Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime }-\frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.644 |
|
|
\[ {}y^{\prime }+a y-c \,{\mathrm e}^{b x} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.124 |
|
|
\[ {}y^{\prime }+a y-b \sin \left (c x \right ) = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.301 |
|
|
\[ {}y^{\prime }+2 x y-x \,{\mathrm e}^{-x^{2}} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.869 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right )-{\mathrm e}^{2 x} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.787 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right )-\frac {\sin \left (2 x \right )}{2} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.807 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right )-{\mathrm e}^{-\sin \left (x \right )} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.965 |
|
|
\[ {}y^{\prime }+y \tan \left (x \right )-\sin \left (2 x \right ) = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.166 |
|
|
\[ {}y^{\prime }-\left (\sin \left (\ln \left (x \right )\right )+\cos \left (\ln \left (x \right )\right )+a \right ) y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.509 |
|
|
\[ {}y^{\prime }+f^{\prime }\left (x \right ) y-f \left (x \right ) f^{\prime }\left (x \right ) = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.095 |
|
|
\[ {}y^{\prime }+f \left (x \right ) y-g \left (x \right ) = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.904 |
|
|
\[ {}y^{\prime }+y^{2}-1 = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.26 |
|
|
\[ {}y^{\prime }+y^{2}-x a -b = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.892 |
|
|
\[ {}y^{\prime }+y^{2}+a \,x^{m} = 0 \] |
1 |
1 |
1 |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
2.517 |
|
|
\[ {}y^{\prime }+y^{2}-2 x^{2} y+x^{4}-2 x -1 = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
1.682 |
|
|
\[ {}y^{\prime }+y^{2}+\left (x y-1\right ) f \left (x \right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.187 |
|
|
\[ {}y^{\prime }-y^{2}-3 y+4 = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.45 |
|
|
\[ {}y^{\prime }-y^{2}-x y-x +1 = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.724 |
|
|
\[ {}y^{\prime }-\left (x +y\right )^{2} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.865 |
|
|
\[ {}y^{\prime }-y^{2}+\left (x^{2}+1\right ) y-2 x = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.007 |
|
|
\[ {}y^{\prime }-y^{2}+y \sin \left (x \right )-\cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.408 |
|
|
\[ {}y^{\prime }-y^{2}-y \sin \left (2 x \right )-\cos \left (2 x \right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.329 |
|
|
\[ {}y^{\prime }+a y^{2}-b = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.371 |
|
|
\[ {}y^{\prime }+a y^{2}-b \,x^{\nu } = 0 \] |
1 |
1 |
1 |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
2.557 |
|
|
\[ {}y^{\prime }+a y^{2}-b \,x^{2 \nu }-c \,x^{\nu -1} = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
37.242 |
|
|
\[ {}y^{\prime }-\left (y A -a \right ) \left (B y-b \right ) = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.909 |
|
|
\[ {}y^{\prime }+a y \left (y-x \right )-1 = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.786 |
|
|
\[ {}y^{\prime }+x y^{2}-x^{3} y-2 x = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.886 |
|
|
\[ {}y^{\prime }-x y^{2}-3 x y = 0 \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.435 |
|
|
\[ {}y^{\prime }+x^{-a -1} y^{2}-x^{a} = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.22 |
|
|
\[ {}y^{\prime }-a \,x^{n} \left (1+y^{2}\right ) = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.48 |
|
|
\[ {}y^{\prime }+y^{2} \sin \left (x \right )-\frac {2 \sin \left (x \right )}{\cos \left (x \right )^{2}} = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.928 |
|
|
\[ {}y^{\prime }-\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}+\frac {g^{\prime }\left (x \right )}{f \left (x \right )} = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.881 |
