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) |
|
\[ {}y^{\prime } = x^{2}+y^{2} \] |
1 |
1 |
1 |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
1.366 |
|
|
\[ {}y^{\prime } = \frac {x}{y} \] |
1 |
1 |
2 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.51 |
|
|
\[ {}y^{\prime } = y+3 y^{\frac {1}{3}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.855 |
|
|
\[ {}y^{\prime } = \sqrt {x -y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
5.955 |
|
|
\[ {}y^{\prime } = \sqrt {x^{2}-y}-x \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
15.783 |
|
|
\[ {}y^{\prime } = \sqrt {1-y^{2}} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.566 |
|
|
\[ {}y^{\prime } = \frac {y+1}{x -y} \] |
1 |
1 |
1 |
homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.871 |
|
|
\[ {}y^{\prime } = \sin \left (y\right )-\cos \left (x \right ) \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
0.949 |
|
|
\[ {}y^{\prime } = 1-\cot \left (y\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.836 |
|
|
\[ {}y^{\prime } = \left (3 x -y\right )^{\frac {1}{3}}-1 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
4.842 |
|
|
\[ {}y^{\prime } = \sin \left (x y\right ) \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.863 |
|
|
\[ {}x y^{\prime }+y = \cos \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.331 |
|
|
\[ {}y^{\prime }+2 y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.125 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }+x y = 2 x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.665 |
|
|
\[ {}y^{\prime } = 1+x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.22 |
|
|
\[ {}y^{\prime } = x +y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.951 |
|
|
\[ {}y^{\prime } = y-x \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.972 |
|
|
\[ {}y^{\prime } = \frac {x}{2}-y+\frac {3}{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.108 |
|
|
\[ {}y^{\prime } = \left (y-1\right )^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.333 |
|
|
\[ {}y^{\prime } = \left (y-1\right ) x \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.275 |
|
|
\[ {}y^{\prime } = x^{2}-y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.157 |
|
|
\[ {}y^{\prime } = \cos \left (x -y\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.09 |
|
|
\[ {}y^{\prime } = -x^{2}+y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.992 |
|
|
\[ {}y^{\prime } = x^{2}+2 x -y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.021 |
|
|
\[ {}y^{\prime } = \frac {y+1}{-1+x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.953 |
|
|
\[ {}y^{\prime } = \frac {x +y}{x -y} \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.728 |
|
|
\[ {}y^{\prime } = 1-x \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.221 |
|
|
\[ {}y^{\prime } = 2 x -y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.009 |
|
|
\[ {}y^{\prime } = y+x^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.995 |
|
|
\[ {}y^{\prime } = -\frac {y}{x} \] |
1 |
1 |
1 |
exact, linear, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.709 |
|
|
\[ {}y^{\prime } = 1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.204 |
|
|
\[ {}y^{\prime } = \frac {1}{x} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.219 |
|
|
\[ {}y^{\prime } = y \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.227 |
|
|
\[ {}y^{\prime } = y^{2} \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.231 |
|
|
\[ {}y^{\prime } = x^{2}-y^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.845 |
|
|
\[ {}y^{\prime } = x +y^{2} \] |
1 |
1 |
1 |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
3.363 |
|
|
\[ {}y^{\prime } = x +y \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.161 |
|
|
\[ {}y^{\prime } = 2 y-2 x^{2}-3 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.28 |
|
|
\[ {}x y^{\prime } = 2 x -y \] |
1 |
1 |
1 |
exact, linear, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.59 |
|
|
\[ {}1+y^{2}+\left (x^{2}+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.477 |
|
|
\[ {}1+y^{2}+x y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.681 |
|
|
\[ {}y^{\prime } \sin \left (x \right )-y \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.12 |
|
|
\[ {}1+y^{2} = x y^{\prime } \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.455 |
|
|
\[ {}x \sqrt {1+y^{2}}+y y^{\prime } \sqrt {x^{2}+1} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.202 |
|
|
\[ {}x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.635 |
|
|
\[ {}{\mathrm e}^{-y} y^{\prime } = 1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.302 |
|
|
\[ {}y \ln \left (y\right )+x y^{\prime } = 1 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
11.04 |
|
|
\[ {}y^{\prime } = a^{x +y} \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.565 |
|
|
\[ {}{\mathrm e}^{y} \left (x^{2}+1\right ) y^{\prime }-2 x \left (1+{\mathrm e}^{y}\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.532 |
|
|
\[ {}2 x \sqrt {1-y^{2}} = \left (x^{2}+1\right ) y^{\prime } \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.034 |
|
|
\[ {}{\mathrm e}^{x} \sin \left (y\right )^{3}+\left ({\mathrm e}^{2 x}+1\right ) \cos \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.908 |
|
|
\[ {}y^{2} \sin \left (x \right )+\cos \left (x \right )^{2} \ln \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.586 |
|
|
\[ {}y^{\prime } = \sin \left (x -y\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.056 |
|
|
\[ {}y^{\prime } = x a +b y+c \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.722 |
|
|
\[ {}\left (x +y\right )^{2} y^{\prime } = a^{2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.625 |
|
|
\[ {}x y^{\prime }+y = a \left (1+x y\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.595 |
|
|
\[ {}a^{2}+y^{2}+2 x \sqrt {x a -x^{2}}\, y^{\prime } = 0 \] |
1 |
0 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
5.954 |
|
|
\[ {}y^{\prime } = \frac {y}{x} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.441 |
|
|
\[ {}\cos \left (y^{\prime }\right ) = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.224 |
|
|
\[ {}{\mathrm e}^{y^{\prime }} = 1 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.209 |
|
|
\[ {}\sin \left (y^{\prime }\right ) = x \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.291 |
|
|
\[ {}\ln \left (y^{\prime }\right ) = x \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.603 |
|
|
\[ {}\tan \left (y^{\prime }\right ) = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.209 |
|
|
\[ {}{\mathrm e}^{y^{\prime }} = x \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.238 |
|
|
\[ {}\tan \left (y^{\prime }\right ) = x \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.312 |
|
|
\[ {}x^{2} y^{\prime } \cos \left (y\right )+1 = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✗ |
6.134 |
|
|
\[ {}x^{2} y^{\prime }+\cos \left (2 y\right ) = 1 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✗ |
10.013 |
|
|
\[ {}x^{3} y^{\prime }-\sin \left (y\right ) = 1 \] |
1 |
1 |
0 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
5.432 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }-\frac {\cos \left (2 y\right )^{2}}{2} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
5.362 |
|
|
\[ {}{\mathrm e}^{y} = {\mathrm e}^{4 y} y^{\prime }+1 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.597 |
|
|
\[ {}\left (1+x \right ) y^{\prime } = y-1 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.071 |
|
|
\[ {}y^{\prime } = 2 x \left (\pi +y\right ) \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.981 |
|
|
\[ {}x^{2} y^{\prime }+\sin \left (2 y\right ) = 1 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✗ |
32.723 |
|
|
\[ {}x y^{\prime } = y+x \cos \left (\frac {y}{x}\right )^{2} \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.314 |
|
|
\[ {}x -y+x y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.218 |
|
|
\[ {}x y^{\prime } = y \left (\ln \left (y\right )-\ln \left (x \right )\right ) \] |
1 |
1 |
1 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
6.031 |
|
|
\[ {}x^{2} y^{\prime } = y^{2}-x y+x^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.745 |
|
|
\[ {}x y^{\prime } = y+\sqrt {-x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
5.697 |
|
|
\[ {}2 x^{2} y^{\prime } = x^{2}+y^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.951 |
|
|
\[ {}4 x -3 y+\left (2 y-3 x \right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
5.026 |
|
|
\[ {}y-x +\left (x +y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.454 |
|
|
\[ {}x +y-2+\left (1-x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.473 |
|
|
\[ {}3 y-7 x +7-\left (3 x -7 y-3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.897 |
|
|
\[ {}x +y-2+\left (-y+4+x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.743 |
|
|
\[ {}x +y+\left (x -y-2\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.56 |
|
|
\[ {}2 x +3 y-5+\left (3 x +2 y-5\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.665 |
|
|
\[ {}8 x +4 y+1+\left (4 x +2 y+1\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, differentialType, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.536 |
|
|
\[ {}x -2 y-1+\left (3 x -6 y+2\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.121 |
|
|
\[ {}x +y+\left (x +y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, differentialType, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.97 |
|
|
\[ {}2 x y^{\prime } \left (x -y^{2}\right )+y^{3} = 0 \] |
1 |
1 |
1 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.533 |
|
|
\[ {}4 y^{6}+x^{3} = 6 x y^{5} y^{\prime } \] |
1 |
1 |
6 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
2.838 |
|
|
\[ {}y \left (1+\sqrt {y^{4} x^{2}+1}\right )+2 x y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.709 |
|
|
\[ {}x +y^{3}+3 \left (y^{3}-x \right ) y^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.108 |
|
|
\[ {}y^{\prime }+2 y = {\mathrm e}^{-x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.133 |
|
|
\[ {}x^{2}-x y^{\prime } = y \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.849 |
