+x^{2}-2 x y+y^{2} = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.846 |
|
|
\[ {}\left (x +y\right )^{2} y^{\prime } = x^{2}-2 x y+5 y^{2} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.983 |
|
|
\[ {}\left (a +b +x +y\right )^{2} y^{\prime } = 2 \left (a +y\right )^{2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
1.479 |
|
|
\[ {}\left (2 x^{2}+4 x y-y^{2}\right ) y^{\prime } = x^{2}-4 x y-2 y^{2} \] |
1 |
1 |
3 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
3.232 |
|
|
\[ {}\left (3 x +y\right )^{2} y^{\prime } = 4 \left (3 x +2 y\right ) y \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.04 |
|
|
\[ {}\left (1-3 x -y\right )^{2} y^{\prime } = \left (1-2 y\right ) \left (3-6 x -4 y\right ) \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
3.851 |
|
|
\[ {}\left (\cot \left (x \right )-2 y^{2}\right ) y^{\prime } = y^{3} \csc \left (x \right ) \sec \left (x \right ) \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
37.644 |
|
|
\[ {}3 y^{2} y^{\prime } = 1+x +a y^{3} \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.469 |
|
|
\[ {}\left (x^{2}-3 y^{2}\right ) y^{\prime }+1+2 x y = 0 \] |
1 |
1 |
3 |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
9.799 |
|
|
\[ {}\left (2 x^{2}+3 y^{2}\right ) y^{\prime }+x \left (3 x +y\right ) = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.871 |
|
|
\[ {}3 \left (x^{2}-y^{2}\right ) y^{\prime }+3 \,{\mathrm e}^{x}+6 x y \left (1+x \right )-2 y^{3} = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.554 |
|
|
\[ {}\left (3 x^{2}+2 x y+4 y^{2}\right ) y^{\prime }+2 x^{2}+6 x y+y^{2} = 0 \] |
1 |
1 |
3 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
1.765 |
|
|
\[ {}\left (1-3 x +2 y\right )^{2} y^{\prime } = \left (4+2 x -3 y\right )^{2} \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
5.158 |
|
|
\[ {}\left (1-3 x^{2} y+6 y^{2}\right ) y^{\prime }+x^{2}-3 x y^{2} = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational] |
✓ |
✓ |
1.418 |
|
|
\[ {}\left (x -6 y\right )^{2} y^{\prime }+a +2 x y-6 y^{2} = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
1.193 |
|
|
\[ {}\left (x^{2}+a y^{2}\right ) y^{\prime } = x y \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.386 |
|
|
\[ {}\left (x^{2}+x y+a y^{2}\right ) y^{\prime } = x^{2} a +x y+y^{2} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.31 |
|
|
\[ {}\left (x^{2} a +2 x y-a y^{2}\right ) y^{\prime }+x^{2}-2 a x y-y^{2} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
10.048 |
|
|
\[ {}\left (x^{2} a +2 b x y+c y^{2}\right ) y^{\prime }+k \,x^{2}+2 a x y+b y^{2} = 0 \] |
1 |
1 |
3 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
3.106 |
|
|
\[ {}x \left (1-y^{2}\right ) y^{\prime } = \left (x^{2}+1\right ) y \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.766 |
|
|
\[ {}x \left (3 x -y^{2}\right ) y^{\prime }+\left (5 x -2 y^{2}\right ) y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.765 |
|
|
\[ {}x \left (x^{2}+y^{2}\right ) y^{\prime } = \left (x^{2}+x^{4}+y^{2}\right ) y \] |
1 |
1 |
1 |
homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational] |
✓ |
✓ |
1.078 |
|
|
\[ {}x \left (1-x^{2}+y^{2}\right ) y^{\prime }+\left (1+x^{2}-y^{2}\right ) y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
5.356 |
|
|
\[ {}x \left (a -x^{2}-y^{2}\right ) y^{\prime }+\left (a +x^{2}+y^{2}\right ) y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
5.602 |
|
|
\[ {}x \left (2 x^{2}+y^{2}\right ) y^{\prime } = \left (2 x^{2}+3 y^{2}\right ) y \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.135 |
|
|
\[ {}\left (x \left (a -x^{2}-y^{2}\right )+y\right ) y^{\prime }+x -\left (a -x^{2}-y^{2}\right ) y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
2.066 |
|
|
\[ {}x \left (a +y\right )^{2} y^{\prime } = b y^{2} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.206 |
|
|
\[ {}x \left (x^{2}-x y+y^{2}\right ) y^{\prime }+\left (x^{2}+x y+y^{2}\right ) y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.157 |
|
|
\[ {}x \left (x^{2}-x y-y^{2}\right ) y^{\prime } = \left (x^{2}+x y-y^{2}\right ) y \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.309 |
|
|
\[ {}x \left (x^{2}+a x y+y^{2}\right ) y^{\prime } = \left (x^{2}+b x y+y^{2}\right ) y \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.513 |
|
|
\[ {}x \left (x^{2}-2 y^{2}\right ) y^{\prime } = \left (2 x^{2}-y^{2}\right ) y \] |
1 |
1 |
6 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
4.329 |
|
|
\[ {}x \left (x^{2}+2 y^{2}\right ) y^{\prime } = \left (2 x^{2}+3 y^{2}\right ) y \] |
1 |
1 |
4 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.257 |
|
|
\[ {}2 x \left (5 x^{2}+y^{2}\right ) y^{\prime } = x^{2} y-y^{3} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.391 |
|
|
\[ {}x \left (x^{2}+a x y+2 y^{2}\right ) y^{\prime } = \left (x a +2 y\right ) y^{2} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.135 |
|
|
\[ {}3 x y^{2} y^{\prime } = 2 x -y^{3} \] |
1 |
1 |
3 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
1.135 |
|
|
\[ {}\left (1-4 x +3 x y^{2}\right ) y^{\prime } = \left (2-y^{2}\right ) y \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.402 |
|
|
\[ {}x \left (x -3 y^{2}\right ) y^{\prime }+\left (2 x -y^{2}\right ) y = 0 \] |
1 |
1 |
3 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
✓ |
3.18 |
|
|
\[ {}3 x \left (x +y^{2}\right ) y^{\prime }+x^{3}-3 x y-2 y^{3} = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.22 |
|
|
\[ {}x \left (x^{3}-3 x^{3} y+4 y^{2}\right ) y^{\prime } = 6 y^{3} \] |
1 |
0 |
1 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✗ |
N/A |
1.036 |
|
|
\[ {}6 x y^{2} y^{\prime }+x +2 y^{3} = 0 \] |
1 |
1 |
3 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.997 |
|
|
\[ {}x \left (x +6 y^{2}\right ) y^{\prime }+x y-3 y^{3} = 0 \] |
1 |
1 |
1 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.339 |
|
|
\[ {}x \left (x^{2}-6 y^{2}\right ) y^{\prime } = 4 \left (x^{2}+3 y^{2}\right ) y \] |
1 |
1 |
2 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.709 |
|
|
\[ {}x \left (3 x -7 y^{2}\right ) y^{\prime }+\left (5 x -3 y^{2}\right ) y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.315 |
|
|
\[ {}x^{2} y^{2} y^{\prime }+1-x +x^{3} = 0 \] |
1 |
1 |
3 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.359 |
|
|
\[ {}\left (1-x^{2} y^{2}\right ) y^{\prime } = x y^{3} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.975 |
|
|
\[ {}\left (1-x^{2} y^{2}\right ) y^{\prime } = \left (1+x y\right ) y^{2} \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.971 |
|
|
\[ {}x \left (1+x y^{2}\right ) y^{\prime }+y = 0 \] |
1 |
1 |
4 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.185 |
|
|
\[ {}x \left (1+x y^{2}\right ) y^{\prime } = \left (2-3 x y^{2}\right ) y \] |
1 |
1 |
2 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.484 |
|
|
\[ {}x^{2} \left (a +y\right )^{2} y^{\prime } = \left (x^{2}+1\right ) \left (y^{2}+a^{2}\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.001 |
|
|
\[ {}\left (x^{2}+1\right ) \left (1+y^{2}\right ) y^{\prime }+2 x y \left (1-y^{2}\right ) = 0 \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.552 |
|
|
\[ {}\left (x^{2}+1\right ) \left (1+y^{2}\right ) y^{\prime }+2 x y \left (1-y\right )^{2} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.2 |
|
|
\[ {}\left (1-x^{3}+6 x^{2} y^{2}\right ) y^{\prime } = \left (6+3 x y-4 y^{3}\right ) x \] |
1 |
1 |
3 |
exact |
[_exact, _rational] |
✓ |
✓ |
1.378 |
|
|
\[ {}x \left (3+5 x -12 x y^{2}+4 x^{2} y\right ) y^{\prime }+\left (3+10 x -8 x y^{2}+6 x^{2} y\right ) y = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational] |
✓ |
✓ |
1.655 |
|
|
\[ {}x^{3} \left (1+y^{2}\right ) y^{\prime }+3 x^{2} y = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.556 |
|
|
\[ {}x \left (1-x y\right )^{2} y^{\prime }+\left (1+x^{2} y^{2}\right ) y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.223 |
|
|
\[ {}\left (1-x^{4} y^{2}\right ) y^{\prime } = x^{3} y^{3} \] |
1 |
1 |
4 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.205 |
|
|
\[ {}\left (3 x -y^{3}\right ) y^{\prime } = x^{2}-3 y \] |
1 |
1 |
1 |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
