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) |
|
\[ {}y^{\prime } = \frac {x^{2}}{y} \] |
1 |
2 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.309 |
|
|
\[ {}y^{\prime } = \frac {x^{2}}{y \left (x^{3}+1\right )} \] |
1 |
2 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.181 |
|
|
\[ {}y^{\prime } = y \sin \left (x \right ) \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.265 |
|
|
\[ {}x y^{\prime } = \sqrt {1-y^{2}} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.757 |
|
|
\[ {}y^{\prime } = \frac {x^{2}}{1+y^{2}} \] |
1 |
3 |
3 |
separable |
[_separable] |
✓ |
✓ |
156.172 |
|
|
\[ {}x y y^{\prime } = \sqrt {1+y^{2}} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.33 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+2 x y^{2} = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.523 |
|
|
\[ {}y^{\prime } = 3 y^{\frac {2}{3}} \] |
1 |
1 |
1 |
separable |
[_quadrature] |
✓ |
✓ |
0.492 |
|
|
\[ {}x y^{\prime }+y = y^{2} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.662 |
|
|
\[ {}2 x^{2} y y^{\prime }+y^{2} = 2 \] |
1 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.775 |
|
|
\[ {}y^{\prime }-x y^{2} = 2 x y \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.595 |
|
|
\[ {}\left (1+z^{\prime }\right ) {\mathrm e}^{-z} = 1 \] |
1 |
1 |
1 |
separable |
[_quadrature] |
✓ |
✓ |
0.446 |
|
|
\[ {}y^{\prime } = \frac {3 x^{2}+4 x +2}{2 y-2} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.828 |
|
|
\[ {}{\mathrm e}^{x}-\left (1+{\mathrm e}^{x}\right ) y y^{\prime } = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
1.022 |
|
|
\[ {}\frac {y}{-1+x}+\frac {x y^{\prime }}{y+1} = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.474 |
|
|
\[ {}x +2 x^{3}+\left (y+2 y^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
separable |
[_separable] |
✓ |
✓ |
0.3 |
|
|
\[ {}\frac {1}{\sqrt {x}}+\frac {y^{\prime }}{\sqrt {y}} = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.27 |
|
|
\[ {}\frac {1}{\sqrt {-x^{2}+1}}+\frac {y^{\prime }}{\sqrt {1-y^{2}}} = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.385 |
|
|
\[ {}2 x \sqrt {1-y^{2}}+y y^{\prime } = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.428 |
|
|
\[ {}y^{\prime } = \left (y-1\right ) \left (1+x \right ) \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.286 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{x -y} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.161 |
|
|
\[ {}y^{\prime } = \frac {\sqrt {y}}{\sqrt {x}} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.244 |
|
|
\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.21 |
|
|
\[ {}z^{\prime } = 10^{x +z} \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.222 |
|
|
\[ {}x^{\prime }+t = 1 \] |
1 |
1 |
1 |
separable |
[_quadrature] |
✓ |
✓ |
0.129 |
|
|
\[ {}y^{\prime } = \cos \left (x -y\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.181 |
|
|
\[ {}y^{\prime }-y = 2 x -3 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.591 |
|
|
\[ {}\left (2 y+x \right ) y^{\prime } = 1 \] |
1 |
1 |
1 |
homogeneousTypeC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.563 |
|
|
\[ {}y^{\prime }+y = 2 x +1 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.585 |
|
|
\[ {}y^{\prime } = \cos \left (x -y-1\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.274 |
|
|
\[ {}y^{\prime }+\sin \left (x +y\right )^{2} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.364 |
|
|
\[ {}y^{\prime } = 2 \sqrt {2 x +y+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
4.118 |
|
|
\[ {}y^{\prime } = \left (1+x +y\right )^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.776 |
|
|
\[ {}y^{2}+x y^{2}+\left (x^{2}-x^{2} y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
0.765 |
|
|
\[ {}\left (1+y^{2}\right ) \left ({\mathrm e}^{2 x}-{\mathrm e}^{y} y^{\prime }\right )-\left (y+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
separable |
[_separable] |
✓ |
✓ |
1.892 |
|
|
\[ {}x -y+\left (x +y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.62 |
|
|
\[ {}y-2 x y+x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.016 |
|
|
\[ {}2 x y^{\prime } = y \left (2 x^{2}-y^{2}\right ) \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.091 |
|
|
\[ {}y^{2}+x^{2} y^{\prime } = x y y^{\prime } \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.435 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime } = 2 x y \] |
1 |
1 |
2 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.681 |
|
|
\[ {}-y+x y^{\prime } = x \tan \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.096 |
|
|
\[ {}x y^{\prime } = y-x \,{\mathrm e}^{\frac {y}{x}} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.381 |
|
|
\[ {}-y+x y^{\prime } = \left (x +y\right ) \ln \left (\frac {x +y}{x}\right ) \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.677 |
|
|
