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) |
|
\[ {}\frac {2 x y+1}{y}+\frac {\left (y-x \right ) y^{\prime }}{y^{2}} = 0 \] |
1 |
1 |
1 |
exact, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.813 |
|
|
\[ {}\frac {y^{2}-2 x^{2}}{x y^{2}-x^{3}}+\frac {\left (2 y^{2}-x^{2}\right ) y^{\prime }}{y^{3}-x^{2} y} = 0 \] |
1 |
1 |
2 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
6.898 |
|
|
\[ {}\frac {1}{\sqrt {x^{2}+y^{2}}}+\left (\frac {1}{y}-\frac {x}{y \sqrt {x^{2}+y^{2}}}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
2.803 |
|
|
\[ {}y+x +x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.278 |
|
|
\[ {}6 x -2 y+1+\left (2 y-2 x -3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.003 |
|
|
\[ {}\sec \left (x \right ) \cos \left (y\right )^{2}-\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
7.841 |
|
|
\[ {}\left (1+x \right ) y^{2}-x^{3} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.849 |
|
|
\[ {}2 \left (1-y^{2}\right ) x y+\left (x^{2}+1\right ) \left (1+y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.162 |
|
|
\[ {}\sin \left (x \right ) \cos \left (y\right )^{2}+\cos \left (x \right )^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.36 |
|
|
\[ {}x \,{\mathrm e}^{\frac {y}{x}}+y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.009 |
|
|
\[ {}2 x^{2} y+3 y^{3}-\left (x^{3}+2 x y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.087 |
|
|
\[ {}y^{2}-x y+x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.946 |
|
|
\[ {}2 x^{2} y+y^{3}-x^{3} y^{\prime } = 0 \] |
1 |
2 |
2 |
bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.338 |
|
|
\[ {}y^{3}+x^{3} y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.187 |
|
|
\[ {}x +y \cos \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.654 |
|
|
\[ {}4 x +3 y+1+\left (1+x +y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.957 |
|
|
\[ {}4 x -y+2+\left (x +y+3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.927 |
|
|
\[ {}2 x +y-\left (4 x +2 y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.305 |
|
|
\[ {}y+2 x y^{2}-x^{2} y^{3}+2 x^{2} y y^{\prime } = 0 \] |
1 |
1 |
2 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
1.801 |
|
|
\[ {}2 y+3 x y^{2}+\left (x +2 x^{2} y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.91 |
|
|
\[ {}y+x y^{2}+\left (x -x^{2} y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.565 |
|
|
\[ {}y^{\prime }+y \cot \left (x \right ) = \sec \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.151 |
|
|
\[ {}x y^{\prime }+\left (1+x \right ) y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.905 |
|
|
\[ {}y^{\prime }-\frac {2 y}{1+x} = \left (1+x \right )^{3} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.865 |
|
|
\[ {}\left (x^{3}+x \right ) y^{\prime }+4 x^{2} y = 2 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.918 |
|
|
\[ {}x^{2} y^{\prime }+\left (1-2 x \right ) y = x^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.898 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }-2 \left (1+x \right ) y = y^{\frac {5}{2}} \] |
1 |
3 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
16.983 |
|
|
\[ {}y y^{\prime }+x y^{2} = x \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.46 |
|
|
\[ {}\sin \left (y\right ) y^{\prime }+\sin \left (x \right ) \cos \left (y\right ) = \sin \left (x \right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
32.255 |
|
|
\[ {}4 x y^{\prime }+3 y+{\mathrm e}^{x} x^{4} y^{5} = 0 \] |
1 |
4 |
4 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.53 |
|
|
\[ {}y^{\prime }-\frac {y+1}{1+x} = \sqrt {y+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.266 |
|
|
\[ {}x^{4} y \left (3 y+2 x y^{\prime }\right )+x^{2} \left (4 y+3 x y^{\prime }\right ) = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
9.448 |
|
|
\[ {}y^{2} \left (3 y-6 x y^{\prime }\right )-x \left (y-2 x y^{\prime }\right ) = 0 \] |
1 |
1 |
3 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.021 |
|
|
\[ {}2 x^{3} y-y^{2}-\left (2 x^{4}+x y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.711 |
|
|
\[ {}y^{2}-x y+x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.939 |
|
|
\[ {}\frac {-y+x y^{\prime }}{\sqrt {x^{2}-y^{2}}} = x y^{\prime } \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
2.869 |
|
|
\[ {}x +y-\left (x -y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.5 |
|
|
\[ {}x^{2}+y^{2}-2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.666 |
|
|
\[ {}x -y^{2}+2 x y y^{\prime } = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.887 |
|
|
\[ {}-y+x y^{\prime } = x^{2}+y^{2} \] |
1 |
1 |
1 |
riccati, exactByInspection, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.434 |
|
|
\[ {}3 x^{2}+6 x y+3 y^{2}+\left (2 x^{2}+3 x y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.418 |
|
|
\[ {}2 x +\left (x^{2}+y^{2}+2 y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.183 |
|
|
\[ {}y^{4}+2 y+\left (x y^{3}+2 y^{4}-4 x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.352 |
|
|
\[ {}x^{3} y-y^{4}+\left (x y^{3}-x^{4}\right ) y^{\prime } = 0 \] |
1 |
1 |
4 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.923 |
|
|
\[ {}y^{2}-x^{2}+2 m x y+\left (m y^{2}-m \,x^{2}-2 x y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
11.212 |
|
|
\[ {}x y^{\prime }-y+2 x^{2} y-x^{3} = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.48 |
|
|
\[ {}\left (x +y\right ) y^{\prime }-1 = 0 \] |
1 |
1 |
1 |
homogeneousTypeC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.92 |
|
|