|
|
\[ {}y^{\prime }-2 x y = 2 x \,{\mathrm e}^{x^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.074 |
|
|
\[ {}y^{\prime }+2 x y = {\mathrm e}^{-x^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.083 |
|
|
\[ {}y^{\prime } \cos \left (x \right )-y \sin \left (x \right ) = 2 x \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.506 |
|
|
\[ {}x y^{\prime }-2 y = x^{3} \cos \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.51 |
|
|
\[ {}y^{\prime }-y \tan \left (x \right ) = \frac {1}{\cos \left (x \right )^{3}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
9.075 |
|
|
\[ {}y^{\prime } x \ln \left (x \right )-y = 3 x^{3} \ln \left (x \right )^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.676 |
|
|
\[ {}\left (2 x -y^{2}\right ) y^{\prime } = 2 y \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.271 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = \cos \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.506 |
|
|
\[ {}y^{\prime } = \frac {y}{2 y \ln \left (y\right )+y-x} \] |
1 |
1 |
1 |
exact, differentialType, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.484 |
|
|
\[ {}\left (\frac {{\mathrm e}^{-y^{2}}}{2}-x y\right ) y^{\prime }-1 = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.204 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x} y = 2 x \,{\mathrm e}^{{\mathrm e}^{x}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.138 |
|
|
\[ {}y^{\prime }+x y \,{\mathrm e}^{x} = {\mathrm e}^{\left (1-x \right ) {\mathrm e}^{x}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.222 |
|
|
\[ {}y^{\prime }-y \ln \left (2\right ) = 2^{\sin \left (x \right )} \left (\cos \left (x \right )-1\right ) \ln \left (2\right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
2.681 |
|
|
\[ {}y^{\prime }-y = -2 \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
3.295 |
|
|
\[ {}y^{\prime } \sin \left (x \right )-y \cos \left (x \right ) = -\frac {\sin \left (x \right )^{2}}{x^{2}} \] |
1 |
0 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✗ |
N/A |
4.23 |
|
|
\[ {}x^{2} y^{\prime } \cos \left (\frac {1}{x}\right )-y \sin \left (\frac {1}{x}\right ) = -1 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
5.698 |
|
|
\[ {}2 x y^{\prime }-y = 1-\frac {2}{\sqrt {x}} \] |
1 |
0 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✗ |
N/A |
2.308 |
|
|
\[ {}2 x y^{\prime }+y = \left (x^{2}+1\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✗ |
4.82 |
|
|
\[ {}x y^{\prime }+y = 2 x \] |
1 |
1 |
1 |
exact, linear, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.009 |
|
|
\[ {}y^{\prime } \sin \left (x \right )+y \cos \left (x \right ) = 1 \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.679 |
|
|
\[ {}y^{\prime } \cos \left (x \right )-y \sin \left (x \right ) = -\sin \left (2 x \right ) \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.099 |
|
|
\[ {}y^{\prime }+2 x y = 2 x y^{2} \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.732 |
|
|
\[ {}3 x y^{2} y^{\prime }-2 y^{3} = x^{3} \] |
1 |
1 |
3 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
2.863 |
|
|
\[ {}\left (x^{3}+{\mathrm e}^{y}\right ) y^{\prime } = 3 x^{2} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.671 |
|
|
\[ {}y^{\prime }+3 x y = y \,{\mathrm e}^{x^{2}} \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.742 |
|
|
\[ {}y^{\prime }-2 \,{\mathrm e}^{x} y = 2 \sqrt {{\mathrm e}^{x} y} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
2.112 |
|
|
\[ {}2 y^{\prime } \ln \left (x \right )+\frac {y}{x} = \frac {\cos \left (x \right )}{y} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
10.419 |
|
|
\[ {}2 y^{\prime } \sin \left (x \right )+y \cos \left (x \right ) = y^{3} \sin \left (x \right )^{2} \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
10.866 |
|
|
\[ {}\left (1+x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
1 |
1 |
4 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.496 |
|
|
\[ {}y^{\prime }-y \cos \left (x \right ) = y^{2} \cos \left (x \right ) \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.376 |
|
|
\[ {}y^{\prime }-\tan \left (y\right ) = \frac {{\mathrm e}^{x}}{\cos \left (y\right )} \] |
1 |
1 |
0 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
3.112 |
|
|
\[ {}y^{\prime } = y \left ({\mathrm e}^{x}+\ln \left (y\right )\right ) \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.427 |
|
|
\[ {}y^{\prime } \cos \left (y\right )+\sin \left (y\right ) = 1+x \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
4.095 |
|
|
\[ {}y y^{\prime }+1 = \left (-1+x \right ) {\mathrm e}^{-\frac {y^{2}}{2}} \] |
1 |
0 |
2 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.293 |
|
|
\[ {}y^{\prime }+x \sin \left (2 y\right ) = 2 x \,{\mathrm e}^{-x^{2}} \cos \left (y\right )^{2} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
3.725 |
|
|
\[ {}x \left (2 x^{2}+y^{2}\right )+y \left (x^{2}+2 y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
3.869 |
|
|
\[ {}3 x^{2}+6 x y^{2}+\left (6 x^{2} y+4 y^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
exact |
[_exact, _rational] |
✓ |
✓ |
1.825 |
|
|
\[ {}\frac {x}{\sqrt {x^{2}+y^{2}}}+\frac {1}{x}+\frac {1}{y}+\left (\frac {y}{\sqrt {x^{2}+y^{2}}}+\frac {1}{y}-\frac {x}{y^{2}}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
30.911 |
|
|
\[ {}3 x^{2} \tan \left (y\right )-\frac {2 y^{3}}{x^{3}}+\left (x^{3} \sec \left (y\right )^{2}+4 y^{3}+\frac {3 y^{2}}{x^{2}}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
51.631 |
|
|
\[ {}2 x +\frac {x^{2}+y^{2}}{x^{2} y} = \frac {\left (x^{2}+y^{2}\right ) y^{\prime }}{x y^{2}} \] |
1 |
1 |
2 |
exact, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _exact, _rational] |
✓ |
✓ |
2.423 |
|
|
\[ {}\frac {\sin \left (2 x \right )}{y}+x +\left (y-\frac {\sin \left (x \right )^{2}}{y^{2}}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_exact] |
✓ |
✓ |
11.772 |
|
|
\[ {}3 x^{2}-2 x -y+\left (2 y-x +3 y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
22.627 |
|
|
\[ {}\frac {x y}{\sqrt {x^{2}+1}}+2 x y-\frac {y}{x}+\left (\sqrt {x^{2}+1}+x^{2}-\ln \left (x \right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
37.151 |
|
|
\[ {}\sin \left (y\right )+y \sin \left (x \right )+\frac {1}{x}+\left (x \cos \left (y\right )-\cos \left (x \right )+\frac {1}{y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
41.57 |
|
|
\[ {}\frac {y+\sin \left (x \right ) \cos \left (x y\right )^{2}}{\cos \left (x y\right )^{2}}+\left (\frac {x}{\cos \left (x y\right )^{2}}+\sin \left (y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
62.519 |
|
|
\[ {}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0 \] |
1 |
1 |
1 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
6.991 |
|
|
\[ {}y \left (x^{2}+y^{2}+a^{2}\right ) y^{\prime }+x \left (x^{2}+y^{2}-a^{2}\right ) = 0 \] |
1 |
1 |
4 |
exact |
[_exact, _rational] |
✓ |
✓ |
2.072 |
|
|
\[ {}3 x^{2} y+y^{3}+\left (x^{3}+3 x y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
4.919 |
|
|
\[ {}1-x^{2} y+x^{2} \left (y-x \right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.543 |
|
|
\[ {}x^{2}+y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.183 |
|
|
\[ {}x +y^{2}-2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.652 |
|
|
\[ {}2 x^{2} y+2 y+5+\left (2 x^{3}+2 x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.444 |
|
|
\[ {}x^{4} \ln \left (x \right )-2 x y^{3}+3 x^{2} y^{2} y^{\prime } = 0 \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.855 |
|
|
\[ {}x +\sin \left (x \right )+\sin \left (y\right )+y^{\prime } \cos \left (y\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
4.541 |
|
|
\[ {}2 x y^{2}-3 y^{3}+\left (7-3 x y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.615 |
|
|
\[ {}3 y^{2}-x +\left (2 y^{3}-6 x y\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
8.741 |
|
|
\[ {}x^{2}+y^{2}+1-2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.676 |
|
|
\[ {}x -x y+\left (y+x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.663 |
|
|
\[ {}4 {y^{\prime }}^{2}-9 x = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.444 |
|
|
\[ {}{y^{\prime }}^{2}-2 y y^{\prime } = y^{2} \left ({\mathrm e}^{2 x}-1\right ) \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
1.556 |
|
|
\[ {}{y^{\prime }}^{2}-2 x y^{\prime }-8 x^{2} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.635 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+3 x y y^{\prime }+2 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
2.367 |
|
|
\[ {}{y^{\prime }}^{2}-\left (y+2 x \right ) y^{\prime }+x^{2}+x y = 0 \] |
2 |
1 |
2 |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
1.46 |
|
|
\[ {}{y^{\prime }}^{3}+\left (2+x \right ) {\mathrm e}^{y} = 0 \] |
3 |
3 |
3 |
first_order_nonlinear_p_but_separable |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
✓ |
3.904 |
|
|
\[ {}{y^{\prime }}^{3} = y {y^{\prime }}^{2}-x^{2} y^{\prime }+x^{2} y \] |
3 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.882 |
|
|
\[ {}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
9.56 |
|
|
\[ {}{y^{\prime }}^{2}-4 x y^{\prime }+2 y+2 x^{2} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.668 |
|
|
\[ {}y = {y^{\prime }}^{2} {\mathrm e}^{y^{\prime }} \] |
0 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.111 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{\frac {y^{\prime }}{y}} \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.197 |
|
|
\[ {}x = \ln \left (y^{\prime }\right )+\sin \left (y^{\prime }\right ) \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✗ |
5.153 |
|
|
\[ {}x = {y^{\prime }}^{2}-2 y^{\prime }+2 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.63 |
|
|
\[ {}y = y^{\prime } \ln \left (y^{\prime }\right ) \] |
0 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.011 |
|
|
\[ {}y = \left (y^{\prime }-1\right ) {\mathrm e}^{y^{\prime }} \] |
0 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.575 |
|
|
\[ {}{y^{\prime }}^{2} x = {\mathrm e}^{\frac {1}{y^{\prime }}} \] |
0 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.648 |