|
|
\[ {}y^{\prime }+y^{2} f \left (x \right )+g \left (x \right ) y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.957 |
|
|
\[ {}y^{\prime }+f \left (x \right ) \left (y^{2}+2 a y+b \right ) = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.998 |
|
|
\[ {}y^{\prime }+y^{3}+a x y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
1.772 |
|
|
\[ {}y^{\prime }-y^{3}-a \,{\mathrm e}^{x} y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
2.759 |
|
|
\[ {}y^{\prime }-a y^{3}-\frac {b}{x^{\frac {3}{2}}} = 0 \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Abel] |
✓ |
✓ |
1.441 |
|
|
\[ {}y^{\prime }-\operatorname {a3} y^{3}-\operatorname {a2} y^{2}-\operatorname {a1} y-\operatorname {a0} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.288 |
|
|
\[ {}y^{\prime }+3 a y^{3}+6 a x y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
1.727 |
|
|
\[ {}y^{\prime }+a x y^{3}+b y^{2} = 0 \] |
1 |
1 |
1 |
abelFirstKind, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
✓ |
9.667 |
|
|
\[ {}y^{\prime }-x \left (2+x \right ) y^{3}-\left (x +3\right ) y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
3.515 |
|
|
\[ {}y^{\prime }+\left (4 x \,a^{2}+3 x^{2} a +b \right ) y^{3}+3 x y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
9.302 |
|
|
\[ {}y^{\prime }+2 a \,x^{3} y^{3}+2 x y = 0 \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.204 |
|
|
\[ {}y^{\prime }+2 \left (a^{2} x^{3}-b^{2} x \right ) y^{3}+3 b y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
7.814 |
|
|
\[ {}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1} = 0 \] |
1 |
1 |
2 |
abelFirstKind |
[_Abel] |
✓ |
✗ |
62.872 |
|
|
\[ {}y^{\prime }-a \left (x^{n}-x \right ) y^{3}-y^{2} = 0 \] |
1 |
0 |
0 |
abelFirstKind |
[_Abel] |
❇ |
N/A |
14.926 |
|
|
\[ {}y^{\prime }-\left (a \,x^{n}+b x \right ) y^{3}-c y^{2} = 0 \] |
1 |
0 |
0 |
abelFirstKind |
[_Abel] |
❇ |
N/A |
10.164 |
|
|
\[ {}y^{\prime }+a \phi ^{\prime }\left (x \right ) y^{3}+6 a \phi \left (x \right ) y^{2}+\frac {\left (2 a +1\right ) y \phi ^{\prime \prime }\left (x \right )}{\phi ^{\prime }\left (x \right )}+2+2 a = 0 \] |
1 |
0 |
0 |
unknown |
[_Abel] |
❇ |
N/A |
0.239 |
|
|
\[ {}y^{\prime }-f_{3} \left (x \right ) y^{3}-f_{2} \left (x \right ) y^{2}-f_{1} \left (x \right ) y-f_{0} \left (x \right ) = 0 \] |
1 |
0 |
0 |
abelFirstKind |
[_Abel] |
❇ |
N/A |
6.798 |
|
|
\[ {}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
7.599 |
|
|
\[ {}y^{\prime }-a y^{n}-b \,x^{\frac {n}{-n +1}} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
✓ |
1.619 |
|
|
\[ {}y^{\prime }-f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.73 |
|
|
\[ {}y^{\prime }-a^{n} f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.378 |
|
|
\[ {}y^{\prime }-f \left (x \right ) y^{n}-g \left (x \right ) y-h \left (x \right ) = 0 \] |
1 |
0 |
0 |
unknown |
[_Chini] |
❇ |
N/A |
1.478 |
|
|
\[ {}y^{\prime }-f \left (x \right ) y^{a}-g \left (x \right ) y^{b} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.91 |
|
|
\[ {}y^{\prime }-\sqrt {{| y|}} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.085 |
|
|
\[ {}y^{\prime }-a \sqrt {y}-b x = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
✓ |
4.431 |
|
|
\[ {}y^{\prime }-a \sqrt {1+y^{2}}-b = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
72.331 |
|
|
\[ {}y^{\prime }-\frac {\sqrt {y^{2}-1}}{\sqrt {x^{2}-1}} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
22.204 |
|
|
\[ {}y^{\prime }-\frac {\sqrt {x^{2}-1}}{\sqrt {y^{2}-1}} = 0 \] |
1 |
1 |
1 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.974 |
|
|
\[ {}y^{\prime }-\frac {y-x^{2} \sqrt {x^{2}-y^{2}}}{x y \sqrt {x^{2}-y^{2}}+x} = 0 \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
3.517 |
|
|
\[ {}y^{\prime }-\frac {1+y^{2}}{{| y+\sqrt {y+1}|} \left (1+x \right )^{\frac {3}{2}}} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
130.102 |
|
|
\[ {}y^{\prime }-\sqrt {\frac {a y^{2}+b y+c}{x^{2} a +b x +c}} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