\[ {}x +y y^{\prime }+y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.106 |
|
|
\[ {}x y^{\prime }-a y+b y^{2} = c \,x^{2 a} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.706 |
|
|
\[ {}x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.969 |
|
|
\[ {}\sqrt {1-y^{2}}+\sqrt {-x^{2}+1}\, y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.859 |
|
|
\[ {}y^{\prime }-x^{2} y = x^{5} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.726 |
|
|
\[ {}\left (y-x \right )^{2} y^{\prime } = 1 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.933 |
|
|
\[ {}x y^{\prime }+y+x^{4} y^{4} {\mathrm e}^{x} = 0 \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.786 |
|
|
\[ {}\left (1-x \right ) y+\left (1-y\right ) x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.22 |
|
|
\[ {}\left (y-x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.315 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.91 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {x^{2}-y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.668 |
|
|
\[ {}x \sin \left (\frac {y}{x}\right )-y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.359 |
|
|
\[ {}x -2 y+5+\left (4+2 x -y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.618 |
|
|
\[ {}y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{\frac {3}{2}}} = \frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
5.35 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime }-x y = a x y^{2} \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.538 |
|
|
\[ {}x y^{2} \left (3 y+x y^{\prime }\right )-2 y+x y^{\prime } = 0 \] |
1 |
1 |
2 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.125 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+y = \arctan \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.999 |
|
|
\[ {}5 x y-3 y^{3}+\left (3 x^{2}-7 x y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.938 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.682 |
|
|
\[ {}x y^{2}+y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.929 |
|
|
\[ {}\left (1-x \right ) y-\left (y+1\right ) x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.126 |
|
|
\[ {}3 x^{2} y+\left (x^{3}+x^{3} y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.99 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right ) = \left (x^{2}+y^{2}+x \right ) \left (-y+x y^{\prime }\right ) \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
1.308 |
|
|
\[ {}2 x +3 y-1+\left (2 x +3 y-5\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.217 |
|
|
\[ {}y^{3}-2 x^{2} y+\left (2 x y^{2}-x^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
4.365 |
|
|
\[ {}2 x^{3} y^{2}-y+\left (2 x^{2} y^{3}-x \right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.448 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right )+\sqrt {1+x^{2}+y^{2}}\, \left (y-x y^{\prime }\right ) = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
2.616 |
|
|
\[ {}1+{\mathrm e}^{\frac {y}{x}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.179 |
|
|
\[ {}x y^{\prime }+y-y^{2} \ln \left (x \right ) = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.163 |
|
|
\[ {}x^{3} y^{4}+x^{2} y^{3}+x y^{2}+y+\left (x^{4} y^{3}-x^{3} y^{2}-x^{3} y+x \right ) y^{\prime } = 0 \] |
1 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
1.693 |
|
|
\[ {}\left (2 \sqrt {x y}-x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.38 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+x y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.577 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.416 |
|
|
\[ {}y^{2}+{y^{\prime }}^{2} = 1 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.601 |
|
|
\[ {}\left (2 x y^{\prime }-y\right )^{2} = 8 x^{3} \] |
2 |
2 |
2 |
linear |
[_linear] |
✓ |
✓ |
0.988 |
|
|
\[ {}\left (x^{2}+1\right ) {y^{\prime }}^{2} = 1 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.446 |
|
|
\[ {}{y^{\prime }}^{3}-\left (y^{2}+2 x \right ) {y^{\prime }}^{2}+\left (x^{2}-y^{2}+2 x y^{2}\right ) y^{\prime }-\left (x^{2}-y^{2}\right ) y^{2} = 0 \] |
3 |
1 |
3 |
linear, quadrature |
[_quadrature] |
✓ |
✓ |
0.855 |
|
|
\[ {}2 x y^{\prime }-y+\ln \left (y^{\prime }\right ) = 0 \] |
0 |
2 |
2 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.593 |
|
|
\[ {}4 x {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.349 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.381 |
|
|
\[ {}y^{\prime }+2 x y = x^{2}+y^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
2.116 |
|
|
\[ {}y = -x y^{\prime }+x^{4} {y^{\prime }}^{2} \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
5.934 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.498 |
|
|
\[ {}x +y^{\prime } y \left (2 {y^{\prime }}^{2}+3\right ) = 0 \] |
3 |
7 |
5 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
140.131 |
|
|
\[ {}a^{2} y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.736 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.397 |
|
|
\[ {}{y^{\prime }}^{3}-4 x y y^{\prime }+8 y^{2} = 0 \] |
3 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
136.206 |
|
|
\[ {}\left (-y+x y^{\prime }\right )^{2} = 1+{y^{\prime }}^{2} \] |
2 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.709 |
|
|
\[ {}4 \,{\mathrm e}^{2 y} {y^{\prime }}^{2}+2 x y^{\prime }-1 = 0 \] |
2 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
4.438 |
|
|
\[ {}4 \,{\mathrm e}^{2 y} {y^{\prime }}^{2}+2 \,{\mathrm e}^{2 x} y^{\prime }-{\mathrm e}^{2 x} = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.338 |
|
|