|
|
\[ {}x \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} = a \] |
6 |
6 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
14.369 |
|
|
\[ {}y^{\frac {2}{5}}+{y^{\prime }}^{\frac {2}{5}} = a^{\frac {2}{5}} \] |
2 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.309 |
|
|
\[ {}x = y^{\prime }+\sin \left (y^{\prime }\right ) \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.668 |
|
|
\[ {}y = y^{\prime } \left (1+y^{\prime } \cos \left (y^{\prime }\right )\right ) \] |
0 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.714 |
|
|
\[ {}y = \arcsin \left (y^{\prime }\right )+\ln \left (1+{y^{\prime }}^{2}\right ) \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✗ |
1.055 |
|
|
\[ {}y = 2 x y^{\prime }+\ln \left (y^{\prime }\right ) \] |
0 |
2 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.528 |
|
|
\[ {}y = x \left (1+y^{\prime }\right )+{y^{\prime }}^{2} \] |
2 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.702 |
|
|
\[ {}y = 2 x y^{\prime }+\sin \left (y^{\prime }\right ) \] |
0 |
2 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
1.829 |
|
|
\[ {}y = {y^{\prime }}^{2} x -\frac {1}{y^{\prime }} \] |
3 |
4 |
3 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
151.895 |
|
|
\[ {}y = \frac {3 x y^{\prime }}{2}+{\mathrm e}^{y^{\prime }} \] |
0 |
2 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
1.561 |
|
|
\[ {}y = x y^{\prime }+\frac {a}{{y^{\prime }}^{2}} \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.572 |
|
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{2} \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.279 |
|
|
\[ {}{y^{\prime }}^{2} x -y y^{\prime }-y^{\prime }+1 = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.306 |
|
|
\[ {}y = x y^{\prime }+a \sqrt {1+{y^{\prime }}^{2}} \] |
2 |
3 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.637 |
|
|
\[ {}x = \frac {y}{y^{\prime }}+\frac {1}{{y^{\prime }}^{2}} \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.299 |
|
|
\[ {}y^{\prime } {\mathrm e}^{-x}+y^{2}-2 \,{\mathrm e}^{x} y = 1-{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.674 |
|
|
\[ {}y^{\prime }+y^{2}-2 y \sin \left (x \right )+\sin \left (x \right )^{2}-\cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.881 |
|
|
\[ {}x y^{\prime }-y^{2}+\left (2 x +1\right ) y = x^{2}+2 x \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
0.905 |
|
|
\[ {}x^{2} y^{\prime } = x^{2} y^{2}+x y+1 \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
0.944 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) y^{2}-4 y y^{\prime }-4 x = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
2.936 |
|
|
\[ {}{y^{\prime }}^{2}-4 y = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.335 |
|
|
\[ {}{y^{\prime }}^{3}-4 x y y^{\prime }+8 y^{2} = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
90.18 |
|
|
\[ {}{y^{\prime }}^{2}-y^{2} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.23 |
|
|
\[ {}y^{\prime } = y^{\frac {2}{3}}+a \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.427 |
|
|
\[ {}\left (x y^{\prime }+y\right )^{2}+3 x^{5} \left (x y^{\prime }-2 y\right ) = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
69.164 |
|
|
\[ {}y \left (y-2 x y^{\prime }\right )^{2} = 2 y^{\prime } \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.633 |
|
|
\[ {}8 {y^{\prime }}^{3}-12 {y^{\prime }}^{2} = 27 y-27 x \] |
3 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.437 |
|
|
\[ {}\left (y^{\prime }-1\right )^{2} = y^{2} \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.35 |
|
|
\[ {}y = {y^{\prime }}^{2}-x y^{\prime }+x \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.348 |
|
|
\[ {}\left (x y^{\prime }+y\right )^{2} = y^{2} y^{\prime } \] |
2 |
3 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.498 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+y^{2} = 1 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.394 |
|
|
\[ {}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.18 |
|
|
\[ {}3 {y^{\prime }}^{2} x -6 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.298 |
|
|
\[ {}y = x y^{\prime }+\sqrt {a^{2} {y^{\prime }}^{2}+b^{2}} \] |
2 |
3 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
2.647 |
|
|
\[ {}y^{\prime } = \left (x -y\right )^{2}+1 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.559 |
|
|
\[ {}x \sin \left (x \right ) y^{\prime }+\left (\sin \left (x \right )-x \cos \left (x \right )\right ) y = \sin \left (x \right ) \cos \left (x \right )-x \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
5.52 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = y^{n} \sin \left (2 x \right ) \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
9.935 |
|
|
\[ {}x^{3}-3 x y^{2}+\left (y^{3}-3 x^{2} y\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
1.65 |
|
|
\[ {}5 x y-4 y^{2}-6 x^{2}+\left (y^{2}-8 x y+\frac {5 x^{2}}{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
1.411 |
|
|
\[ {}3 x y^{2}-x^{2}+\left (3 x^{2} y-6 y^{2}-1\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational] |
✓ |
✓ |
1.541 |
|
|
\[ {}y-x y^{2} \ln \left (x \right )+x y^{\prime } = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.984 |
|
|
\[ {}2 x y \,{\mathrm e}^{x^{2}}-x \sin \left (x \right )+{\mathrm e}^{x^{2}} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.996 |
|
|
\[ {}y^{\prime } = \frac {1}{2 x -y^{2}} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
✓ |
0.829 |
|
|
\[ {}x^{2}+x y^{\prime } = 3 x +y^{\prime } \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.163 |
|
|
\[ {}x y y^{\prime }-y^{2} = x^{4} \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.998 |
|
|
\[ {}\frac {1}{y^{2}-x y+x^{2}} = \frac {y^{\prime }}{2 y^{2}-x y} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
16.427 |
|
|
\[ {}\left (2 x -1\right ) y^{\prime }-2 y = \frac {1-4 x}{x^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.589 |
|
|
\[ {}x -y+3+\left (3 x +y+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.276 |
|
|
\[ {}y^{\prime }+\cos \left (\frac {x}{2}+\frac {y}{2}\right ) = \cos \left (\frac {x}{2}-\frac {y}{2}\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.551 |
|
|
\[ {}y^{\prime } \left (3 x^{2}-2 x \right )-y \left (6 x -2\right ) = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.583 |
|
|
\[ {}x y^{2} y^{\prime }-y^{3} = \frac {x^{4}}{3} \] |
1 |
1 |
3 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.174 |
|
|
\[ {}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
✓ |
7.802 |
|
|
\[ {}x^{2}+y^{2}-x y y^{\prime } = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.013 |
|
|
\[ {}x -y+2+\left (x -y+3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.623 |
|
|
\[ {}x y^{2}+y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.939 |
|
|
\[ {}x^{2}+y^{2}+2 x +2 y y^{\prime } = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.345 |
|
|
\[ {}\left (-1+x \right ) \left (y^{2}-y+1\right ) = \left (y-1\right ) \left (x^{2}+x +1\right ) y^{\prime } \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.444 |
|
|
\[ {}\left (x -2 x y-y^{2}\right ) y^{\prime }+y^{2} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.728 |
|
|
\[ {}y \cos \left (x \right )+\left (2 y-\sin \left (x \right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.917 |
|
|
\[ {}y^{\prime }-1 = {\mathrm e}^{2 y+x} \] |
1 |
1 |
1 |
first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.79 |
|
|
\[ {}2 x^{5}+4 x^{3} y-2 x y^{2}+\left (y^{2}+2 x^{2} y-x^{4}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.443 |
|
|
\[ {}x^{2} y^{n} y^{\prime } = 2 x y^{\prime }-y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.296 |
|
|
\[ {}\left (3 x +3 y+a^{2}\right ) y^{\prime } = 4 x +4 y+b^{2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.179 |
|
|
\[ {}x -y^{2}+2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.971 |
|
|
\[ {}x y^{\prime }+y = y^{2} \ln \left (x \right ) \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.35 |
|
|
\[ {}\sin \left (\ln \left (x \right )\right )-\cos \left (\ln \left (y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
59.475 |
|
|
\[ {}y^{\prime } = \sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}} \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.1 |
|
|
\[ {}\left (5 x -7 y+1\right ) y^{\prime }+x +y-1 = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.893 |
|
|
\[ {}x +y+1+\left (2 x +2 y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.923 |
|
|
\[ {}y^{3}+2 \left (x^{2}-x y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.227 |
|
|
\[ {}y^{\prime } = \frac {2 \left (y+2\right )^{2}}{\left (x +y-1\right )^{2}} \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
1.625 |
|
|
\[ {}4 x^{2} {y^{\prime }}^{2}-y^{2} = x y^{3} \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
45.149 |
|
|
\[ {}y^{\prime }+{y^{\prime }}^{2} x -y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.367 |
|
|
\[ {}y^{\prime \prime }+y = 2 \cos \left (x \right )+2 \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.612 |
|
|
\[ {}x y^{\prime \prime \prime } = 2 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _quadrature]] |
✓ |
✓ |
0.227 |
|
|
\[ {}y^{\prime \prime } = {y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.329 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime } = 1 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.053 |
|
|
\[ {}{y^{\prime }}^{4} = 1 \] |
4 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.375 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.675 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 2 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.315 |
|
|
\[ {}y^{\prime \prime } = \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} \] |
2 |
2 |
3 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.97 |
|
|
\[ {}{y^{\prime }}^{2}+y y^{\prime \prime } = 1 \] |
1 |
1 |
2 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
6.869 |
|
|
\[ {}y^{\prime \prime \prime \prime } = x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _quadrature]] |
✓ |
✓ |
0.132 |
|
|