293.068 |
|
|
\[ {}y^{\prime }-\sqrt {\frac {y^{3}+1}{x^{3}+1}} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
43.512 |
|
|
\[ {}y^{\prime }-\frac {\sqrt {{| y \left (y-1\right ) \left (-1+a y\right )|}}}{\sqrt {{| x \left (-1+x \right ) \left (x a -1\right )|}}} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.278 |
|
|
\[ {}y^{\prime }-\frac {\sqrt {1-y^{4}}}{\sqrt {-x^{4}+1}} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.066 |
|
|
\[ {}y^{\prime }-\sqrt {\frac {a y^{4}+b y^{2}+1}{a \,x^{4}+b \,x^{2}+1}} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
5.464 |
|
|
\[ {}y^{\prime }-\sqrt {\left (b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0} \right ) \left (a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0} \right )} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✗ |
57.172 |
|
|
\[ {}y^{\prime }-\sqrt {\frac {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✗ |
47.033 |
|
|
\[ {}y^{\prime }-\sqrt {\frac {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}{a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✗ |
43.478 |
|
|
\[ {}y^{\prime }-\operatorname {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right ) \operatorname {R2} \left (y, \sqrt {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.336 |
|
|
\[ {}y^{\prime }-\left (\frac {a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0}}\right )^{\frac {2}{3}} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
6.01 |
|
|
\[ {}y^{\prime }-f \left (x \right ) \left (y-g \left (x \right )\right ) \sqrt {\left (y-a \right ) \left (y-b \right )} = 0 \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
1.742 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x -y}+{\mathrm e}^{x} = 0 \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.88 |
|
|
\[ {}y^{\prime }-a \cos \left (y\right )+b = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.566 |
|
|
\[ {}y^{\prime }-\cos \left (b x +a y\right ) = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
78.877 |
|
|
\[ {}y^{\prime }+a \sin \left (\alpha y+\beta x \right )+b = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
101.241 |
|
|
\[ {}y^{\prime }+f \left (x \right ) \cos \left (a y\right )+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) = 0 \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
3.526 |
|
|
\[ {}y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )-f^{\prime }\left (x \right )-1 = 0 \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.924 |
|
|
\[ {}y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )-1 = 0 \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.215 |
|
|
\[ {}y^{\prime }-a \left (1+\tan \left (y\right )^{2}\right )+\tan \left (y\right ) \tan \left (x \right ) = 0 \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
4.063 |
|
|
\[ {}y^{\prime }-\tan \left (x y\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.944 |
|
|
\[ {}y^{\prime }-f \left (x a +b y\right ) = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.006 |
|
|
\[ {}y^{\prime }-x^{a -1} y^{-b +1} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.996 |
|
|
\[ {}y^{\prime }-\frac {y-x f \left (x^{2}+a y^{2}\right )}{x +a y f \left (x^{2}+a y^{2}\right )} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
2.263 |
|
|
\[ {}y^{\prime }-\frac {y a f \left (x^{c} y\right )+c \,x^{a} y^{b}}{x b f \left (x^{c} y\right )-x^{a} y^{b}} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
3.453 |
|
|
\[ {}2 y^{\prime }-3 y^{2}-4 a y-b -c \,{\mathrm e}^{-2 x a} = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.635 |
|
|
\[ {}x y^{\prime }-\sqrt {a^{2}-x^{2}} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.408 |
|
|
\[ {}x y^{\prime }+y-x \sin \left (x \right ) = 0 \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.964 |
|
|
\[ {}x y^{\prime }-y-\frac {x}{\ln \left (x \right )} = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.069 |
|
|
\[ {}x y^{\prime }-y-x^{2} \sin \left (x \right ) = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.191 |