1.135 |
|
|
\[ {}\left (x^{3}-y^{3}\right ) y^{\prime }+x^{2} y = 0 \] |
1 |
1 |
10 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.447 |
|
|
\[ {}\left (x^{3}+y^{3}\right ) y^{\prime }+x^{2} \left (x a +3 y\right ) = 0 \] |
1 |
1 |
1 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
4.654 |
|
|
\[ {}\left (x -x^{2} y-y^{3}\right ) y^{\prime } = x^{3}-y+x y^{2} \] |
1 |
1 |
1 |
exact |
[_exact, _rational] |
✓ |
✓ |
1.349 |
|
|
\[ {}\left (x \,a^{2}+y \left (x^{2}-y^{2}\right )\right ) y^{\prime }+x \left (x^{2}-y^{2}\right ) = a^{2} y \] |
1 |
1 |
0 |
exactByInspection |
[_rational] |
✓ |
✓ |
1.569 |
|
|
\[ {}\left (a +x^{2}+y^{2}\right ) y y^{\prime } = x \left (a -x^{2}-y^{2}\right ) \] |
1 |
1 |
4 |
exact |
[_exact, _rational] |
✓ |
✓ |
1.224 |
|
|
\[ {}\left (y^{2}+3 x^{2}\right ) y y^{\prime }+x \left (x^{2}+3 y^{2}\right ) = 0 \] |
1 |
1 |
4 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
1.694 |
|
|
\[ {}\left (a -3 x^{2}-y^{2}\right ) y y^{\prime }+x \left (-x^{2}+y^{2}+a \right ) = 0 \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.15 |
|
|
\[ {}2 y^{3} y^{\prime } = x^{3}-x y^{2} \] |
1 |
1 |
6 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
6.335 |
|
|
\[ {}y \left (2 y^{2}+1\right ) y^{\prime } = x \left (2 x^{2}+1\right ) \] |
1 |
1 |
4 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.901 |
|
|
\[ {}\left (3 x^{2}+2 y^{2}\right ) y y^{\prime }+x^{3} = 0 \] |
1 |
1 |
4 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
5.286 |
|
|
\[ {}\left (5 x^{2}+2 y^{2}\right ) y y^{\prime }+x \left (x^{2}+5 y^{2}\right ) = 0 \] |
1 |
1 |
4 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
1.744 |
|
|
\[ {}\left (x^{2}-x^{3}+3 x y^{2}+2 y^{3}\right ) y^{\prime }+2 x^{3}+3 x^{2} y+y^{2}-y^{3} = 0 \] |
1 |
0 |
3 |
unknown |
[_rational] |
✗ |
N/A |
1.039 |
|
|
\[ {}\left (3 x^{3}+6 x^{2} y-3 x y^{2}+20 y^{3}\right ) y^{\prime }+4 x^{3}+9 x^{2} y+6 x y^{2}-y^{3} = 0 \] |
1 |
1 |
1 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
3.194 |
|
|
\[ {}\left (x^{3}+a y^{3}\right ) y^{\prime } = x^{2} y \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.381 |
|
|
\[ {}x y^{3} y^{\prime } = \left (-x^{2}+1\right ) \left (1+y^{2}\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.384 |
|
|
\[ {}x \left (x -y^{3}\right ) y^{\prime } = \left (3 x +y^{3}\right ) y \] |
1 |
1 |
3 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.442 |
|
|
\[ {}x \left (2 x^{3}+y^{3}\right ) y^{\prime } = \left (2 x^{3}-x^{2} y+y^{3}\right ) y \] |
1 |
1 |
3 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.92 |
|
|
\[ {}x \left (2 x^{3}-y^{3}\right ) y^{\prime } = \left (x^{3}-2 y^{3}\right ) y \] |
1 |
1 |
3 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
5.336 |
|
|
\[ {}x \left (x^{3}+3 x^{2} y+y^{3}\right ) y^{\prime } = \left (y^{2}+3 x^{2}\right ) y^{2} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.106 |
|
|
\[ {}x \left (x^{3}-2 y^{3}\right ) y^{\prime } = \left (2 x^{3}-y^{3}\right ) y \] |
1 |
1 |
3 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.612 |
|
|
\[ {}x \left (x^{4}-2 y^{3}\right ) y^{\prime }+\left (2 x^{4}+y^{3}\right ) y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
5.208 |
|
|
\[ {}x \left (x +y+2 y^{3}\right ) y^{\prime } = \left (x -y\right ) y \] |
1 |
1 |
1 |
exactByInspection |
[_rational] |
✓ |
✓ |
1.237 |
|
|
\[ {}\left (5 x -y-7 x y^{3}\right ) y^{\prime }+5 y-y^{4} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.248 |
|
|
\[ {}x \left (1-2 x y^{3}\right ) y^{\prime }+\left (1-2 x^{3} y\right ) y = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.28 |
|
|
\[ {}x \left (2-x y^{2}-2 x y^{3}\right ) y^{\prime }+1+2 y = 0 \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.363 |
|
|
\[ {}\left (2-10 y^{3} x^{2}+3 y^{2}\right ) y^{\prime } = x \left (1+5 y^{4}\right ) \] |
1 |
1 |
1 |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.213 |
|
|
\[ {}x \left (a +b x y^{3}\right ) y^{\prime }+\left (a +c \,x^{3} y\right ) y = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.53 |
|
|
\[ {}x \left (1-2 y^{3} x^{2}\right ) y^{\prime }+\left (1-2 x^{3} y^{2}\right ) y = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.236 |
|
|
\[ {}x \left (1-x y\right ) \left (1-x^{2} y^{2}\right ) y^{\prime }+\left (1+x y\right ) \left (1+x^{2} y^{2}\right ) y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.201 |
|
|
\[ {}\left (x^{2}-y^{4}\right ) y^{\prime } = x y \] |
1 |
1 |
4 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.335 |
|
|
\[ {}\left (x^{3}-y^{4}\right ) y^{\prime } = 3 x^{2} y \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.968 |
|
|
\[ {}\left (a^{2} x^{2}+\left (x^{2}+y^{2}\right )^{2}\right ) y^{\prime } = a^{2} x y \] |
1 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
6.356 |
|
|
\[ {}2 \left (x -y^{4}\right ) y^{\prime } = y \] |
1 |
1 |
4 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.197 |
|
|
\[ {}\left (4 x -x y^{3}-2 y^{4}\right ) y^{\prime } = \left (2+y^{3}\right ) y \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.21 |
|
|
\[ {}\left (a \,x^{3}+\left (x a +b y\right )^{3}\right ) y y^{\prime }+x \left (\left (x a +b y\right )^{3}+b y^{3}\right ) = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
5.631 |
|
|
\[ {}\left (x +2 y+2 y^{3} x^{2}+y^{4} x \right ) y^{\prime }+\left (1+y^{4}\right ) y = 0 \] |
1 |
0 |
3 |
unknown |
[_rational] |
✗ |
N/A |
1.309 |
|
|
\[ {}2 x \left (x^{3}+y^{4}\right ) y^{\prime } = \left (x^{3}+2 y^{4}\right ) y \] |
1 |
1 |
8 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.709 |
|
|
\[ {}x \left (1-y^{4} x^{2}\right ) y^{\prime }+y = 0 \] |
1 |
1 |
4 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.585 |
|
|
\[ {}\left (x^{2}-y^{5}\right ) y^{\prime } = 2 x y \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.212 |
|
|
\[ {}x \left (x^{3}+y^{5}\right ) y^{\prime } = \left (x^{3}-y^{5}\right ) y \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.368 |
|
|
\[ {}x^{3} \left (1+5 x^{3} y^{7}\right ) y^{\prime }+\left (3 x^{5} y^{5}-1\right ) y^{3} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.467 |
|
|
\[ {}\left (1+a \left (x +y\right )\right )^{n} y^{\prime }+a \left (x +y\right )^{n} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.24 |
|
|
\[ {}x \left (a +x y^{n}\right ) y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.418 |
|
|
\[ {}f \left (x \right ) y^{m} y^{\prime }+g \left (x \right ) y^{1+m}+h \left (x \right ) y^{n} = 0 \] |
1 |
0 |
1 |
unknown |
[_Bernoulli] |
✗ |
N/A |
1.502 |
|
|
\[ {}y^{\prime } \sqrt {b^{2}+y^{2}} = \sqrt {a^{2}+x^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.374 |
|
|
\[ {}y^{\prime } \sqrt {b^{2}-y^{2}} = \sqrt {a^{2}-x^{2}} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.537 |
|
|
\[ {}y^{\prime } \sqrt {y} = \sqrt {x} \] |
1 |
1 |
1 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
166.565 |
|
|
\[ {}\left (1+\sqrt {x +y}\right ) y^{\prime }+1 = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.79 |
|
|
\[ {}y^{\prime } \sqrt {x y}+x -y = \sqrt {x y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
9.322 |
|
|
\[ {}\left (x -2 \sqrt {x y}\right ) y^{\prime } = y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
5.217 |
|
|
\[ {}\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{\frac {3}{2}} y^{\prime } = 1+y^{2} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
8.955 |
|
|
\[ {}\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{\frac {3}{2}} y^{\prime } = 1+y^{2} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
7.553 |
|
|
\[ {}\left (x -\sqrt {x^{2}+y^{2}}\right ) y^{\prime } = y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.154 |
|
|
\[ {}x \left (1-\sqrt {x^{2}-y^{2}}\right ) y^{\prime } = y \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.405 |
|
|
\[ {}x \left (x +\sqrt {x^{2}+y^{2}}\right ) y^{\prime }+y \sqrt {x^{2}+y^{2}} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
✓ |
78.474 |
|
|
\[ {}x y \left (x +\sqrt {x^{2}-y^{2}}\right ) y^{\prime } = x y^{2}-\left (x^{2}-y^{2}\right )^{\frac {3}{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.333 |
|
|