\[ {}y^{\prime \prime \prime } = x +\cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _quadrature]] |
✓ |
✓ |
0.182 |
|
|
\[ {}y^{\prime \prime } \left (2+x \right )^{5} = 1 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.948 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.001 |
|
|
\[ {}y^{\prime \prime } = 2 x \ln \left (x \right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.646 |
|
|
\[ {}x y^{\prime \prime } = y^{\prime } \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.02 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.675 |
|
|
\[ {}x y^{\prime \prime } = \left (2 x^{2}+1\right ) y^{\prime } \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.651 |
|
|
\[ {}x y^{\prime \prime } = y^{\prime }+x^{2} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.184 |
|
|
\[ {}x \ln \left (x \right ) y^{\prime \prime } = y^{\prime } \] |
1 |
1 |
1 |
second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.678 |
|
|
\[ {}x y = y^{\prime } \ln \left (\frac {y^{\prime }}{x}\right ) \] |
0 |
2 |
2 |
separable |
[_separable] |
✓ |
✓ |
1.428 |
|
|
\[ {}2 y^{\prime \prime } = \frac {y^{\prime }}{x}+\frac {x^{2}}{y^{\prime }} \] |
1 |
2 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
0.941 |
|
|
\[ {}y^{\prime \prime \prime } = \sqrt {1-{y^{\prime \prime }}^{2}} \] |
1 |
3 |
0 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
❇ |
N/A |
0.0 |
|
|
\[ {}x y^{\prime \prime \prime }-y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.553 |
|
|
\[ {}y^{\prime \prime } = \sqrt {1+{y^{\prime }}^{2}} \] |
2 |
2 |
3 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.484 |
|
|
\[ {}y^{\prime \prime } = {y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.346 |
|
|
\[ {}y^{\prime \prime } = \sqrt {1-{y^{\prime }}^{2}} \] |
2 |
2 |
3 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.649 |
|
|
\[ {}y^{\prime \prime } = 1+{y^{\prime }}^{2} \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.721 |
|
|
\[ {}y^{\prime \prime } = \sqrt {1+y^{\prime }} \] |
2 |
3 |
2 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.595 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } \ln \left (y^{\prime }\right ) \] |
0 |
1 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.464 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+2 = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.143 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } \left (1+y^{\prime }\right ) \] |
1 |
1 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y, second_order_nonlinear_solved_by_mainardi_lioville_method |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.342 |
|
|
\[ {}3 y^{\prime \prime } = \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} \] |
2 |
2 |
3 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.937 |
|
|
\[ {}y^{\prime \prime \prime }+{y^{\prime \prime }}^{2} = 0 \] |
1 |
3 |
1 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y y^{\prime \prime } = {y^{\prime }}^{2} \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.493 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
0 |
1 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.768 |
|
|
\[ {}3 y^{\prime } y^{\prime \prime } = 2 y \] |
1 |
3 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.805 |
|
|
\[ {}2 y^{\prime \prime } = 3 y^{2} \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
4.26 |
|
|
\[ {}{y^{\prime }}^{2}+y y^{\prime \prime } = 0 \] |
1 |
2 |
3 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.703 |
|
|
\[ {}y y^{\prime \prime } = y^{\prime }+{y^{\prime }}^{2} \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.682 |
|
|
\[ {}y y^{\prime \prime } = 1+{y^{\prime }}^{2} \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
3.768 |
|
|
\[ {}2 y y^{\prime \prime } = 1+{y^{\prime }}^{2} \] |
1 |
2 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.599 |
|
|
\[ {}y^{3} y^{\prime \prime } = -1 \] |
1 |
1 |
0 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✗ |
0.992 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{2} y^{\prime } \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.674 |
|
|
\[ {}y^{\prime \prime } = {\mathrm e}^{2 y} \] |
1 |
1 |
1 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
71.557 |
|
|
\[ {}2 y y^{\prime \prime }-3 {y^{\prime }}^{2} = 4 y^{2} \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.588 |
|
|
\[ {}y^{\prime \prime \prime } = 3 y y^{\prime } \] |
1 |
3 |
1 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.529 |
|
|
\[ {}3 y^{\prime \prime }-2 y^{\prime }-8 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.217 |
|
|
\[ {}y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.217 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.25 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.396 |
|
|
\[ {}y^{\prime \prime \prime }+6 y^{\prime \prime }+11 y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.198 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.228 |
|
|
\[ {}y^{\left (6\right )}+2 y^{\left (5\right )}+y^{\prime \prime \prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.279 |
|
|
\[ {}4 y^{\prime \prime }-8 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.309 |
|
|
\[ {}y^{\prime \prime \prime }-8 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.368 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+10 y^{\prime \prime }+12 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.409 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.461 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.696 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+4 y^{\prime \prime }-2 y^{\prime }-5 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.322 |
|
|
\[ {}y^{\left (5\right )}+4 y^{\prime \prime \prime \prime }+5 y^{\prime \prime \prime }-6 y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.357 |
|
|
\[ {}y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.201 |
|
|
\[ {}y^{\prime \prime \prime }-2 y^{\prime \prime }+2 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.244 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.253 |
|
|
\[ {}y^{\left (5\right )} = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _quadrature]] |
✓ |
✓ |
0.184 |
|
|
\[ {}y^{\prime \prime \prime }-3 y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.242 |
|
|
\[ {}2 y^{\prime \prime \prime }-3 y^{\prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.198 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.296 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = 3 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.917 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime } = \left (-1+x \right )^{2} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.098 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.07 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime } = {\mathrm e}^{-7 x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.112 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = \left (1-x \right ) {\mathrm e}^{4 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.448 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = {\mathrm e}^{5 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.397 |
|
|
\[ {}4 y^{\prime \prime }-3 y^{\prime } = x \,{\mathrm e}^{\frac {3 x}{4}} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.201 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime } = x \,{\mathrm e}^{4 x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.064 |
|
|
\[ {}y^{\prime \prime }+25 y = \cos \left (5 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.498 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )-\cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime \prime }+16 y = \sin \left (4 x +\alpha \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.692 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+8 y = {\mathrm e}^{2 x} \left (\sin \left (2 x \right )+\cos \left (2 x \right )\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.261 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+8 y = {\mathrm e}^{2 x} \left (\sin \left (2 x \right )-\cos \left (2 x \right )\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.008 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+13 y = {\mathrm e}^{-3 x} \cos \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.918 |
|
|
\[ {}y^{\prime \prime }+k^{2} y = k \sin \left (k x +\alpha \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.767 |
|
|
\[ {}y^{\prime \prime }+k^{2} y = k \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.478 |
|
|
\[ {}y^{\prime \prime \prime }+y = x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.342 |
|
|
\[ {}y^{\prime \prime \prime }+6 y^{\prime \prime }+11 y^{\prime }+6 y = 1 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.644 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime } = 2 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.754 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime } = 3 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.157 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y = 1 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
1.75 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y^{\prime } = 2 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