\[ {}{\mathrm e}^{2 y} {y^{\prime }}^{3}+\left ({\mathrm e}^{2 x}+{\mathrm e}^{3 x}\right ) y^{\prime }-{\mathrm e}^{3 x} = 0 \] |
3 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
204.26 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }+x = 0 \] |
2 |
2 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
10.509 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) \left (1+y^{\prime }\right )^{2}-2 \left (x +y\right ) \left (1+y^{\prime }\right ) \left (x +y y^{\prime }\right )+\left (x +y y^{\prime }\right )^{2} = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.897 |
|
|
\[ {}y = 2 x y^{\prime }+y^{2} {y^{\prime }}^{3} \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
133.768 |
|
|
\[ {}a^{2} y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.736 |
|
|
\[ {}\left (x -y^{\prime }-y\right )^{2} = x^{2} \left (2 x y-x^{2} y^{\prime }\right ) \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
5.846 |
|
|
\[ {}y^{2} \left (1+{y^{\prime }}^{2}\right ) = a^{2} \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.152 |
|
|
\[ {}y y^{\prime } = \left (-b +x \right ) {y^{\prime }}^{2}+a \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.422 |
|
|
\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+1 = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
6.652 |
|
|
\[ {}3 x {y^{\prime }}^{2}-6 y y^{\prime }+x +2 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.42 |
|
|
\[ {}y = {y^{\prime }}^{2} \left (1+x \right ) \] |
2 |
3 |
3 |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
1.473 |
|
|
\[ {}\left (-y+x y^{\prime }\right ) \left (x +y y^{\prime }\right ) = a^{2} y^{\prime } \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
72.443 |
|
|
\[ {}{y^{\prime }}^{2}+2 y^{\prime } y \cot \left (x \right ) = y^{2} \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
6.134 |
|
|
\[ {}\left (x^{2}+1\right ) {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}-1 = 0 \] |
2 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.564 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 \left (x y+2 y^{\prime }\right ) y^{\prime }+y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
0.957 |
|
|
\[ {}y = x y^{\prime }+\frac {y {y^{\prime }}^{2}}{x^{2}} \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
8.293 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2} = x^{2} y^{2}+x^{4} \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.427 |
|
|
\[ {}y = x y^{\prime }+\frac {1}{y^{\prime }} \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.363 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.373 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 \left (x y-2\right ) y^{\prime }+y^{2} = 0 \] |
2 |
4 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _Clairaut] |
✓ |
✓ |
0.776 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-\left (-1+x \right )^{2} = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.37 |
|
|
\[ {}8 \left (1+y^{\prime }\right )^{3} = 27 \left (x +y\right ) \left (1-y^{\prime }\right )^{3} \] |
3 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
473.104 |
|
|
\[ {}4 {y^{\prime }}^{2} = 9 x \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.282 |
|
|
\[ {}y \left (3-4 y\right )^{2} {y^{\prime }}^{2} = 4-4 y \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.886 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.229 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.593 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.228 |
|
|
\[ {}y^{\prime \prime \prime }-2 y^{\prime \prime }-y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.223 |
|
|
\[ {}4 y^{\prime \prime \prime }-3 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.281 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.253 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-2 y^{\prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.119 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.272 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.122 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.531 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime }-2 y^{\prime } = {\mathrm e}^{-x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{{\mathrm e}^{x}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.498 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y = 2 \,{\mathrm e}^{-x}-x^{2} {\mathrm e}^{-x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.234 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{x}}{\left (1-x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.519 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.42 |
|
|
\[ {}y^{\prime \prime \prime }-3 y^{\prime \prime }-y^{\prime }+3 y = x^{2} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.583 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.739 |
|
|
\[ {}y^{\prime \prime \prime }-4 y^{\prime \prime }+5 y^{\prime }-2 y = x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.612 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.461 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.758 |
|
|
\[ {}y^{\prime \prime }+4 y = x^{2}+\cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.291 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 \,{\mathrm e}^{2 x} x -\sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.616 |
|
|
\[ {}y^{\prime \prime }+y = 2 \,{\mathrm e}^{x}+x^{3}-x \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 3 \,{\mathrm e}^{2 x}-\cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.894 |
|
|
\[ {}y^{\prime \prime \prime }-y = x^{2} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.954 |