\[ {}x y^{\prime } = y \cos \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.194 |
|
|
\[ {}y+\sqrt {x y}-x y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.49 |
|
|
\[ {}x y^{\prime }-\sqrt {x^{2}-y^{2}}-y = 0 \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.134 |
|
|
\[ {}x +y-\left (x -y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.603 |
|
|
\[ {}x^{2}+2 x y-y^{2}+\left (y^{2}+2 x y-x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.068 |
|
|
\[ {}-y+x y^{\prime } = y y^{\prime } \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.46 |
|
|
\[ {}y^{2}+\left (x^{2}-x y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.436 |
|
|
\[ {}x^{2}+x y+y^{2} = x^{2} y^{\prime } \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
0.477 |
|
|
\[ {}\frac {1}{x^{2}-x y+y^{2}} = \frac {y^{\prime }}{2 y^{2}-x y} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
12.835 |
|
|
\[ {}y^{\prime } = \frac {2 x y}{3 x^{2}-y^{2}} \] |
1 |
1 |
3 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.686 |
|
|
\[ {}y^{\prime } = \frac {x}{y}+\frac {y}{x} \] |
1 |
1 |
2 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.636 |
|
|
\[ {}x y^{\prime } = y+\sqrt {-x^{2}+y^{2}} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.382 |
|
|
\[ {}y+\left (2 \sqrt {x y}-x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.884 |
|
|
\[ {}x y^{\prime } = y \ln \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.492 |
|
|
\[ {}y^{\prime } \left (y^{\prime }+y\right ) = x \left (x +y\right ) \] |
2 |
1 |
1 |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
0.649 |
|
|
\[ {}\left (x y^{\prime }+y\right )^{2} = y^{2} y^{\prime } \] |
2 |
2 |
6 |
homogeneous |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
5.827 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-3 x y y^{\prime }+2 y^{2} = 0 \] |
2 |
2 |
2 |
homogeneous |
[_separable] |
✓ |
✓ |
0.859 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {x^{2}+y^{2}} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.688 |
|
|
\[ {}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
2 |
7 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.971 |
|
|
\[ {}y^{\prime }+\frac {2 y+x}{x} = 0 \] |
1 |
1 |
1 |
homogeneous |
[_linear] |
✓ |
✓ |
0.663 |
|
|
\[ {}y^{\prime } = \frac {y}{x +y} \] |
1 |
1 |
1 |
homogeneous |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.464 |
|
|
\[ {}x y^{\prime } = x +\frac {y}{2} \] |
1 |
0 |
1 |
homogeneous |
[_linear] |
✗ |
N/A |
0.697 |
|
|
\[ {}y^{\prime } = \frac {x +y-2}{y-x -4} \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.264 |
|
|
\[ {}2 x -4 y+6+\left (x +y-2\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.81 |
|
|
\[ {}y^{\prime } = \frac {2 y-x +5}{2 x -y-4} \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.946 |
|
|
\[ {}y^{\prime } = -\frac {4 x +3 y+15}{2 x +y+7} \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.898 |
|
|
\[ {}y^{\prime } = \frac {x +3 y-5}{x -y-1} \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.607 |
|
|
\[ {}y^{\prime } = \frac {2 \left (y+2\right )^{2}}{\left (1+x +y\right )^{2}} \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
1.73 |
|
|
\[ {}2 x +y+1-\left (4 x +2 y-3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.326 |
|
|
\[ {}x -y-1+\left (-x +y+2\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
polynomial |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.306 |
|
|
\[ {}\left (x +4 y\right ) y^{\prime } = 2 x +3 y-5 \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.892 |
|
|
\[ {}y+2 = \left (-4+2 x +y\right ) y^{\prime } \] |
1 |
1 |
2 |
homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.622 |
|
|
\[ {}\left (1+y^{\prime }\right ) \ln \left (\frac {x +y}{x +3}\right ) = \frac {x +y}{x +3} \] |
1 |
1 |
1 |
exact, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _dAlembert] |
✓ |
✓ |
6.257 |
|
|
\[ {}y^{\prime } = \frac {x -2 y+5}{y-2 x -4} \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.926 |
|
|
\[ {}y^{\prime } = \frac {3 x -y+1}{2 x +y+4} \] |
1 |
1 |
1 |
polynomial |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.9 |
|
|
\[ {}2 x y^{\prime }+\left (y^{4} x^{2}+1\right ) y = 0 \] |
1 |
4 |
4 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.036 |
|
|
\[ {}2 x y^{\prime } \left (x -y^{2}\right )+y^{3} = 0 \] |
1 |
1 |
1 |
isobaric |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.254 |
|
|
\[ {}x^{3} \left (y^{\prime }-x \right ) = y^{2} \] |
1 |
1 |
1 |
isobaric |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
0.309 |
|
|
\[ {}2 x^{2} y^{\prime } = y^{3}+x y \] |
1 |
1 |
2 |
isobaric |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.221 |
|
|
\[ {}y+x \left (2 x y+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
isobaric |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.223 |
|
|
\[ {}2 y^{\prime }+x = 4 \sqrt {y} \] |
1 |
1 |
1 |
isobaric |
[[_1st_order, _with_linear_symmetries], _Chini] |