2.664 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y^{\prime \prime } = 3 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.184 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y^{\prime \prime \prime } = 4 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.191 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+4 y^{\prime \prime } = 1 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.201 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+y^{\prime \prime } = {\mathrm e}^{4 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.204 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+y^{\prime \prime } = {\mathrm e}^{-x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.207 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+y^{\prime \prime } = x \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.232 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime }+4 y = \sin \left (2 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.296 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime }+4 y = \cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.25 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime }+4 y = x \sin \left (2 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.345 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 n^{2} y^{\prime \prime }+n^{4} y = a \sin \left (n x +\alpha \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.625 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 n^{2} y^{\prime \prime }+n^{4} y = \cos \left (n x +\alpha \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.297 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+6 y^{\prime \prime }+4 y^{\prime }+y = \sin \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.24 |
|
|
\[ {}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.229 |
|
|
\[ {}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+y = x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.237 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = -2 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.548 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = -2 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.509 |
|
|
\[ {}y^{\prime \prime }+9 y = 9 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.54 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime } = 1 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.153 |
|
|
\[ {}5 y^{\prime \prime \prime }-7 y^{\prime \prime } = 3 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.168 |
|
|
\[ {}y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime } = -6 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.184 |
|
|
\[ {}3 y^{\prime \prime \prime \prime }+y^{\prime \prime \prime } = 2 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.184 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+2 y^{\prime \prime }-2 y^{\prime }+y = 1 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
2.198 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.653 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime } = 8 x \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.697 |
|
|
\[ {}y^{\prime \prime }-2 k y^{\prime }+k^{2} y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.674 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 8 \,{\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.663 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = 9 \,{\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.578 |
|
|
\[ {}7 y^{\prime \prime }-y^{\prime } = 14 x \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.685 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = 3 x \,{\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.924 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 10 \left (1-x \right ) {\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.618 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 1+x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.779 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \left (x^{2}+x \right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.029 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-2 y = 8 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.922 |
|
|
\[ {}y^{\prime \prime }+y = 4 x \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.885 |
|
|
\[ {}y^{\prime \prime }-2 m y^{\prime }+m^{2} y = \sin \left (n x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.987 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = {\mathrm e}^{-x} \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.977 |
|
|
\[ {}y^{\prime \prime }+a^{2} y = 2 \cos \left (m x \right )+3 \sin \left (m x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.224 |
|
|
\[ {}y^{\prime \prime }-y^{\prime } = {\mathrm e}^{x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.687 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 4 \left (\cos \left (x \right )+\sin \left (x \right )\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.516 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 10 \,{\mathrm e}^{-2 x} \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.906 |
|
|
\[ {}4 y^{\prime \prime }+8 y^{\prime } = x \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
5.262 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = x \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = x^{2} {\mathrm e}^{4 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.611 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \left (x^{2}+x \right ) {\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = x^{2}+x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.436 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+2 y^{\prime \prime }-2 y^{\prime }+y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
1.724 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = x^{3} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.653 |
|
|
\[ {}y^{\prime \prime \prime \prime }+y^{\prime \prime } = x^{2}+x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.234 |
|
|
\[ {}y^{\prime \prime }+y = x^{2} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.147 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = x^{2} {\mathrm e}^{-x} \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.021 |
|
|
\[ {}y^{\prime \prime \prime }-y = \sin \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.694 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime }+y = \cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.234 |
|
|
\[ {}y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y = {\mathrm e}^{x} \cos \left (2 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.241 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = {\mathrm e}^{2 x} \left (\sin \left (x \right )+2 \cos \left (x \right )\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.991 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = {\mathrm e}^{x}+{\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.628 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = x +{\mathrm e}^{-4 x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.099 |
|
|
\[ {}y^{\prime \prime }-y = x +\sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.757 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = \left (\sin \left (x \right )+1\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.906 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime } = 1+{\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.187 |
|
|
\[ {}y^{\prime \prime \prime }+4 y^{\prime } = {\mathrm e}^{2 x}+\sin \left (2 x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
1.992 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (x \right ) \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.778 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime } = 2 \cos \left (4 x \right )^{2} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
5.691 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 4 x -2 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = 18 x -10 \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.339 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2+{\mathrm e}^{x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.815 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = \left (5 x +4\right ) {\mathrm e}^{x}+{\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.892 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 4 \,{\mathrm e}^{-x}+17 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.145 |
|
|
|
|||||||||
|
\[ {}2 y^{\prime \prime }-3 y^{\prime }-2 y = 5 \,{\mathrm e}^{x} \cosh \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.052 |
|
|
\[ {}y^{\prime \prime }+4 y = x \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.514 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+2 y^{\prime \prime }+2 y^{\prime }+y = x \,{\mathrm e}^{x}+\frac {\cos \left (x \right )}{2} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.379 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = \cos \left (x \right )^{2}+{\mathrm e}^{x}+x^{2} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
10.406 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime } = {\mathrm e}^{x}+3 \sin \left (2 x \right )+1 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.35 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 10 \sin \left (x \right )+17 \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.356 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x^{2}-{\mathrm e}^{-x}+{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.491 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 2 x +{\mathrm e}^{-x}-2 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.839 |
|
|
\[ {}y^{\prime \prime }+4 y = {\mathrm e}^{x}+4 \sin \left (2 x \right )+2 \cos \left (x \right )^{2}-1 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.427 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 6 x \,{\mathrm e}^{-x} \left (1-{\mathrm e}^{-x}\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.811 |
|
|
\[ {}y^{\prime \prime }+y = \cos \left (2 x \right )^{2}+\sin \left (\frac {x}{2}\right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.928 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 1+8 \cos \left (x \right )+{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.629 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = {\mathrm e}^{x} \sin \left (\frac {x}{2}\right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.671 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = 1+{\mathrm e}^{x}+\cos \left (x \right )+\sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.404 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = {\mathrm e}^{x} \left (1-2 \sin \left (x \right )^{2}\right )+10 x +1 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.007 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 4 x +\sin \left (x \right )+\sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.831 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 1+2 \cos \left (x \right )+\cos \left (2 x \right )-\sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.813 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y+1 = \sin \left (x \right )+x +x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.04 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 18 \,{\mathrm e}^{-3 x}+8 \sin \left (x \right )+6 \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.781 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+1 = 3 \sin \left (2 x \right )+\cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.422 |