\[ {}\left (x \sqrt {1+x^{2}+y^{2}}-y \left (x^{2}+y^{2}\right )\right ) y^{\prime } = x \left (x^{2}+y^{2}\right )+y \sqrt {1+x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.963 |
|
|
\[ {}y^{\prime } \cos \left (y\right ) \left (\cos \left (y\right )-\sin \left (A \right ) \sin \left (x \right )\right )+\cos \left (x \right ) \left (\cos \left (x \right )-\sin \left (A \right ) \sin \left (y\right )\right ) = 0 \] |
1 |
1 |
1 |
exact |
unknown |
✓ |
✓ |
4.592 |
|
|
\[ {}\left (a \cos \left (b x +a y\right )-b \sin \left (x a +b y\right )\right ) y^{\prime }+b \cos \left (b x +a y\right )-a \sin \left (x a +b y\right ) = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
2.376 |
|
|
\[ {}\left (x +\cos \left (x \right ) \sec \left (y\right )\right ) y^{\prime }+\tan \left (y\right )-y \sin \left (x \right ) \sec \left (y\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[NONE] |
✓ |
✓ |
18.316 |
|
|
\[ {}\left (1+\left (x +y\right ) \tan \left (y\right )\right ) y^{\prime }+1 = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.122 |
|
|
\[ {}x \left (x -y \tan \left (\frac {y}{x}\right )\right ) y^{\prime }+\left (x +y \tan \left (\frac {y}{x}\right )\right ) y = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.917 |
|
|
\[ {}\left ({\mathrm e}^{x}+x \,{\mathrm e}^{y}\right ) y^{\prime }+{\mathrm e}^{x} y+{\mathrm e}^{y} = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
1.407 |
|
|
\[ {}\left (1-2 x -\ln \left (y\right )\right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.09 |
|
|
\[ {}\left (\sinh \left (x \right )+x \cosh \left (y\right )\right ) y^{\prime }+y \cosh \left (x \right )+\sinh \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
35.838 |
|
|
\[ {}y^{\prime } \left (1+\sinh \left (x \right )\right ) \sinh \left (y\right )+\cosh \left (x \right ) \left (\cosh \left (y\right )-1\right ) = 0 \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.651 |
|
|
\[ {}{y^{\prime }}^{2} = a \,x^{n} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.273 |
|
|
\[ {}{y^{\prime }}^{2} = y \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.28 |
|
|
\[ {}{y^{\prime }}^{2} = x -y \] |
2 |
2 |
1 |
dAlembert, first_order_nonlinear_p_but_linear_in_x_y |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.408 |
|
|
\[ {}{y^{\prime }}^{2} = y+x^{2} \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.932 |
|
|
\[ {}{y^{\prime }}^{2}+x^{2} = 4 y \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.751 |
|
|
\[ {}{y^{\prime }}^{2}+3 x^{2} = 8 y \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.812 |
|
|
\[ {}{y^{\prime }}^{2}+x^{2} a +b y = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.79 |
|
|
\[ {}{y^{\prime }}^{2} = 1+y^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.377 |
|
|
\[ {}{y^{\prime }}^{2} = 1-y^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.512 |
|
|
\[ {}{y^{\prime }}^{2} = a^{2}-y^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.484 |
|
|
\[ {}{y^{\prime }}^{2} = y^{2} a^{2} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.398 |
|
|
\[ {}{y^{\prime }}^{2} = a +b y^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.543 |
|
|
\[ {}{y^{\prime }}^{2} = x^{2} y^{2} \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.364 |
|
|
\[ {}{y^{\prime }}^{2} = \left (y-1\right ) y^{2} \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.499 |
|
|
\[ {}{y^{\prime }}^{2} = \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) \] |
2 |
2 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.505 |
|
|
\[ {}{y^{\prime }}^{2} = a^{2} y^{n} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.709 |
|
|
\[ {}{y^{\prime }}^{2} = a^{2} \left (1-\ln \left (y\right )^{2}\right ) y^{2} \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.108 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right ) = 0 \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.375 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) = 0 \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
0.869 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) = 0 \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.198 |
|
|
\[ {}{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right ) = 0 \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
4.577 |
|
|
\[ {}{y^{\prime }}^{2} = f \left (x \right )^{2} \left (y-a \right ) \left (y-b \right ) \left (y-c \right )^{2} \] |
2 |
2 |
2 |
first_order_nonlinear_p_but_separable |
[_separable] |
✓ |
✓ |
5.11 |
|
|
\[ {}{y^{\prime }}^{2} = f \left (x \right )^{2} \left (y-u \left (x \right )\right )^{2} \left (y-a \right ) \left (y-b \right ) \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
2.323 |
|
|
\[ {}{y^{\prime }}^{2}+2 y^{\prime }+x = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.255 |
|
|
\[ {}{y^{\prime }}^{2}-2 y^{\prime }+a \left (x -y\right ) = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.232 |
|
|
\[ {}{y^{\prime }}^{2}-2 y^{\prime }-y^{2} = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.415 |
|
|
\[ {}{y^{\prime }}^{2}-5 y^{\prime }+6 = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.228 |
|
|
\[ {}{y^{\prime }}^{2}-7 y^{\prime }+12 = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.191 |
|
|
\[ {}{y^{\prime }}^{2}+a y^{\prime }+b = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.222 |
|
|
\[ {}{y^{\prime }}^{2}+a y^{\prime }+b x = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.295 |
|
|
\[ {}{y^{\prime }}^{2}+a y^{\prime }+b y = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.654 |
|
|
\[ {}{y^{\prime }}^{2}+x y^{\prime }+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.368 |
|
|
\[ {}{y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.206 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.213 |
|
|
\[ {}{y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.34 |
|
|
\[ {}{y^{\prime }}^{2}+x y^{\prime }+x -y = 0 \] |
2 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.307 |
|
|
\[ {}{y^{\prime }}^{2}+\left (1-x \right ) y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.199 |
|
|
\[ {}{y^{\prime }}^{2}-\left (1+x \right ) y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.222 |
|
|
\[ {}{y^{\prime }}^{2}-\left (2-x \right ) y^{\prime }+1-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.239 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x +a \right ) y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.212 |
|
|
\[ {}{y^{\prime }}^{2}-2 x y^{\prime }+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.351 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-3 x^{2} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.238 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.315 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.279 |
|
|
\[ {}{y^{\prime }}^{2}-2 x y^{\prime }+2 y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.225 |
|
|
\[ {}{y^{\prime }}^{2}-\left (2 x +1\right ) y^{\prime }-x \left (1-x \right ) = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.326 |
|
|
\[ {}{y^{\prime }}^{2}+2 \left (1-x \right ) y^{\prime }-2 x +2 y = 0 \] |
2 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.299 |
|
|
\[ {}{y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.36 |
|
|
\[ {}{y^{\prime }}^{2}-4 \left (1+x \right ) y^{\prime }+4 y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.201 |
|
|
\[ {}{y^{\prime }}^{2}+a x y^{\prime } = b c \,x^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.256 |
|
|
\[ {}{y^{\prime }}^{2}-a x y^{\prime }+a y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.234 |
|
|
\[ {}{y^{\prime }}^{2}+a x y^{\prime }+b \,x^{2}+c y = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.085 |
|
|
\[ {}{y^{\prime }}^{2}+\left (b x +a \right ) y^{\prime }+c = b y \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.229 |
|
|
\[ {}{y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y^{\prime } = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.228 |
|
|
\[ {}{y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
9.748 |
|
|
\[ {}{y^{\prime }}^{2}-2 a \,x^{3} y^{\prime }+4 a \,x^{2} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.714 |
|
|
\[ {}{y^{\prime }}^{2}+4 x^{5} y^{\prime }-12 x^{4} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.308 |
