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) |
|
\[ {}\left (1+x \right ) y+x \left (1-y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.963 |
|
|
\[ {}y^{2}+x y^{2}+\left (x^{2}-x^{2} y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.661 |
|
|
\[ {}x y \left (x^{2}+1\right ) y^{\prime }-1-y^{2} = 0 \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
6.803 |
|
|
\[ {}1+y^{2}-\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{\frac {3}{2}} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
23.264 |
|
|
\[ {}\sin \left (x \right ) \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.344 |
|
|
\[ {}\sec \left (x \right )^{2} \tan \left (y\right )+\sec \left (y\right )^{2} \tan \left (x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
40.663 |
|
|
\[ {}\left (y-x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.105 |
|
|
\[ {}\left (2 \sqrt {x y}-x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.198 |
|
|
\[ {}x y^{\prime }-y-\sqrt {x^{2}+y^{2}} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.44 |
|
|
\[ {}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.442 |
|
|
\[ {}8 y+10 x +\left (5 y+7 x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.527 |
|
|
\[ {}2 x -y+1+\left (2 y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.587 |
|
|
\[ {}3 y-7 x +7+\left (7 y-3 x +3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.013 |
|
|
\[ {}y^{\prime }+\frac {x y}{x^{2}+1} = \frac {1}{2 \left (x^{2}+1\right ) x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.835 |
|
|
\[ {}x \left (-x^{2}+1\right ) y^{\prime }+\left (2 x^{2}-1\right ) y = a \,x^{3} \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.347 |
|
|
\[ {}y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{\frac {3}{2}}} = \frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
4.732 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.525 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+y = \arctan \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.905 |
|
|
\[ {}\left (-x^{2}+1\right ) z^{\prime }-x z = a x z^{2} \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.239 |
|
|
\[ {}3 z^{2} z^{\prime }-a z^{3} = 1+x \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.86 |
|
|
\[ {}z^{\prime }+2 x z = 2 a \,x^{3} z^{3} \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.997 |
|
|
\[ {}z^{\prime }+z \cos \left (x \right ) = z^{n} \sin \left (2 x \right ) \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
32.015 |
|
|
\[ {}x y^{\prime }+y = y^{2} \ln \left (x \right ) \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.004 |
|
|
\[ {}x^{3}+3 x y^{2}+\left (y^{3}+3 x^{2} y\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
0.334 |
|
|
\[ {}1+\frac {y^{2}}{x^{2}}-\frac {2 y y^{\prime }}{x} = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.256 |
|
|
\[ {}\frac {3 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.381 |
|
|
\[ {}x +y y^{\prime }+\frac {x y^{\prime }}{x^{2}+y^{2}}-\frac {y}{x^{2}+y^{2}} = 0 \] |
1 |
1 |
1 |
exact |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
✓ |
0.434 |
|
|
\[ {}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
✓ |
0.387 |
|
|
\[ {}{\mathrm e}^{x} \left (x^{2}+y^{2}+2 x \right )+2 y \,{\mathrm e}^{x} y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class D‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.313 |
|
|
\[ {}n \cos \left (n x +m y\right )-m \sin \left (m x +n y\right )+\left (m \cos \left (n x +m y\right )-n \sin \left (m x +n y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.605 |
|
|
\[ {}\frac {x}{\sqrt {1+x^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y}{x^{2}+y^{2}}-\frac {x y^{\prime }}{x^{2}+y^{2}} = 0 \] |
1 |
1 |
1 |
exact |
[[_1st_order, _with_linear_symmetries], _exact] |
✓ |
✓ |
0.865 |
|
|
\[ {}\frac {x^{n} y^{\prime }}{b y^{2}-c \,x^{2 a}}-\frac {a y x^{a -1}}{b y^{2}-c \,x^{2 a}}+x^{a -1} = 0 \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
40.685 |
|
|
\[ {}2 x y+\left (y^{2}-2 x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.365 |
|
|
\[ {}\frac {1}{x}+\frac {y^{\prime }}{y}+\frac {2}{y}-\frac {2 y^{\prime }}{x} = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.227 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.613 |
|
|
\[ {}8 y+10 x +\left (5 y+7 x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.67 |
|
|