|
|
\[ {}y^{\prime \prime \prime }-2 y^{\prime \prime }+y^{\prime } = 2 x +{\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.169 |
|
|
\[ {}y^{\prime \prime }+y = 2 \sin \left (x \right ) \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.769 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }-2 y^{\prime } = 4 x +3 \sin \left (x \right )+\cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.71 |
|
|
\[ {}y^{\prime \prime \prime }-4 y^{\prime } = {\mathrm e}^{2 x} x +\sin \left (x \right )+x^{2} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
1.459 |
|
|
\[ {}y^{\left (5\right )}-y^{\prime \prime \prime \prime } = x \,{\mathrm e}^{x}-1 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.17 |
|
|
\[ {}y^{\left (5\right )}-y^{\prime \prime \prime } = x +2 \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.174 |
|
|
\[ {}y^{\prime \prime }+y = 2-2 x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 9 x^{2}-12 x +2 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.638 |
|
|
\[ {}y^{\prime \prime }+9 y = 36 \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.662 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 2 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.657 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = \left (12 x -7\right ) {\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.584 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = {\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.381 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 10 \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.787 |
|
|
\[ {}y^{\prime \prime }+y = 2 \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.646 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.77 |
|
|
\[ {}y^{\prime \prime }+y = 4 x \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.681 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 2 x^{2} {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.778 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 16 \,{\mathrm e}^{-x}+9 x -6 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.668 |
|
|
\[ {}y^{\prime \prime }-y^{\prime } = -5 \,{\mathrm e}^{-x} \left (\cos \left (x \right )+\sin \left (x \right )\right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.992 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 4 \,{\mathrm e}^{x} \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.786 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime } = -2 x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.539 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y = 8 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
1.296 |
|
|
\[ {}y^{\prime \prime \prime }-y = 2 x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.396 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y = 8 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
1.104 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.538 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 4 \cos \left (2 x \right )+\sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.633 |
|
|
\[ {}y^{\prime \prime }-y = 1 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.75 |
|
|
\[ {}y^{\prime \prime }-y = -2 \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.386 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 4 \,{\mathrm e}^{-x} \] |
1 |
0 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.413 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = 8 \,{\mathrm e}^{x}+9 \] |
1 |
0 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.382 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-5 y = 1 \] |
1 |
0 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
0.471 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 2 \,{\mathrm e}^{x} \left (\sin \left (x \right )+7 \cos \left (x \right )\right ) \] |
1 |
0 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.661 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 2 \,{\mathrm e}^{-2 x} \left (9 \sin \left (2 x \right )+4 \cos \left (2 x \right )\right ) \] |
1 |
0 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.591 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = {\mathrm e}^{-x} \left (9 x^{2}+5 x -12\right ) \] |
1 |
0 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.493 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] |
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]] |
✓ |
✓ |
1.964 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = 0 \] |
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 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.899 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+6 y = 0 \] |
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]] |
✓ |
✓ |
1.697 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.774 |
|
|
\[ {}\left (2+x \right )^{2} y^{\prime \prime }+3 \left (2+x \right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.79 |
|
|
\[ {}\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.802 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-3 x y^{\prime \prime }+3 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.47 |
|
|
\[ {}x^{2} y^{\prime \prime \prime } = 2 y^{\prime } \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.345 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime \prime }-12 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.564 |
|
|
\[ {}\left (2 x +1\right )^{2} y^{\prime \prime \prime }+2 \left (2 x +1\right ) y^{\prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
1.044 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = x \left (6-\ln \left (x \right )\right ) \] |
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 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.521 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = \sin \left (\ln \left (x \right )\right ) \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.589 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-3 y = -\frac {16 \ln \left (x \right )}{x} \] |
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_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.35 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-2 y = x^{2}-2 x +2 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.255 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = x^{m} \] |
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]] |
✓ |
✓ |
2.67 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = 2 \ln \left (x \right )^{2}+12 x \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.657 |
|
|
\[ {}\left (1+x \right )^{3} y^{\prime \prime }+3 \left (1+x \right )^{2} y^{\prime }+\left (1+x \right ) y = 6 \ln \left (1+x \right ) \] |
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]] |
✓ |
✓ |
4.355 |
|
|
\[ {}\left (-2+x \right )^{2} y^{\prime \prime }-3 \left (-2+x \right ) y^{\prime }+4 y = x \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.848 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime \prime }+\left (-2+4 x \right ) y^{\prime }-8 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.714 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }+\left (2 x -3\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[_Jacobi] |
✓ |
✓ |
1.088 |
|
|
\[ {}\left (2 x^{2}+3 x \right ) y^{\prime \prime }-6 \left (1+x \right ) y^{\prime }+6 y = 6 \] |
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]] |
✓ |
✓ |
1.181 |
|
|
\[ {}x^{2} \left (-1+\ln \left (x \right )\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.287 |
|
|
\[ {}y^{\prime \prime }+\left (\tan \left (x \right )-2 \cot \left (x \right )\right ) y^{\prime }+2 \cot \left (x \right )^{2} y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.844 |
|
|
\[ {}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.562 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-y = 1 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.324 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-3 y = 5 x^{4} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.283 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = \left (-1+x \right )^{2} {\mathrm e}^{x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.352 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+{\mathrm e}^{-2 x} y = {\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.536 |
|
|
\[ {}\left (x^{4}-x^{3}\right ) y^{\prime \prime }+\left (2 x^{3}-2 x^{2}-x \right ) y^{\prime }-y = \frac {\left (-1+x \right )^{2}}{x} \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.374 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+{\mathrm e}^{2 x} y = {\mathrm e}^{2 x} x -1 \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.623 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+2 y = x^{2} \left (2 x -3\right ) \] |
1 |
1 |
1 |
reduction_of_order |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.359 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\sin \left (x \right )} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.5 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = \frac {1}{1+{\mathrm e}^{x}} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.326 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\cos \left (x \right )^{3}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\sqrt {\sin \left (x \right )^{5} \cos \left (x \right )}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.762 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{x}}{x^{2}+1} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.472 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = \frac {{\mathrm e}^{-x}}{\sin \left (x \right )} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.582 |