|
|
\[ {}{y^{\prime }}^{2}-2 y^{\prime } \cosh \left (x \right )+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.265 |
|
|
\[ {}{y^{\prime }}^{2}+y y^{\prime } = x \left (x +y\right ) \] |
2 |
1 |
2 |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
0.347 |
|
|
\[ {}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.498 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+x y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.279 |
|
|
\[ {}{y^{\prime }}^{2}-2 y y^{\prime }-2 x = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.417 |
|
|
\[ {}{y^{\prime }}^{2}+\left (1+2 y\right ) y^{\prime }+y \left (y-1\right ) = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
142.175 |
|
|
\[ {}{y^{\prime }}^{2}-2 \left (x -y\right ) y^{\prime }-4 x y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.291 |
|
|
\[ {}{y^{\prime }}^{2}-\left (4 y+1\right ) y^{\prime }+\left (4 y+1\right ) y = 0 \] |
2 |
2 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.342 |
|
|
\[ {}{y^{\prime }}^{2}-2 \left (-3 y+1\right ) y^{\prime }-\left (4-9 y\right ) y = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.604 |
|
|
\[ {}{y^{\prime }}^{2}+\left (a +6 y\right ) y^{\prime }+y \left (3 a +b +9 y\right ) = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.172 |
|
|
\[ {}{y^{\prime }}^{2}+a y y^{\prime }-x a = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.776 |
|
|
\[ {}{y^{\prime }}^{2}-a y y^{\prime }-x a = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.467 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x a +b y\right ) y^{\prime }+a b x y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.336 |
|
|
\[ {}{y^{\prime }}^{2}-x y y^{\prime }+y^{2} \ln \left (a y\right ) = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
4.219 |
|
|
\[ {}{y^{\prime }}^{2}-\left (2 x y+1\right ) y^{\prime }+2 x y = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.297 |
|
|
\[ {}{y^{\prime }}^{2}-\left (4+y^{2}\right ) y^{\prime }+4+y^{2} = 0 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.833 |
|
|
\[ {}{y^{\prime }}^{2}-\left (x -y\right ) y y^{\prime }-x y^{3} = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_separable] |
✓ |
✓ |
0.333 |
|
|
|
|||||||||
|
\[ {}{y^{\prime }}^{2}+x y^{2} y^{\prime }+y^{3} = 0 \] |
2 |
1 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
75.362 |
|
|
\[ {}{y^{\prime }}^{2}-2 x^{3} y^{2} y^{\prime }-4 y^{3} x^{2} = 0 \] |
2 |
1 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
18.745 |
|
|
\[ {}{y^{\prime }}^{2}-x y \left (x^{2}+y^{2}\right ) y^{\prime }+x^{4} y^{4} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.554 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{3} y^{\prime }+y^{4} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.079 |
|
|
\[ {}{y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right )-y^{2} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
4.494 |
|
|
\[ {}{y^{\prime }}^{2}-3 x y^{\frac {2}{3}} y^{\prime }+9 y^{\frac {5}{3}} = 0 \] |
2 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
6.318 |
|
|
\[ {}{y^{\prime }}^{2} = {\mathrm e}^{4 x -2 y} \left (y^{\prime }-1\right ) \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.48 |
|
|
\[ {}2 {y^{\prime }}^{2}+x y^{\prime }-2 y = 0 \] |
2 |
3 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.306 |
|
|
\[ {}2 {y^{\prime }}^{2}-\left (1-x \right ) y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.204 |
|
|
\[ {}2 {y^{\prime }}^{2}-2 x^{2} y^{\prime }+3 x y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
7.216 |
|
|
\[ {}2 {y^{\prime }}^{2}+2 \left (6 y-1\right ) y^{\prime }+3 y \left (6 y-1\right ) = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.972 |
|
|
\[ {}3 {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.305 |
|
|
\[ {}3 {y^{\prime }}^{2}+4 x y^{\prime }+x^{2}-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.894 |
|
|
\[ {}4 {y^{\prime }}^{2} = 9 x \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.227 |
|
|
\[ {}4 {y^{\prime }}^{2}+2 x \,{\mathrm e}^{-2 y} y^{\prime }-{\mathrm e}^{-2 y} = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
11.534 |
|
|
\[ {}4 {y^{\prime }}^{2}+2 \,{\mathrm e}^{2 x -2 y} y^{\prime }-{\mathrm e}^{2 x -2 y} = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.481 |
|
|
\[ {}5 {y^{\prime }}^{2}+3 x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.383 |
|
|
\[ {}5 {y^{\prime }}^{2}+6 x y^{\prime }-2 y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.349 |
|
|
\[ {}9 {y^{\prime }}^{2}+3 x y^{4} y^{\prime }+y^{5} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
82.548 |
|
|
\[ {}x {y^{\prime }}^{2} = a \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.234 |
|
|
\[ {}x {y^{\prime }}^{2} = -x^{2}+a \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.574 |
|
|
\[ {}x {y^{\prime }}^{2} = y \] |
2 |
3 |
3 |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.632 |
|
|
\[ {}x {y^{\prime }}^{2}+x -2 y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.435 |
|
|
\[ {}x {y^{\prime }}^{2}+y^{\prime } = y \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.362 |
|
|
\[ {}x {y^{\prime }}^{2}+2 y^{\prime }-y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.366 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y^{\prime }-y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.329 |
|
|
\[ {}x {y^{\prime }}^{2}+4 y^{\prime }-2 y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.349 |
|
|
\[ {}x {y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.375 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (x^{2}+1\right ) y^{\prime }+x = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.241 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }+a = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
✓ |
0.438 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }+a = 0 \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.263 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }+x a = 0 \] |
2 |
2 |
1 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.434 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }+x^{3} = 0 \] |
2 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
8.079 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }+a y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.448 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }-y^{4} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.631 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (-y+a \right ) y^{\prime }+b = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.294 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }+1-y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.258 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (a +x -y\right ) y^{\prime }-y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.279 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (3 x -y\right ) y^{\prime }+y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.56 |
|
|
\[ {}x {y^{\prime }}^{2}+a +b x -y-b y = 0 \] |
2 |
3 |
1 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
1.33 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+a = 0 \] |
2 |
2 |
3 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
0.435 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+x a = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.293 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.315 |
|
|
\[ {}x {y^{\prime }}^{2}-3 y y^{\prime }+9 x^{2} = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
5.328 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (2 x +3 y\right ) y^{\prime }+6 y = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.382 |
|
|
\[ {}x {y^{\prime }}^{2}-a y y^{\prime }+b = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
54.294 |
|
|
\[ {}x {y^{\prime }}^{2}+a y y^{\prime }+b x = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.714 |
|
|
\[ {}x {y^{\prime }}^{2}-\left (1+x y\right ) y^{\prime }+y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.249 |
|
|
\[ {}x {y^{\prime }}^{2}+y \left (1-x \right ) y^{\prime }-y^{2} = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.321 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (1-x^{2} y\right ) y^{\prime }-x y = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.332 |
