Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}x^{2}-y^{2}+x y y^{\prime } = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.035 |
|
|
\[ {}x^{2} y^{\prime }-2 x y-2 y^{2} = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.959 |
|
|
\[ {}x^{2} y^{\prime } = 3 \left (x^{2}+y^{2}\right ) \arctan \left (\frac {y}{x}\right )+x y \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
5.107 |
|
|
\[ {}x \sin \left (\frac {y}{x}\right ) y^{\prime } = y \sin \left (\frac {y}{x}\right )+x \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.435 |
|
|
\[ {}x y^{\prime } = y+2 \,{\mathrm e}^{-\frac {y}{x}} \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
0.848 |
|
|
\[ {}y^{\prime } = \left (x +y\right )^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.807 |
|
|
\[ {}y^{\prime } = \sin \left (x -y+1\right )^{2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.507 |
|
|
\[ {}y^{\prime } = \frac {x +y+4}{x -y-6} \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.974 |
|
|
\[ {}y^{\prime } = \frac {x +y+4}{x +y-6} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.003 |
|
|
\[ {}\left (x +\frac {2}{y}\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.217 |
|
|
\[ {}\sin \left (x \right ) \tan \left (y\right )+1+\cos \left (x \right ) \sec \left (y\right )^{2} y^{\prime } = 0 \] |
1 |
1 |
0 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
33.48 |
|
|
\[ {}y-x^{3}+\left (x +y^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
1.237 |
|
|
\[ {}2 y^{2}-4 x +5 = \left (4-2 y+4 x y\right ) y^{\prime } \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.05 |
|
|
\[ {}y+y \cos \left (x y\right )+\left (x +x \cos \left (x y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, linear, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.069 |
|
|
\[ {}\cos \left (x \right ) \cos \left (y\right )^{2}+2 \sin \left (x \right ) \sin \left (y\right ) \cos \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
38.22 |
|
|
\[ {}\left (\sin \left (x \right ) \sin \left (y\right )-x \,{\mathrm e}^{y}\right ) y^{\prime } = {\mathrm e}^{y}+\cos \left (x \right ) \cos \left (y\right ) \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
7.582 |
|
|
\[ {}-\frac {\sin \left (\frac {x}{y}\right )}{y}+\frac {x \sin \left (\frac {x}{y}\right ) y^{\prime }}{y^{2}} = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.973 |
|
|
\[ {}1+y+\left (1-x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.994 |
|
|
\[ {}2 x y^{3}+y \cos \left (x \right )+\left (3 x^{2} y^{2}+\sin \left (x \right )\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
31.98 |
|
|
\[ {}1 = \frac {y}{1-x^{2} y^{2}}+\frac {x y^{\prime }}{1-x^{2} y^{2}} \] |
1 |
1 |
1 |
exact, riccati |
[_exact, _rational, _Riccati] |
✓ |
✓ |
1.38 |
|
|
\[ {}\left (3 x^{2}-y^{2}\right ) y^{\prime }-2 x y = 0 \] |
1 |
1 |
3 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.34 |
|
|
\[ {}x y-1+\left (x^{2}-x y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.131 |
|
|
\[ {}\left (x +3 x^{3} y^{4}\right ) y^{\prime }+y = 0 \] |
1 |
1 |
4 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.339 |
|
|
\[ {}\left (x -1-y^{2}\right ) y^{\prime }-y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
0.852 |
|
|
\[ {}y-\left (x +x y^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.707 |
|
|
\[ {}x y^{\prime } = x^{5}+x^{3} y^{2}+y \] |
1 |
1 |
1 |
riccati, exactByInspection, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.711 |
|
|
\[ {}\left (x +y\right ) y^{\prime } = y-x \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.237 |
|
|
\[ {}x y^{\prime } = y+x^{2}+9 y^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.45 |
|
|
\[ {}x y^{\prime }-3 y = x^{4} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.685 |
|
|
\[ {}y^{\prime }+y = \frac {1}{{\mathrm e}^{2 x}+1} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.835 |
|
|
\[ {}2 x y+\left (x^{2}+1\right ) y^{\prime } = \cot \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.515 |
|
|
\[ {}y^{\prime }+y = 2 x \,{\mathrm e}^{-x}+x^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.165 |
|
|
\[ {}y^{\prime }+y \cot \left (x \right ) = 2 x \csc \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.387 |
|
|
\[ {}2 y-x^{3} = x y^{\prime } \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.095 |
|
|
\[ {}\left (1-x y\right ) y^{\prime } = y^{2} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.646 |
|
|
\[ {}2 x +3 y+1+\left (2 y-3 x +5\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.294 |
|
|
\[ {}x y^{\prime } = \sqrt {x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
11.182 |
|
|
\[ {}y^{2} = \left (x^{3}-x y\right ) y^{\prime } \] |
1 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
4.336 |
|
|
\[ {}x^{2}+y^{3}+y = \left (x^{3} y^{2}-x \right ) y^{\prime } \] |
1 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
1.843 |
|
|
\[ {}x y^{\prime }+y = x \cos \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.263 |
|
|
\[ {}\left (x y-x^{2}\right ) y^{\prime } = y^{2} \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.762 |
|
|
\[ {}\left ({\mathrm e}^{x}-3 x^{2} y^{2}\right ) y^{\prime }+{\mathrm e}^{x} y = 2 x y^{3} \] |
1 |
1 |
3 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.849 |
|
|
\[ {}y+x^{2} = x y^{\prime } \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.75 |
|
|
\[ {}x y^{\prime }+y = x^{2} \cos \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.962 |
|
|
\[ {}6 x +4 y+3+\left (3 x +2 y+2\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.287 |
|
|
\[ {}\cos \left (x +y\right )-x \sin \left (x +y\right ) = x \sin \left (x +y\right ) y^{\prime } \] |
1 |
1 |
1 |
exact, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _exact] |
✓ |
✓ |
3.343 |
|
|
\[ {}y^{2} {\mathrm e}^{x y}+\cos \left (x \right )+\left ({\mathrm e}^{x y}+x y \,{\mathrm e}^{x y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
32.26 |
|
|
\[ {}y^{\prime } \ln \left (x -y\right ) = 1+\ln \left (x -y\right ) \] |
1 |
1 |
1 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _dAlembert] |
✓ |
✓ |
1.388 |
|
|
\[ {}y^{\prime }+2 x y = {\mathrm e}^{-x^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.701 |
|
|
\[ {}y^{2}-3 x y-2 x^{2} = \left (x^{2}-x y\right ) y^{\prime } \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.286 |
|
|
\[ {}2 x y+\left (x^{2}+1\right ) y^{\prime } = 4 x^{3} \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.955 |
|
|
\[ {}{\mathrm e}^{x} \sin \left (y\right )-y \sin \left (x y\right )+\left ({\mathrm e}^{x} \cos \left (y\right )-x \sin \left (x y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
33.056 |
|
|
\[ {}\left (x \,{\mathrm e}^{y}+y-x^{2}\right ) y^{\prime } = 2 x y-{\mathrm e}^{y}-x \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
1.879 |
|
|
\[ {}{\mathrm e}^{x} \left (1+x \right ) = \left (x \,{\mathrm e}^{x}-y \,{\mathrm e}^{y}\right ) y^{\prime } \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.777 |
|
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