|
|
\[ {}y^{\prime \prime }+y = \frac {2}{\sin \left (x \right )^{3}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.625 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = {\mathrm e}^{2 x} \cos \left ({\mathrm e}^{x}\right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.529 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime } = \frac {-1+x}{x^{3}} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.315 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime } = 4 x^{3} {\mathrm e}^{x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.488 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime } = 1 \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.468 |
|
|
\[ {}x \ln \left (x \right ) y^{\prime \prime }-y^{\prime } = \ln \left (x \right )^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.908 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x -1\right ) y^{\prime } = -4 x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.764 |
|
|
\[ {}y^{\prime \prime }+\tan \left (x \right ) y^{\prime } = \cot \left (x \right ) \cos \left (x \right ) \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.911 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = 1 \] |
1 |
0 |
0 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.32 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = \frac {6+x}{x^{2}} \] |
1 |
0 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
2.353 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime } = \frac {1}{x^{2}+1} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.061 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (-1+x \right )^{2} {\mathrm e}^{x} \] |
1 |
0 |
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]] |
✗ |
N/A |
1.839 |
|
|
\[ {}2 x^{2} \left (2-\ln \left (x \right )\right ) y^{\prime \prime }+x \left (4-\ln \left (x \right )\right ) y^{\prime }-y = \frac {\left (2-\ln \left (x \right )\right )^{2}}{\sqrt {x}} \] |
1 |
0 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.892 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-y = 4 \,{\mathrm e}^{x} \] |
1 |
0 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
8.398 |
|
|
\[ {}x^{3} \left (-1+\ln \left (x \right )\right ) y^{\prime \prime }-x^{2} y^{\prime }+x y = 2 \ln \left (x \right ) \] |
1 |
0 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.788 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-2 \left (1-x \right ) y = 2 x -2 \] |
1 |
0 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.733 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.384 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+6 x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.363 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.253 |
|
|
\[ {}x^{\prime \prime }+{x^{\prime }}^{2}+x = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.035 |
|
|
\[ {}x^{\prime \prime }-2 {x^{\prime }}^{2}+x^{\prime }-2 x = 0 \] |
1 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.378 |
|
|
\[ {}x^{\prime \prime }-x \,{\mathrm e}^{x^{\prime }} = 0 \] |
0 |
1 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
0.585 |
|
|
\[ {}x^{\prime \prime }+{\mathrm e}^{-x^{\prime }}-x = 0 \] |
0 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.542 |
|
|
\[ {}x^{\prime \prime }+x {x^{\prime }}^{2} = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.524 |
|
|
\[ {}x^{\prime \prime }+\left (x+2\right ) x^{\prime } = 0 \] |
1 |
1 |
1 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.703 |
|
|
\[ {}x^{\prime \prime }-x^{\prime }+x-x^{2} = 0 \] |
1 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.385 |
|
|
\[ {}y^{\prime \prime }+\lambda y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.363 |
|
|
\[ {}y^{\prime \prime }+\lambda y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.679 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.101 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
0 |
0 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
1.0 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
1 |
1 |
1 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
8.125 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.275 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.601 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.421 |
|
|
\[ {}y^{\prime \prime }+\alpha y^{\prime } = 0 \] |
1 |
0 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.14 |
|
|
\[ {}y^{\prime \prime }+\alpha ^{2} y = 1 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
111.799 |
|
|
\[ {}y^{\prime \prime }+y = 1 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.641 |
|
|
\[ {}y^{\prime \prime }+\lambda ^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.908 |
|
|
\[ {}y^{\prime \prime }+\lambda ^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.921 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }-y^{\prime }-y = 0 \] |
1 |
1 |
1 |
unknown |
[[_3rd_order, _missing_x]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime \prime \prime }-\lambda ^{4} y = 0 \] |
1 |
1 |
1 |
unknown |
[[_high_order, _missing_x]] |
✗ |
N/A |
0.0 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.819 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }+4 x y^{\prime \prime \prime }+2 y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.757 |
|
|
\[ {}x^{3} y^{\prime \prime \prime \prime }+6 x^{2} y^{\prime \prime \prime }+6 x y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.667 |
|
|
\[ {}y^{\prime } = 1-x y \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
1.467 |
|
|
\[ {}y^{\prime } = \frac {y-x}{x +y} \] |
1 |
1 |
1 |
first order ode series method. Taylor series method |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.197 |
|
|
\[ {}y^{\prime } = y \sin \left (x \right ) \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
1.372 |
|
|
\[ {}y^{\prime \prime }+x y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.148 |
|
|
\[ {}y^{\prime \prime }-y^{\prime } \sin \left (x \right ) = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.501 |
|
|
\[ {}x y^{\prime \prime }+y \sin \left (x \right ) = x \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.568 |
|
|
\[ {}\ln \left (x \right ) y^{\prime \prime }-y \sin \left (x \right ) = 0 \] |
1 |
1 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
40.846 |
|
|
\[ {}y^{\prime \prime \prime }+x \sin \left (y\right ) = 0 \] |
1 |
1 |
1 |
unknown |
[NONE] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime }-2 x y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
1.181 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.511 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+y = 1 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.207 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2}+1\right ) y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.238 |
|
|
\[ {}y^{\prime \prime } = x^{2} y-y^{\prime } \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.415 |
|
|
\[ {}y^{\prime \prime }-{\mathrm e}^{x} y = 0 \] |
1 |
2 |
1 |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.629 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{y}+x y \] |
1 |
1 |
1 |
first order ode series method. Taylor series method |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.368 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.831 |
|
|
\[ {}\left (1+x \right ) y^{\prime }-n y = 0 \] |
1 |
2 |
1 |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.543 |
|
|
\[ {}9 x \left (1-x \right ) y^{\prime \prime }-12 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
second order series method. Regular singular point. Difference not integer |
[_Jacobi] |
✓ |
✓ |
1.122 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (4 x^{2}-\frac {1}{9}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.582 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.802 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+\frac {y}{9} = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.517 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+4 y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.52 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+4 \left (x^{4}-1\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.675 |
|
|
\[ {}x y^{\prime \prime }+\frac {y^{\prime }}{2}+\frac {y}{4} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.894 |
|
|
\[ {}y^{\prime \prime }+\frac {5 y^{\prime }}{x}+y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[_Lienard] |
✓ |
✓ |
0.542 |
|
|
\[ {}y^{\prime \prime }+\frac {3 y^{\prime }}{x}+4 y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.556 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.67 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \pi ^{2}-x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.481 |
|
|
\[ {}y^{\prime \prime }-4 y = \cos \left (\pi x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.514 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \arcsin \left (\sin \left (x \right )\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.8 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (x \right )^{3} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.135 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime }=-2 t x_{1}^{2} \\ x_{2}^{\prime }=\frac {x_{2}+t}{t} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.119 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime }={\mathrm e}^{t -x_{1}} \\ x_{2}^{\prime }=2 \,{\mathrm e}^{x_{1}} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.246 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=\frac {y^{2}}{x} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.122 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime }=\frac {x_{1}^{2}}{x_{2}} \\ x_{2}^{\prime }=x_{2}-x_{1} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.129 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\frac {{\mathrm e}^{-x}}{t} \\ y^{\prime }=\frac {x \,{\mathrm e}^{-y}}{t} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.267 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\frac {t +y}{x+y} \\ y^{\prime }=\frac {x-t}{x+y} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.262 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\frac {t -y}{-x+y} \\ y^{\prime }=\frac {x-t}{-x+y} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.257 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\frac {t +y}{x+y} \\ y^{\prime }=\frac {t +x}{x+y} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.237 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-9 y \\ y^{\prime }=x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.349 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=t +y \\ y^{\prime }=x-t \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.595 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+3 x+4 y=0 \\ y^{\prime }+2 x+5 y=0 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.286 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+5 y \\ y^{\prime }=-x-3 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.332 |