|
|
\[ {}\left (1+x \right ) {y^{\prime }}^{2} = y \] |
2 |
3 |
3 |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.651 |
|
|
\[ {}\left (1+x \right ) {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.295 |
|
|
\[ {}\left (a -x \right ) {y^{\prime }}^{2}+y y^{\prime }-b = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.297 |
|
|
\[ {}2 x {y^{\prime }}^{2}+\left (2 x -y\right ) y^{\prime }+1-y = 0 \] |
2 |
3 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.555 |
|
|
\[ {}3 x {y^{\prime }}^{2}-6 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.287 |
|
|
\[ {}\left (1+3 x \right ) {y^{\prime }}^{2}-3 \left (y+2\right ) y^{\prime }+9 = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.311 |
|
|
\[ {}\left (3 x +5\right ) {y^{\prime }}^{2}-\left (3+3 y\right ) y^{\prime }+y = 0 \] |
2 |
3 |
2 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
1.172 |
|
|
\[ {}4 x {y^{\prime }}^{2} = \left (a -3 x \right )^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.289 |
|
|
\[ {}4 x {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.264 |
|
|
\[ {}4 x {y^{\prime }}^{2}-3 y y^{\prime }+3 = 0 \] |
2 |
2 |
4 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
0.332 |
|
|
\[ {}4 x {y^{\prime }}^{2}+4 y y^{\prime } = 1 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
0.421 |
|
|
\[ {}4 x {y^{\prime }}^{2}+4 y y^{\prime }-y^{4} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.632 |
|
|
\[ {}4 \left (2-x \right ) {y^{\prime }}^{2}+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.267 |
|
|
\[ {}16 x {y^{\prime }}^{2}+8 y y^{\prime }+y^{6} = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.387 |
|
|
\[ {}x^{2} {y^{\prime }}^{2} = a^{2} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.255 |
|
|
\[ {}x^{2} {y^{\prime }}^{2} = y^{2} \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.419 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+x^{2}-y^{2} = 0 \] |
2 |
4 |
1 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.098 |
|
|
\[ {}x^{2} {y^{\prime }}^{2} = \left (x -y\right )^{2} \] |
2 |
1 |
2 |
linear |
[_linear] |
✓ |
✓ |
0.527 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+y^{2}-y^{4} = 0 \] |
2 |
2 |
5 |
first_order_nonlinear_p_but_separable |
[_separable] |
✓ |
✓ |
0.937 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-x y^{\prime }+y \left (1-y\right ) = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.51 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+2 a x y^{\prime }+a^{2}+x^{2}-2 a y = 0 \] |
2 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
1.644 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x +y \left (y+1\right ) = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
3.608 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{4}+\left (-x^{2}+1\right ) y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.108 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-\left (2 x y+1\right ) y^{\prime }+1+y^{2} = 0 \] |
2 |
4 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.423 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-\left (a +2 x y\right ) y^{\prime }+y^{2} = 0 \] |
2 |
4 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.525 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-x \left (x -2 y\right ) y^{\prime }+y^{2} = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.512 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+2 x \left (y+2 x \right ) y^{\prime }-4 a +y^{2} = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
4.254 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+x \left (x^{3}-2 y\right ) y^{\prime }-\left (2 x^{3}-y\right ) y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
9.777 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+3 x y y^{\prime }+2 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.542 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-3 x y y^{\prime }+x^{3}+2 y^{2} = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
16.458 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+4 x y y^{\prime }-5 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.556 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-4 x \left (y+2\right ) y^{\prime }+4 \left (y+2\right ) y = 0 \] |
2 |
2 |
5 |
separable |
[_separable] |
✓ |
✓ |
3.259 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-5 x y y^{\prime }+6 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.499 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+x \left (x^{2}+x y-2 y\right ) y^{\prime }+\left (1-x \right ) \left (x^{2}-y\right ) y = 0 \] |
2 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
7.028 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+\left (y+2 x \right ) y y^{\prime }+y^{2} = 0 \] |
2 |
3 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.978 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+\left (2 x -y\right ) y y^{\prime }+y^{2} = 0 \] |
2 |
3 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.935 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+\left (a +b \,x^{2} y^{3}\right ) y^{\prime }+a b y^{3} = 0 \] |
2 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.434 |
|
|
\[ {}\left (-x^{2}+1\right ) {y^{\prime }}^{2} = 1-y^{2} \] |
2 |
2 |
4 |
first_order_nonlinear_p_but_separable |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.449 |
|
|
\[ {}\left (-x^{2}+1\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+4 x^{2} = 0 \] |
2 |
0 |
3 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
143.288 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) {y^{\prime }}^{2} = b^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.341 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+b^{2} = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.347 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2} = b^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.466 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2} = x^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.342 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+x^{2} = 0 \] |
2 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
90.2 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}-2 x y y^{\prime }-y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.523 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) {y^{\prime }}^{2}-2 x y y^{\prime }+b +y^{2} = 0 \] |
2 |
6 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
2.242 |
|
|
\[ {}4 x^{2} {y^{\prime }}^{2}-4 x y y^{\prime } = 8 x^{3}-y^{2} \] |
2 |
2 |
2 |
linear |
[_linear] |
✓ |
✓ |
0.633 |
|
|
\[ {}a \,x^{2} {y^{\prime }}^{2}-2 a x y y^{\prime }+a \left (1-a \right ) x^{2}+y^{2} = 0 \] |
2 |
8 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.332 |
|
|
\[ {}\left (-a^{2}+1\right ) x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-a^{2} x^{2}+y^{2} = 0 \] |
2 |
8 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
240.952 |
|
|
\[ {}x^{3} {y^{\prime }}^{2} = a \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.247 |
|
|
\[ {}x^{3} {y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
2 |
2 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
16.725 |
|
|
\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+a = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
5.416 |
|
|
\[ {}x \left (-x^{2}+1\right ) {y^{\prime }}^{2}-2 \left (-x^{2}+1\right ) y y^{\prime }+x \left (1-y^{2}\right ) = 0 \] |
2 |
0 |
3 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
6.466 |
|
|
\[ {}4 x \left (a -x \right ) \left (-x +b \right ) {y^{\prime }}^{2} = \left (a b -2 x \left (a +b \right )+2 x^{2}\right )^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.593 |
|
|
\[ {}x^{4} {y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.308 |
|
|
\[ {}x^{4} {y^{\prime }}^{2}+2 x^{3} y y^{\prime }-4 = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.475 |
|
|
\[ {}x^{4} {y^{\prime }}^{2}+x y^{2} y^{\prime }-y^{3} = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
38.073 |
|
|
\[ {}x^{2} \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.56 |
|
|