✓ |
✓ |
0.579 |
|
|
\[ {}y^{\prime } = y^{2}-\frac {2}{x^{2}} \] |
1 |
1 |
1 |
isobaric |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
0.358 |
|
|
\[ {}2 x y^{\prime }+y = y^{2} \sqrt {x -x^{2} y^{2}} \] |
1 |
1 |
1 |
isobaric |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
0.716 |
|
|
\[ {}\frac {2 x y y^{\prime }}{3} = \sqrt {x^{6}-y^{4}}+y^{2} \] |
1 |
1 |
1 |
isobaric |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.222 |
|
|
\[ {}2 y+\left (x^{2} y+1\right ) x y^{\prime } = 0 \] |
1 |
1 |
1 |
isobaric |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.233 |
|
|
\[ {}y \left (1+x y\right )+x \left (1-x y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
isobaric |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.231 |
|
|
\[ {}y \left (1+x^{2} y^{2}\right )+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0 \] |
1 |
1 |
1 |
isobaric |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.245 |
|
|
\[ {}\left (x^{2}-y^{4}\right ) y^{\prime }-x y = 0 \] |
1 |
1 |
4 |
isobaric |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.489 |
|
|
\[ {}y \left (1+\sqrt {y^{4} x^{2}-1}\right )+2 x y^{\prime } = 0 \] |
1 |
1 |
1 |
isobaric |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
0.726 |
|
|
\[ {}x \left (2-9 x y^{2}\right )+y \left (4 y^{2}-6 x^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
exact |
[_exact, _rational] |
✓ |
✓ |
1.365 |
|
|
\[ {}\frac {y}{x}+\left (y^{3}+\ln \left (x \right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.298 |
|
|
\[ {}2 x +3+\left (2 y-2\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.096 |
|
|
\[ {}2 x +4 y+\left (2 x -2 y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.745 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.289 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.218 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.136 |
|
|
\[ {}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+y \cos \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
3.764 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+x^{2} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.967 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.171 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.717 |
|
|
\[ {}y^{\prime \prime \prime }-2 x y^{\prime \prime }+4 x^{2} y^{\prime }+8 x^{3} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.249 |
|
|
\[ {}y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+{\mathrm e}^{x} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.732 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.605 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }-x^{2} y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
73.313 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.75 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+y = 2 x \,{\mathrm e}^{x}-1 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.059 |
|
|
\[ {}x y^{\prime \prime }+x y^{\prime }-y = x^{2}+2 x \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.22 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = x^{2}+2 x \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.867 |
|
|
\[ {}x^{3} y^{\prime \prime }+x y^{\prime }-y = \cos \left (\frac {1}{x}\right ) \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.593 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }-y = x +\frac {1}{x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.536 |
|
|
\[ {}2 x y^{\prime \prime }+\left (-2+x \right ) y^{\prime }-y = x^{2}-1 \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.181 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y = x +\frac {1}{x} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
10.455 |
|
|
\[ {}x^{2} \left (-1+\ln \left (x \right )\right ) y^{\prime \prime }-x y^{\prime }+y = x \left (1-\ln \left (x \right )\right )^{2} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.549 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = \sec \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.359 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\frac {y}{4} = -\frac {x^{2}}{2}+\frac {1}{2} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
6.195 |
|
|
\[ {}\left (\cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }+\left (\cos \left (x \right )-\sin \left (x \right )\right ) y = \left (\cos \left (x \right )+\sin \left (x \right )\right )^{2} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
51.21 |
|
|
\[ {}\left (\cos \left (x \right )-\sin \left (x \right )\right ) y^{\prime \prime }-2 \sin \left (x \right ) y^{\prime }+\left (\cos \left (x \right )+\sin \left (x \right )\right ) y = \left (\cos \left (x \right )-\sin \left (x \right )\right )^{2} \] |
1 |
0 |
0 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
76.833 |
|
|
|
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|
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