|
|
\[ {}y^{\prime \prime \prime }-2 y^{\prime \prime }-3 y^{\prime } = 3 x^{2}+\sin \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.908 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime }+y = {\mathrm e}^{x}+4 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.211 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = {\mathrm e}^{2 x}+1 \] |
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]] |
✓ |
✓ |
1.929 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y = \cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.004 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+x y^{\prime }-y = x \ln \left (x \right ) \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.353 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+2 y = 10 x +\frac {10}{x} \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.912 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \frac {1}{\left (1-x \right )^{2}} \] |
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, _nonhomogeneous]] |
✓ |
✓ |
3.465 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-\left (1+x \right ) y^{\prime }+6 y = x \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.039 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = \cos \left (x \right )-{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.721 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y = {\mathrm e}^{x} \cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.065 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 2 x^{3}-x \,{\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.601 |
|
|
\[ {}y^{\prime \prime \prime }-4 y^{\prime } = x^{2}-3 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.587 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime }+y = \cos \left (x \right ) \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.226 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \left (1+\ln \left (x \right )\right )^{2} \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.239 |
|
|
\[ {}y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime } = x^{2}-x \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.198 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.099 |
|
|
\[ {}y^{\prime \prime }+4 y = \sec \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.996 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y = {\mathrm e}^{3 x} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.212 |
|
|
\[ {}y^{\prime \prime }+y = x \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.797 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }-x y^{\prime }+y = \frac {1}{x} \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.335 |
|
|
\[ {}y^{\prime \prime \prime }-y = x \,{\mathrm e}^{x}+\cos \left (x \right )^{2} \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
8.678 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+x y = x \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.58 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y = x^{2}-x -1 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.702 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.725 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1-x \right )^{2} \] |
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]] |
✓ |
✓ |
1.987 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }+2 \cos \left (x \right ) y^{\prime }+3 \sin \left (x \right ) y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.663 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-\left (a^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.949 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.31 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-x y = 2 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.154 |
|
|
\[ {}y^{\prime \prime }+\left (2 \,{\mathrm e}^{x}-1\right ) y^{\prime }+{\mathrm e}^{2 x} y = {\mathrm e}^{4 x} \] |
1 |
1 |
1 |
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 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.931 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.29 |
|
|
\[ {}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.284 |
|
|
\[ {}x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y = \frac {1}{x^{2}} \] |
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 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.721 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime }-8 x^{3} y = 4 x^{3} {\mathrm e}^{-x^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.59 |
|
|
\[ {}x y^{\prime \prime }-\left (x +3\right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[_Laguerre] |
✓ |
✓ |
2.465 |
|
|
\[ {}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.355 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.87 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.629 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.741 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.934 |
|
|
\[ {}\left (2 x^{3}-1\right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 x y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.394 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = x^{3} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.515 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 n x \left (1+x \right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.427 |
|
|
\[ {}x^{4} y^{\prime \prime }+2 x^{3} \left (1+x \right ) y^{\prime }+n^{2} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.026 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.408 |