|
|
\[ {}\left [\begin {array}{c} 4 x^{\prime }-y^{\prime }+3 x=\sin \left (t \right ) \\ x^{\prime }+y=\cos \left (t \right ) \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.878 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-y+z \\ y^{\prime }=z \\ z^{\prime }=-x+z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.589 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y+z \\ y^{\prime }=z+x \\ z^{\prime }=x+y \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.402 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }=y \\ y^{\prime \prime }=x \end {array}\right ] \] |
1 |
1 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
|
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }+y^{\prime }+x=0 \\ x^{\prime }+y^{\prime \prime }=0 \end {array}\right ] \] |
1 |
1 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
|
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }=3 x+y \\ y^{\prime }=-2 x \end {array}\right ] \] |
1 |
1 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
|
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }=x^{2}+y \\ y^{\prime }=-2 x x^{\prime }+x \end {array}\right ] \] |
1 |
1 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
|
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x^{2}+y^{2} \\ y^{\prime }=2 x y \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.244 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-\frac {1}{y} \\ y^{\prime }=\frac {1}{x} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.251 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\frac {x}{y} \\ y^{\prime }=\frac {y}{x} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.242 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\frac {y}{x-y} \\ y^{\prime }=\frac {x}{x-y} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.267 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\sin \left (x\right ) \cos \left (y\right ) \\ y^{\prime }=\cos \left (x\right ) \sin \left (y\right ) \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.266 |
|
|
\[ {}\left [\begin {array}{c} {\mathrm e}^{t} x^{\prime }=\frac {1}{y} \\ {\mathrm e}^{t} y^{\prime }=\frac {1}{x} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.269 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\cos \left (x\right )^{2} \cos \left (y\right )^{2}+\sin \left (x\right )^{2} \cos \left (y\right )^{2} \\ y^{\prime }=-\frac {\sin \left (2 x\right ) \sin \left (2 y\right )}{2} \end {array}\right ] \] |
1 |
0 |
0 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.259 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=8 y-x \\ y^{\prime }=x+y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.293 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x-y \\ y^{\prime }=-x+y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.247 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x+y \\ y^{\prime }=x-3 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.373 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+y \\ y^{\prime }=-2 x+4 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.274 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=4 x-5 y \\ y^{\prime }=x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.301 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y+z-x \\ y^{\prime }=x-y+z \\ z^{\prime }=x+y-z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.412 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x-y+z \\ y^{\prime }=x+2 y-z \\ z^{\prime }=x-y+2 z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.385 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x-y+z \\ y^{\prime }=z+x \\ z^{\prime }=y-2 z-3 x \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.353 |
|
|
|
|||||||||
|
\[ {}\left [\begin {array}{c} x^{\prime }+2 x-y=-{\mathrm e}^{2 t} \\ y^{\prime }+3 x-2 y=6 \,{\mathrm e}^{2 t} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.626 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+y-\cos \left (t \right ) \\ y^{\prime }=-y-2 x+\cos \left (t \right )+\sin \left (t \right ) \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.777 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y+\tan \left (t \right )^{2}-1 \\ y^{\prime }=\tan \left (t \right )-x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
2.716 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-4 x-2 y+\frac {2}{{\mathrm e}^{t}-1} \\ y^{\prime }=6 x+3 y-\frac {3}{{\mathrm e}^{t}-1} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.033 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=-x+\frac {1}{\cos \left (t \right )} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.616 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=-x+1 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.65 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=3-2 y \\ y^{\prime }=2 x-2 t \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.899 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-y+\sin \left (t \right ) \\ y^{\prime }=x+\cos \left (t \right ) \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.75 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+y+{\mathrm e}^{t} \\ y^{\prime }=x+y-{\mathrm e}^{t} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.556 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=4 x-5 y+4 t -1 \\ y^{\prime }=x-2 y+t \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.471 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y-x+{\mathrm e}^{t} \\ y^{\prime }=x-y+{\mathrm e}^{t} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.476 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+y=t^{2} \\ y^{\prime }-x=t \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.76 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+y^{\prime }+y={\mathrm e}^{-t} \\ 2 x^{\prime }+y^{\prime }+2 y=\sin \left (t \right ) \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.443 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x+y-2 z+2-t \\ y^{\prime }=-x+1 \\ z^{\prime }=x+y-z+1-t \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.952 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+x+2 y=2 \,{\mathrm e}^{-t} \\ y^{\prime }+y+z=1 \\ z^{\prime }+z=1 \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.564 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=5 x+4 y \\ y^{\prime }=x+2 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.283 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=6 x+y \\ y^{\prime }=4 x+3 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.282 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x-4 y+1 \\ y^{\prime }=-x+5 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.615 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=3 x+y+{\mathrm e}^{t} \\ y^{\prime }=x+3 y-{\mathrm e}^{t} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.546 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x+4 y+\cos \left (t \right ) \\ y^{\prime }=-x-2 y+\sin \left (t \right ) \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.385 |
|
|
\[ {}x^{\prime }+3 x = {\mathrm e}^{-2 t} \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.407 |
|
|
\[ {}x^{\prime }-3 x = 3 t^{3}+3 t^{2}+2 t +1 \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.382 |
|
|
\[ {}x^{\prime }-x = \cos \left (t \right )-\sin \left (t \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.471 |
|
|
\[ {}2 x^{\prime }+6 x = t \,{\mathrm e}^{-3 t} \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.428 |
|
|
\[ {}x^{\prime }+x = 2 \sin \left (t \right ) \] |
1 |
1 |
1 |
first_order_laplace |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.45 |
|
|
\[ {}x^{\prime \prime } = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.21 |
|
|
\[ {}x^{\prime \prime } = 1 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.259 |
|
|
\[ {}x^{\prime \prime } = \cos \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.361 |
|
|
\[ {}x^{\prime \prime }+x^{\prime } = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.23 |
|
|
\[ {}x^{\prime \prime }+x^{\prime } = 0 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.222 |
|
|
\[ {}x^{\prime \prime }-x^{\prime } = 1 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.283 |
|
|
\[ {}x^{\prime \prime }+x = t \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.284 |
|
|
\[ {}x^{\prime \prime }+6 x^{\prime } = 12 t +2 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.303 |
|
|
\[ {}x^{\prime \prime }-2 x^{\prime }+2 x = 2 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.335 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+4 x = 4 \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.359 |
|
|
\[ {}2 x^{\prime \prime }-2 x^{\prime } = \left (t +1\right ) {\mathrm e}^{t} \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.398 |
|
|
\[ {}x^{\prime \prime }+x = 2 \cos \left (t \right ) \] |
1 |
1 |
1 |
second_order_laplace |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.482 |
|
|
|
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