\[ {}3 x^{4} {y^{\prime }}^{2}-x y-y = 0 \] |
2 |
2 |
5 |
first_order_nonlinear_p_but_separable |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
0.737 |
|
|
\[ {}4 x^{5} {y^{\prime }}^{2}+12 x^{4} y y^{\prime }+9 = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
5.716 |
|
|
\[ {}x^{6} {y^{\prime }}^{2}-2 x y^{\prime }-4 y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.532 |
|
|
\[ {}x^{8} {y^{\prime }}^{2}+3 x y^{\prime }+9 y = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
4.646 |
|
|
\[ {}y {y^{\prime }}^{2} = a \] |
2 |
2 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.413 |
|
|
\[ {}y {y^{\prime }}^{2} = x \,a^{2} \] |
2 |
5 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.892 |
|
|
\[ {}y {y^{\prime }}^{2} = {\mathrm e}^{2 x} \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
4.884 |
|
|
\[ {}y {y^{\prime }}^{2}+2 a x y^{\prime }-a y = 0 \] |
2 |
5 |
5 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.69 |
|
|
\[ {}y {y^{\prime }}^{2}-4 a^{2} x y^{\prime }+a^{2} y = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.92 |
|
|
\[ {}y {y^{\prime }}^{2}+a x y^{\prime }+b y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
16.224 |
|
|
\[ {}y {y^{\prime }}^{2}-\left (-2 b x +a \right ) y^{\prime }-b y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.815 |
|
|
\[ {}y {y^{\prime }}^{2}+x^{3} y^{\prime }-x^{2} y = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
6.483 |
|
|
\[ {}y {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }-x = 0 \] |
2 |
2 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.442 |
|
|
\[ {}y {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y = 0 \] |
2 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.855 |
|
|
\[ {}y {y^{\prime }}^{2}-\left (1+x y\right ) y^{\prime }+x = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.249 |
|
|
\[ {}y {y^{\prime }}^{2}+\left (x -y^{2}\right ) y^{\prime }-x y = 0 \] |
2 |
2 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.379 |
|
|
\[ {}y {y^{\prime }}^{2}+y = a \] |
2 |
2 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.975 |
|
|
\[ {}\left (x +y\right ) {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.695 |
|
|
\[ {}\left (2 x -y\right ) {y^{\prime }}^{2}-2 \left (1-x \right ) y^{\prime }+2-y = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.658 |
|
|
\[ {}2 y {y^{\prime }}^{2}+\left (5-4 x \right ) y^{\prime }+2 y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.544 |
|
|
\[ {}9 y {y^{\prime }}^{2}+4 x^{3} y^{\prime }-4 x^{2} y = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
6.322 |
|
|
\[ {}\left (1-a y\right ) {y^{\prime }}^{2} = a y \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.126 |
|
|
\[ {}\left (x^{2}-a y\right ) {y^{\prime }}^{2}-2 x y y^{\prime } = 0 \] |
2 |
1 |
2 |
quadrature, first_order_ode_lie_symmetry_calculated |
[_quadrature] |
✓ |
✓ |
0.816 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+1 = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.343 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.481 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x^{2}-y^{2}\right ) y^{\prime }-x y = 0 \] |
2 |
1 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.465 |
|
|
\[ {}x y {y^{\prime }}^{2}-\left (x^{2}-y^{2}\right ) y^{\prime }-x y = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.448 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (a +x^{2}-y^{2}\right ) y^{\prime }-x y = 0 \] |
2 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
233.874 |
|
|
\[ {}x y {y^{\prime }}^{2}-\left (a -b \,x^{2}+y^{2}\right ) y^{\prime }-b x y = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
66.336 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (3 x^{2}-2 y^{2}\right ) y^{\prime }-6 x y = 0 \] |
2 |
1 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.702 |
|
|
\[ {}x \left (x -2 y\right ) {y^{\prime }}^{2}-2 x y y^{\prime }-2 x y+y^{2} = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.077 |
|
|
\[ {}x \left (x -2 y\right ) {y^{\prime }}^{2}+6 x y y^{\prime }-2 x y+y^{2} = 0 \] |
2 |
9 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
221.604 |
|
|
\[ {}y^{2} {y^{\prime }}^{2} = a^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.414 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-a^{2}+y^{2} = 0 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.568 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-3 x y^{\prime }+y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
4.924 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.02 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-4 a y y^{\prime }+4 a^{2}-4 x a +y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
2.564 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-\left (1+x \right ) y y^{\prime }+x = 0 \] |
2 |
2 |
4 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.453 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }+x^{2} = 0 \] |
2 |
4 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.355 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }+a -y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
8.827 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{2}+2 y^{2} = 0 \] |
2 |
6 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.358 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+a -x^{2}+2 y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
2.385 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (a -1\right ) b +x^{2} a +\left (1-a \right ) y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
3.305 |
|
|
\[ {}\left (1-y^{2}\right ) {y^{\prime }}^{2} = 1 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.752 |
|
|
\[ {}\left (a^{2}-y^{2}\right ) {y^{\prime }}^{2} = y^{2} \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.974 |
|
|
\[ {}\left (a^{2}-2 a x y+y^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
4.385 |
|
|
\[ {}\left (\left (1-a \right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}+2 a x y y^{\prime }+x^{2}+\left (1-a \right ) y^{2} = 0 \] |
2 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.087 |
|
|
\[ {}\left (\left (-4 a^{2}+1\right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}-8 a^{2} x y y^{\prime }+x^{2}+\left (-4 a^{2}+1\right ) y^{2} = 0 \] |
2 |
8 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
183.437 |
|
|
\[ {}\left (\left (-a^{2}+1\right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+x^{2}+\left (-a^{2}+1\right ) y^{2} = 0 \] |
2 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.123 |
|
|
\[ {}\left (x +y\right )^{2} {y^{\prime }}^{2} = y^{2} \] |
2 |
1 |
3 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.948 |
|
|
\[ {}\left (x +y\right )^{2} {y^{\prime }}^{2}-\left (x^{2}-x y-2 y^{2}\right ) y^{\prime }-\left (x -y\right ) y = 0 \] |
2 |
1 |
4 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.136 |
|
|
\[ {}\left (a^{2}-\left (x -y\right )^{2}\right ) {y^{\prime }}^{2}+2 a^{2} y^{\prime }+a^{2}-\left (x -y\right )^{2} = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
11.89 |
|
|
\[ {}2 y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }-1+x^{2}+y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
74.107 |
|
|
\[ {}3 y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{2}+4 y^{2} = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.616 |
|
|
\[ {}4 y^{2} {y^{\prime }}^{2}+2 \left (1+3 x \right ) x y y^{\prime }+3 x^{3} = 0 \] |
2 |
2 |
4 |
separable |
[_separable] |
✓ |
✓ |
0.59 |
|
|
\[ {}\left (x^{2}-4 y^{2}\right ) {y^{\prime }}^{2}+6 x y y^{\prime }-4 x^{2}+y^{2} = 0 \] |
2 |
1 |
2 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.957 |
|
|
\[ {}9 y^{2} {y^{\prime }}^{2}-3 x y^{\prime }+y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
3.781 |
|
|
\[ {}\left (2-3 y\right )^{2} {y^{\prime }}^{2} = 4-4 y \] |
2 |
2 |
7 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.474 |
|
|
\[ {}\left (-a^{2}+1\right ) y^{2} {y^{\prime }}^{2}-3 a^{2} x y y^{\prime }-a^{2} x^{2}+y^{2} = 0 \] |
2 |
8 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
11.996 |
|
|
\[ {}\left (-b +a \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }-a b -b \,x^{2}+a y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
