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) |
|
\[ {}y^{\prime } = 2 x y \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.505 |
|
|
\[ {}y^{\prime } = \frac {y^{2}}{x^{2}+1} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.513 |
|
|
\[ {}{\mathrm e}^{x +y} y^{\prime }-1 = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.407 |
|
|
\[ {}y^{\prime } = \frac {y}{x \ln \left (x \right )} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.674 |
|
|
\[ {}y-\left (-2+x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.615 |
|
|
\[ {}y^{\prime } = \frac {2 x \left (y-1\right )}{x^{2}+3} \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.744 |
|
|
\[ {}y-x y^{\prime } = 3-2 x^{2} y^{\prime } \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.659 |
|
|
\[ {}y^{\prime } = \frac {\cos \left (x -y\right )}{\sin \left (x \right ) \sin \left (y\right )}-1 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
36.244 |
|
|
\[ {}y^{\prime } = \frac {x \left (y^{2}-1\right )}{2 \left (-2+x \right ) \left (-1+x \right )} \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.073 |
|
|
\[ {}y^{\prime } = \frac {x^{2} y-32}{-x^{2}+16}+32 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.852 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }-y+c = 0 \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.233 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+y^{2} = -1 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.069 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+x y = x a \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.399 |
|
|
\[ {}y^{\prime } = 1-\frac {\sin \left (x +y\right )}{\sin \left (y\right ) \cos \left (x \right )} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
36.764 |
|
|
\[ {}y^{\prime } = y^{3} \sin \left (x \right ) \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.524 |
|
|
\[ {}y^{\prime }-y = {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.464 |
|
|
\[ {}x^{2} y^{\prime }-4 x y = x^{7} \sin \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.71 |
|
|
\[ {}y^{\prime }+2 x y = 2 x^{3} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.51 |
|
|
\[ {}y^{\prime }+\frac {2 x y}{x^{2}+1} = 4 x \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.722 |
|
|
\[ {}y^{\prime }+\frac {2 x y}{x^{2}+1} = \frac {4}{\left (x^{2}+1\right )^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.65 |
|
|
\[ {}2 \cos \left (x \right )^{2} y^{\prime }+y \sin \left (2 x \right ) = 4 \cos \left (x \right )^{4} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.946 |
|
|
\[ {}y^{\prime }+\frac {y}{x \ln \left (x \right )} = 9 x^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.669 |
|
|
\[ {}y^{\prime }-y \tan \left (x \right ) = 8 \sin \left (x \right )^{3} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.744 |
|
|
\[ {}t x^{\prime }+2 x = 4 \,{\mathrm e}^{t} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.585 |
|
|
\[ {}y^{\prime } = \sin \left (x \right ) \left (y \sec \left (x \right )-2\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.757 |
|
|
\[ {}1-y \sin \left (x \right )-\cos \left (x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.935 |
|
|
\[ {}y^{\prime }-\frac {y}{x} = 2 \ln \left (x \right ) x^{2} \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.595 |
|
|
\[ {}y^{\prime }+\alpha y = {\mathrm e}^{\beta x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime }+\frac {m}{x} = \ln \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.135 |
|
|
\[ {}\left (3 x -y\right ) y^{\prime } = 3 y \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.94 |
|
|
\[ {}y^{\prime } = \frac {\left (x +y\right )^{2}}{2 x^{2}} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.915 |
|
|
\[ {}\sin \left (\frac {y}{x}\right ) \left (-y+x y^{\prime }\right ) = x \cos \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.482 |
|
|
\[ {}x y^{\prime } = \sqrt {16 x^{2}-y^{2}}+y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.104 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {9 x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.286 |
|
|
\[ {}x \left (x^{2}-y^{2}\right )-x \left (x^{2}+y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.763 |
|
|
\[ {}x y^{\prime }+y \ln \left (x \right ) = y \ln \left (y\right ) \] |
1 |
1 |
1 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.011 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+2 x y-2 x^{2}}{x^{2}-x y+y^{2}} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.046 |
|
|
\[ {}2 x y y^{\prime }-x^{2} {\mathrm e}^{-\frac {y^{2}}{x^{2}}}-2 y^{2} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘]] |
✓ |
✓ |
0.896 |
|
|
\[ {}x^{2} y^{\prime } = y^{2}+3 x y+x^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.714 |
|
|
\[ {}y y^{\prime } = \sqrt {x^{2}+y^{2}}-x \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.407 |
|
|
\[ {}2 x \left (y+2 x \right ) y^{\prime } = y \left (4 x -y\right ) \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.148 |
|
|
\[ {}x y^{\prime } = x \tan \left (\frac {y}{x}\right )+y \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.953 |
|
|
\[ {}y^{\prime } = \frac {x \sqrt {x^{2}+y^{2}}+y^{2}}{x y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.997 |
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