\[ {}x^{2}+2 x y-y^{2}+\left (y^{2}+2 x y-x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.637 |
|
|
\[ {}y^{2}+\left (x y+x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.676 |
|
|
\[ {}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y+\left (x \cos \left (\frac {y}{x}\right )-y \sin \left (\frac {y}{x}\right )\right ) x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.647 |
|
|
\[ {}\left (x^{2} y^{2}+x y\right ) y+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.265 |
|
|
\[ {}\left (x^{3} y^{3}+x^{2} y^{2}+x y+1\right ) y+\left (x^{3} y^{3}-x^{2} y^{2}-x y+1\right ) x y^{\prime } = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.514 |
|
|
\[ {}x^{2}+y^{2}+2 x +2 y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.229 |
|
|
\[ {}x^{2}+y^{2}-2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.204 |
|
|
\[ {}2 x y+\left (y^{2}-3 x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.243 |
|
|
\[ {}y+\left (2 y-x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.33 |
|
|
\[ {}x y^{\prime }-a y+y^{2} = x^{-2 a} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
0.6 |
|
|
\[ {}x y^{\prime }-a y+y^{2} = x^{-\frac {2 a}{3}} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.211 |
|
|
\[ {}u^{\prime }+u^{2} = \frac {c}{x^{\frac {4}{3}}} \] |
1 |
1 |
1 |
riccati |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
0.856 |
|
|
\[ {}u^{\prime }+b u^{2} = \frac {c}{x^{4}} \] |
1 |
1 |
1 |
riccati |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
0.313 |
|
|
\[ {}u^{\prime }-u^{2} = \frac {2}{x^{\frac {8}{3}}} \] |
1 |
1 |
1 |
riccati |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
1.391 |
|
|
\[ {}\frac {\sqrt {f \,x^{4}+c \,x^{3}+c \,x^{2}+b x +a}\, y^{\prime }}{\sqrt {a +b y+c y^{2}+c y^{3}+f y^{4}}} = -1 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
96.521 |
|
|
\[ {}{y^{\prime }}^{2}-5 y^{\prime }+6 = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.249 |
|
|
\[ {}{y^{\prime }}^{2}-\frac {a^{2}}{x^{2}} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.291 |
|
|
\[ {}{y^{\prime }}^{2} = \frac {1-x}{x} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.854 |
|
|
\[ {}{y^{\prime }}^{2}+\frac {2 x y^{\prime }}{y}-1 = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.858 |
|
|
\[ {}y = a y^{\prime }+b {y^{\prime }}^{2} \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.165 |
|
|
\[ {}x = a y^{\prime }+b {y^{\prime }}^{2} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.504 |
|
|
\[ {}y = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.484 |
|
|
\[ {}x = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.289 |
|
|
\[ {}y^{\prime }-\frac {\sqrt {1+{y^{\prime }}^{2}}}{x} = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.679 |
|
|
\[ {}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0 \] |
6 |
6 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
5.812 |
|
|
\[ {}1+{y^{\prime }}^{2} = \frac {\left (x +a \right )^{2}}{2 x a +x^{2}} \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.461 |
|
|
\[ {}y = x y^{\prime }+y^{\prime }-{y^{\prime }}^{2} \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.26 |
|
|
\[ {}y = x y^{\prime }+\sqrt {b^{2}-a^{2} {y^{\prime }}^{2}} \] |
2 |
3 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
4.908 |
|
|
\[ {}y = x y^{\prime }+x \sqrt {1+{y^{\prime }}^{2}} \] |
1 |
2 |
2 |
bernoulli, dAlembert, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
5.283 |
|
|
\[ {}y = x y^{\prime }+a x \sqrt {1+{y^{\prime }}^{2}} \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.856 |
|
|
\[ {}x -y y^{\prime } = a {y^{\prime }}^{2} \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
90.835 |
|
|
\[ {}x +y y^{\prime } = a \sqrt {1+{y^{\prime }}^{2}} \] |
2 |
4 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
38.324 |
|
|
\[ {}y y^{\prime } = x +y^{2}-y^{2} {y^{\prime }}^{2} \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
3.847 |
|
|
\[ {}y-\frac {1}{\sqrt {1+{y^{\prime }}^{2}}} = x +\frac {y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}} \] |
2 |
3 |
1 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
91.131 |
|
|
\[ {}y-2 x y^{\prime } = x {y^{\prime }}^{2} \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.322 |
|
|
\[ {}\frac {y-x y^{\prime }}{y^{2}+y^{\prime }} = \frac {y-x y^{\prime }}{1+x^{2} y^{\prime }} \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.897 |
|
|
|
|||||||||
|
|
|||||||||