3.382 |
|
|
\[ {}a^{2} \left (b^{2}-\left (c x -a y\right )^{2}\right ) {y^{\prime }}^{2}+2 a \,b^{2} c y^{\prime }+c^{2} \left (b^{2}-\left (c x -a y\right )^{2}\right ) = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
19.815 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }+x \,a^{2} = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
5.448 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}+\left (a -x^{3}-y^{3}\right ) y^{\prime }+x^{2} y = 0 \] |
2 |
0 |
9 |
unknown |
[_rational] |
✗ |
N/A |
7.855 |
|
|
\[ {}2 x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }-a = 0 \] |
2 |
2 |
8 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.208 |
|
|
\[ {}4 x^{2} y^{2} {y^{\prime }}^{2} = \left (x^{2}+y^{2}\right )^{2} \] |
2 |
1 |
4 |
homogeneousTypeD2 |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.309 |
|
|
\[ {}4 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
5.898 |
|
|
\[ {}3 x y^{4} {y^{\prime }}^{2}-y^{5} y^{\prime }+1 = 0 \] |
2 |
1 |
12 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.103 |
|
|
\[ {}9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }-a = 0 \] |
2 |
1 |
12 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.038 |
|
|
\[ {}9 \left (-x^{2}+1\right ) y^{4} {y^{\prime }}^{2}+6 x y^{5} y^{\prime }+4 x^{2} = 0 \] |
2 |
0 |
9 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.399 |
|
|
\[ {}{y^{\prime }}^{3} = b x +a \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.587 |
|
|
\[ {}{y^{\prime }}^{3} = a \,x^{n} \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.45 |
|
|
\[ {}{y^{\prime }}^{3}+x -y = 0 \] |
3 |
4 |
3 |
dAlembert, first_order_nonlinear_p_but_linear_in_x_y |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.728 |
|
|
\[ {}{y^{\prime }}^{3} = \left (a +b y+c y^{2}\right ) f \left (x \right ) \] |
3 |
3 |
3 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.733 |
|
|
\[ {}{y^{\prime }}^{3} = \left (y-a \right )^{2} \left (y-b \right )^{2} \] |
3 |
3 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.214 |
|
|
\[ {}{y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2} = 0 \] |
3 |
3 |
3 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.263 |
|
|
\[ {}{y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2} = 0 \] |
3 |
3 |
3 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
188.713 |
|
|
\[ {}{y^{\prime }}^{3}+y^{\prime }+a -b x = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.565 |
|
|
\[ {}{y^{\prime }}^{3}+y^{\prime }-y = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.901 |
|
|
\[ {}{y^{\prime }}^{3}+y^{\prime } = {\mathrm e}^{y} \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.155 |
|
|
\[ {}{y^{\prime }}^{3}-7 y^{\prime }+6 = 0 \] |
3 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.286 |
|
|
\[ {}{y^{\prime }}^{3}-x y^{\prime }+a y = 0 \] |
3 |
4 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
169.119 |
|
|
\[ {}{y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
4 |
4 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
3.142 |
|
|
\[ {}{y^{\prime }}^{3}-2 x y^{\prime }-y = 0 \] |
3 |
4 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
99.775 |
|
|
\[ {}{y^{\prime }}^{3}-a x y^{\prime }+x^{3} = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.445 |
|
|
\[ {}{y^{\prime }}^{3}+a x y^{\prime }-a y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.434 |
|
|
\[ {}{y^{\prime }}^{3}-\left (b x +a \right ) y^{\prime }+b y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.503 |
|
|
\[ {}{y^{\prime }}^{3}-2 y y^{\prime }+y^{2} = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.445 |
|
|
\[ {}{y^{\prime }}^{3}-a x y y^{\prime }+2 a y^{2} = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
16.117 |
|
|
\[ {}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0 \] |
3 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
17.514 |
|
|
\[ {}{y^{\prime }}^{3}+{\mathrm e}^{3 x -2 y} \left (y^{\prime }-1\right ) = 0 \] |
3 |
2 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.973 |
|
|
\[ {}{y^{\prime }}^{3}+{\mathrm e}^{-2 y} \left ({\mathrm e}^{2 x}+{\mathrm e}^{3 x}\right ) y^{\prime }-{\mathrm e}^{3 x -2 y} = 0 \] |
3 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
104.449 |
|
|
\[ {}{y^{\prime }}^{3}+{y^{\prime }}^{2}-y = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.296 |
|
|
|
|||||||||
|
\[ {}{y^{\prime }}^{3}-{y^{\prime }}^{2}+y^{2} = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.492 |
|
|
\[ {}{y^{\prime }}^{3}-{y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.56 |
|
|
\[ {}{y^{\prime }}^{3}-a {y^{\prime }}^{2}+b y+a b x = 0 \] |
3 |
4 |
1 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
167.053 |
|
|
\[ {}{y^{\prime }}^{3}+\operatorname {a0} {y^{\prime }}^{2}+\operatorname {a1} y^{\prime }+\operatorname {a2} +\operatorname {a3} y = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.897 |
|
|
\[ {}{y^{\prime }}^{3}+\left (-3 x +1\right ) {y^{\prime }}^{2}-x \left (-3 x +1\right ) y^{\prime }-1-x^{3} = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.723 |
|
|
\[ {}{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2} = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.787 |
|
|
\[ {}{y^{\prime }}^{3}+\left (\cos \left (x \right ) \cot \left (x \right )-y\right ) {y^{\prime }}^{2}-\left (1+y \cos \left (x \right ) \cot \left (x \right )\right ) y^{\prime }+y = 0 \] |
3 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.78 |
|
|
\[ {}{y^{\prime }}^{3}+\left (2 x -y^{2}\right ) {y^{\prime }}^{2}-2 x y^{2} y^{\prime } = 0 \] |
3 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.313 |
|
|
\[ {}{y^{\prime }}^{3}-\left (y^{2}+2 x \right ) {y^{\prime }}^{2}+\left (x^{2}-y^{2}+2 x y^{2}\right ) y^{\prime }-\left (x^{2}-y^{2}\right ) y^{2} = 0 \] |
3 |
1 |
3 |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
0.622 |
|
|
\[ {}{y^{\prime }}^{3}-\left (x^{2}+x y+y^{2}\right ) {y^{\prime }}^{2}+x y \left (x^{2}+x y+y^{2}\right ) y^{\prime }-x^{3} y^{3} = 0 \] |
3 |
1 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.481 |
|
|
\[ {}{y^{\prime }}^{3}-\left (x^{2}+x y^{2}+y^{4}\right ) {y^{\prime }}^{2}+x y^{2} \left (x^{2}+x y^{2}+y^{4}\right ) y^{\prime }-x^{3} y^{6} = 0 \] |
3 |
1 |
5 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.987 |
|
|
\[ {}2 {y^{\prime }}^{3}+x y^{\prime }-2 y = 0 \] |
3 |
4 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
100.448 |
|
|
\[ {}2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.233 |
|
|
\[ {}3 {y^{\prime }}^{3}-x^{4} y^{\prime }+2 x^{3} y = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
86.383 |
|
|
\[ {}4 {y^{\prime }}^{3}+4 y^{\prime } = x \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.579 |
|
|
\[ {}8 {y^{\prime }}^{3}+12 {y^{\prime }}^{2} = 27 x +27 y \] |
3 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.527 |
|
|
\[ {}x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+a = 0 \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.745 |
|
|
\[ {}x {y^{\prime }}^{3}-\left (x +x^{2}+y\right ) {y^{\prime }}^{2}+\left (x^{2}+y+x y\right ) y^{\prime }-x y = 0 \] |
3 |
1 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.448 |
|
|
\[ {}x {y^{\prime }}^{3}-2 y {y^{\prime }}^{2}+4 x^{2} = 0 \] |
3 |
1 |
11 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
93.679 |
|
|
\[ {}2 x {y^{\prime }}^{3}-3 y {y^{\prime }}^{2}-x = 0 \] |
3 |
4 |
5 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.613 |
|
|
\[ {}4 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}-x +3 y = 0 \] |
3 |
6 |
5 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
13.632 |
|
|
\[ {}8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y = 0 \] |
3 |
6 |
5 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
23.288 |
|
|
\[ {}x^{2} {y^{\prime }}^{3}-2 x y {y^{\prime }}^{2}+y^{2} y^{\prime }+1 = 0 \] |
3 |
8 |
5 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
3.786 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{3}+b x \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}-y^{\prime }-b x = 0 \] |
3 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.54 |
|
|