|
|
\[ {}\left (x y^{\prime \prime \prime }-y^{\prime \prime }\right )^{2} = {y^{\prime \prime \prime }}^{2}+1 \] |
1 |
1 |
3 |
unknown |
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime } = x \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.628 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] |
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]] |
✓ |
✓ |
0.692 |
|
|
\[ {}\left (y^{\prime }-x y^{\prime \prime }\right )^{2} = 1+{y^{\prime \prime }}^{2} \] |
2 |
4 |
3 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.21 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2}-y^{2} y^{\prime } = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.53 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2}+1 = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
7.733 |
|
|
\[ {}2 y^{\prime \prime } = {\mathrm e}^{y} \] |
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]] |
✓ |
✓ |
3.118 |
|
|
\[ {}y y^{\prime \prime }+2 y^{\prime }-{y^{\prime }}^{2} = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.545 |
|
|
\[ {}\left (x^{2}-2 x +2\right ) y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.09 |
|
|
\[ {}x y^{\prime \prime \prime }-y^{\prime \prime }-x y^{\prime }+y = -x^{2}+1 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.133 |
|
|
\[ {}\left (2+x \right )^{2} y^{\prime \prime \prime }+\left (2+x \right ) y^{\prime \prime }+y^{\prime } = 1 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
1.587 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = x \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.149 |
|
|
\[ {}\left (-1+x \right )^{2} y^{\prime \prime }+4 \left (-1+x \right ) y^{\prime }+2 y = \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, 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_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.064 |
|
|
\[ {}\left (x^{3}-x \right ) y^{\prime \prime \prime }+\left (8 x^{2}-3\right ) y^{\prime \prime }+14 x y^{\prime }+4 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _fully, _exact, _linear]] |
✗ |
N/A |
0.408 |
|
|
\[ {}2 x^{3} y y^{\prime \prime \prime }+6 x^{3} y^{\prime } y^{\prime \prime }+18 x^{2} y y^{\prime \prime }+18 x^{2} {y^{\prime }}^{2}+36 x y y^{\prime }+6 y^{2} = 0 \] |
1 |
1 |
3 |
unknown |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}x^{5} y^{\prime \prime }+\left (2 x^{4}-x \right ) y^{\prime }-\left (2 x^{3}-1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.261 |
|
|
\[ {}x^{2} \left (-x^{3}+1\right ) y^{\prime \prime }-x^{3} y^{\prime }-2 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
73.061 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-5 x y^{\prime \prime }+\left (4 x^{4}+5\right ) y^{\prime }-8 x^{3} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.345 |
|
|
\[ {}y^{\prime \prime }+2 \cot \left (x \right ) y^{\prime }+2 \tan \left (x \right ) {y^{\prime }}^{2} = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.154 |
|
|
\[ {}x^{2} y y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{2} = 0 \] |
1 |
0 |
3 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.097 |
|
|
\[ {}x^{3} y^{\prime \prime }-\left (-y+x y^{\prime }\right )^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.108 |
|
|
\[ {}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{2} \ln \left (y\right )-x^{2} y^{2} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.097 |
|
|
\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-2 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.161 |
|
|
\[ {}y^{\prime \prime } = 1+{y^{\prime }}^{2} \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.786 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime } = 2 \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.341 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
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], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.943 |
|
|
\[ {}\left (x^{3}+1\right ) y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+18 x y^{\prime }+6 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _fully, _exact, _linear]] |
✗ |
N/A |
0.476 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }+\left (4 x +2\right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.688 |
|
|
\[ {}y \left (1-\ln \left (y\right )\right ) y^{\prime \prime }+\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2} = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.129 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x} = 0 \] |
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]] |
✓ |
✓ |
1.316 |
|
|
\[ {}x \left (2 y+x \right ) y^{\prime \prime }+2 x {y^{\prime }}^{2}+4 \left (x +y\right ) y^{\prime }+2 y+x^{2} = 0 \] |
1 |
1 |
2 |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.501 |
|
|
\[ {}y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.327 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-\frac {y^{\prime }}{x}+x^{2} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.49 |
|
|
\[ {}4 x^{2} y^{\prime \prime \prime }+8 x y^{\prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.784 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }-\cos \left (x \right ) y^{\prime }+2 \sin \left (x \right ) y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.695 |
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\[ {}\left [\begin {array}{c} 3 x^{\prime }+3 x+2 y={\mathrm e}^{t} \\ 4 x-3 y^{\prime }+3 y=3 t \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.904 |
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