\[ {}x {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+x \left (x^{5}+3 y^{2}\right ) y^{\prime }-2 x^{5} y-y^{3} = 0 \] |
3 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
315.692 |
|
|
\[ {}2 x^{3} {y^{\prime }}^{3}+6 x^{2} y {y^{\prime }}^{2}-\left (1-6 x y\right ) y y^{\prime }+2 y^{3} = 0 \] |
3 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
21.441 |
|
|
\[ {}x^{4} {y^{\prime }}^{3}-x^{3} y {y^{\prime }}^{2}-x^{2} y^{2} y^{\prime }+x y^{3} = 1 \] |
3 |
1 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
18.007 |
|
|
\[ {}x^{6} {y^{\prime }}^{3}-x y^{\prime }-y = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
100.911 |
|
|
\[ {}y {y^{\prime }}^{3}-3 x y^{\prime }+3 y = 0 \] |
3 |
4 |
4 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
173.355 |
|
|
\[ {}2 y {y^{\prime }}^{3}-3 x y^{\prime }+2 y = 0 \] |
3 |
7 |
7 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
154.286 |
|
|
\[ {}\left (2 y+x \right ) {y^{\prime }}^{3}+3 \left (x +y\right ) {y^{\prime }}^{2}+\left (y+2 x \right ) y^{\prime } = 0 \] |
3 |
1 |
4 |
quadrature, homogeneousTypeD2 |
[_quadrature] |
✓ |
✓ |
1.062 |
|
|
\[ {}y^{2} {y^{\prime }}^{3}-x y^{\prime }+y = 0 \] |
3 |
1 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
81.401 |
|
|
\[ {}y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
87.318 |
|
|
\[ {}4 y^{2} {y^{\prime }}^{3}-2 x y^{\prime }+y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
47.616 |
|
|
\[ {}16 y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
86.735 |
|
|
\[ {}x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-x^{2} y = 0 \] |
3 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
175.365 |
|
|
\[ {}y^{3} {y^{\prime }}^{3}-\left (-3 x +1\right ) y^{2} {y^{\prime }}^{2}+3 x^{2} y y^{\prime }+x^{3}-y^{2} = 0 \] |
3 |
0 |
10 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
134.116 |
|
|
\[ {}y^{4} {y^{\prime }}^{3}-6 x y^{\prime }+2 y = 0 \] |
3 |
1 |
10 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
143.295 |
|
|
\[ {}{y^{\prime }}^{4} = \left (y-a \right )^{3} \left (y-b \right )^{2} \] |
4 |
4 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
4.075 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{2} = 0 \] |
4 |
4 |
4 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
4.665 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} = 0 \] |
4 |
4 |
4 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
12.732 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} \left (y-c \right )^{2} = 0 \] |
4 |
4 |
1 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
668.064 |
|
|
\[ {}{y^{\prime }}^{4}+x y^{\prime }-3 y = 0 \] |
4 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.38 |
|
|
\[ {}{y^{\prime }}^{4}-4 x^{2} y {y^{\prime }}^{2}+16 x y^{2} y^{\prime }-16 y^{3} = 0 \] |
4 |
0 |
3 |
unknown |
[[_homogeneous, ‘class G‘]] |
✗ |
N/A |
1.309 |
|
|
\[ {}{y^{\prime }}^{4}+4 y {y^{\prime }}^{3}+6 y^{2} {y^{\prime }}^{2}-\left (1-4 y^{3}\right ) y^{\prime }-\left (3-y^{3}\right ) y = 0 \] |
4 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.856 |
|
|
\[ {}2 {y^{\prime }}^{4}-y y^{\prime }-2 = 0 \] |
4 |
1 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.622 |
|
|
\[ {}x {y^{\prime }}^{4}-2 y {y^{\prime }}^{3}+12 x^{3} = 0 \] |
4 |
0 |
5 |
unknown |
[[_1st_order, _with_linear_symmetries]] |
✗ |
N/A |
2.131 |
|
|
\[ {}3 {y^{\prime }}^{5}-y y^{\prime }+1 = 0 \] |
5 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.233 |
|
|
\[ {}{y^{\prime }}^{6} = \left (y-a \right )^{4} \left (y-b \right )^{3} \] |
6 |
6 |
8 |
quadrature |
[_quadrature] |
✓ |
✓ |
91.425 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{4} \left (y-b \right )^{3} = 0 \] |
6 |
6 |
1 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
99.567 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{3} = 0 \] |
6 |
6 |
1 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
213.078 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{4} = 0 \] |
6 |
6 |
1 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
778.25 |
|
|
\[ {}x^{2} \left ({y^{\prime }}^{6}+3 y^{4}+3 y^{2}+1\right ) = a^{2} \] |
6 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
15.077 |
|
|
\[ {}2 \sqrt {a y^{\prime }}+x y^{\prime }-y = 0 \] |
2 |
2 |
1 |
clairaut |
[[_homogeneous, ‘class G‘], _Clairaut] |
✓ |
✓ |
2.11 |
|
|
\[ {}\left (x -y\right ) \sqrt {y^{\prime }} = a \left (1+y^{\prime }\right ) \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
3.405 |
|
|
\[ {}2 \left (y+1\right )^{\frac {3}{2}}+3 x y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
8.423 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } = x \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.527 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } = y \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.367 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}} = x y^{\prime } \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.731 |
|
|
\[ {}\sqrt {a^{2}+b^{2} {y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
2 |
3 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
4.38 |
|
|
\[ {}a \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
2 |
3 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
2.182 |
|
|
\[ {}a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.764 |
|
|
\[ {}\sqrt {\left (x^{2} a +y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )}-y y^{\prime }-x a = 0 \] |
2 |
8 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
246.119 |
|
|
\[ {}a \left (1+{y^{\prime }}^{3}\right )^{\frac {1}{3}}+x y^{\prime }-y = 0 \] |
3 |
7 |
1 |
clairaut |
[_Clairaut] |
✓ |
✓ |
183.375 |
|
|
\[ {}\cos \left (y^{\prime }\right )+x y^{\prime } = y \] |
0 |
2 |
2 |
clairaut |
[_Clairaut] |
✓ |
✓ |
0.425 |
|
|
\[ {}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.26 |
|
|
\[ {}\sin \left (y^{\prime }\right )+y^{\prime } = x \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.269 |
|
|
\[ {}y^{\prime } \sin \left (y^{\prime }\right )+\cos \left (y^{\prime }\right ) = y \] |
0 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.821 |
|
|
\[ {}{y^{\prime }}^{2} \left (x +\sin \left (y^{\prime }\right )\right ) = y \] |
0 |
3 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.898 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \sin \left (-y+x y^{\prime }\right )^{2} = 1 \] |
0 |
6 |
6 |
clairaut |
[_Clairaut] |
✓ |
✓ |
3.925 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+x a \right )+y^{\prime } = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✗ |
1.71 |
|
|
\[ {}{\mathrm e}^{y^{\prime }-y}-{y^{\prime }}^{2}+1 = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.203 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.567 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a = y \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.879 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a +b y = 0 \] |
0 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
3.114 |
|
|
\[ {}\ln \left (y^{\prime }\right )+4 x y^{\prime }-2 y = 0 \] |
0 |
2 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.312 |
|
|
\[ {}\ln \left (y^{\prime }\right )+a \left (-y+x y^{\prime }\right ) = 0 \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.828 |
|
|
\[ {}a \left (\ln \left (y^{\prime }\right )-y^{\prime }\right )-x +y = 0 \] |
0 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.364 |
|
|
\[ {}y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-x y = 0 \] |
0 |
1 |
1 |
separable, homogeneousTypeD2 |
[_separable] |
✓ |
✓ |
3.098 |
|
|
\[ {}y^{\prime } \ln \left (y^{\prime }\right )-\left (1+x \right ) y^{\prime }+y = 0 \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.868 |
|
|
\[ {}y^{\prime } \ln \left (y^{\prime }+\sqrt {a +{y^{\prime }}^{2}}\right )-\sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }+y = 0 \] |
0 |
2 |
0 |
clairaut |
[_Clairaut] |
✓ |
✓ |
7.899 |
|
|
\[ {}\ln \left (\cos \left (y^{\prime }\right )\right )+y^{\prime } \tan \left (y^{\prime }\right ) = y \] |
0 |
0 |
2 |
unknown |
[_dAlembert] |
✗ |
N/A |
1.551 |
|
|
|
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|
|
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|
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