|
|
\[ {}x y^{\prime }-y-\frac {x \cos \left (\ln \left (\ln \left (x \right )\right )\right )}{\ln \left (x \right )} = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.878 |
|
|
\[ {}x y^{\prime }+a y+b \,x^{n} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.228 |
|
|
\[ {}x y^{\prime }+y^{2}+x^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.49 |
|
|
\[ {}x y^{\prime }-y^{2}+1 = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.039 |
|
|
\[ {}x y^{\prime }+a y^{2}-y+b \,x^{2} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
1.899 |
|
|
\[ {}x y^{\prime }+a y^{2}-b y+c \,x^{2 b} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.335 |
|
|
\[ {}x y^{\prime }+a y^{2}-b y-c \,x^{\beta } = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.734 |
|
|
\[ {}x y^{\prime }+x y^{2}+a = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
1.539 |
|
|
\[ {}x y^{\prime }+x y^{2}-y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.065 |
|
|
\[ {}x y^{\prime }+x y^{2}-y-a \,x^{3} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
3.173 |
|
|
\[ {}x y^{\prime }+x y^{2}-\left (2 x^{2}+1\right ) y-x^{3} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
3.859 |
|
|
\[ {}x y^{\prime }+a x y^{2}+2 y+b x = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.866 |
|
|
\[ {}x y^{\prime }+a x y^{2}+b y+c x +d = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
6.303 |
|
|
\[ {}x y^{\prime }+x^{a} y^{2}+\frac {\left (-b +a \right ) y}{2}+x^{b} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.518 |
|
|
\[ {}x y^{\prime }+a \,x^{\alpha } y^{2}+b y-c \,x^{\beta } = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
4.059 |
|
|
\[ {}x y^{\prime }-y^{2} \ln \left (x \right )+y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.306 |
|
|
\[ {}x y^{\prime }-y \left (2 y \ln \left (x \right )-1\right ) = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.316 |
|
|
\[ {}x y^{\prime }+f \left (x \right ) \left (-x^{2}+y^{2}\right ) = 0 \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
1.879 |
|
|
\[ {}x y^{\prime }+y^{3}+3 x y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
1.629 |
|
|
\[ {}x y^{\prime }-\sqrt {x^{2}+y^{2}}-y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.572 |
|
|
\[ {}x y^{\prime }+a \sqrt {x^{2}+y^{2}}-y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
4.931 |
|
|
\[ {}x y^{\prime }-x \sqrt {x^{2}+y^{2}}-y = 0 \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.835 |
|
|
\[ {}x y^{\prime }-x \left (y-x \right ) \sqrt {x^{2}+y^{2}}-y = 0 \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
2.317 |
|
|
\[ {}x y^{\prime }-x \,{\mathrm e}^{\frac {y}{x}}-y-x = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.657 |
|
|
\[ {}x y^{\prime }-y \ln \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.248 |
|
|
\[ {}x y^{\prime }-y \left (\ln \left (x y\right )-1\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.928 |
|
|
\[ {}x y^{\prime }-y \left (x \ln \left (\frac {x^{2}}{y}\right )+2\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.674 |
|
|
\[ {}x y^{\prime }-\sin \left (x -y\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
2.619 |
|
|
\[ {}x y^{\prime }+\left (\sin \left (y\right )-3 x^{2} \cos \left (y\right )\right ) \cos \left (y\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
3.234 |
|
|
\[ {}x y^{\prime }-x \sin \left (\frac {y}{x}\right )-y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.385 |
|
|
\[ {}x y^{\prime }+x \cos \left (\frac {y}{x}\right )-y+x = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.705 |
|
|
\[ {}x y^{\prime }+x \tan \left (\frac {y}{x}\right )-y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.679 |
|
|
\[ {}x y^{\prime }-y f \left (x y\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.169 |
|
|
\[ {}x y^{\prime }-y f \left (x^{a} y^{b}\right ) = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.609 |
|
|
\[ {}x y^{\prime }+a y-f \left (x \right ) g \left (x^{a} y\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.882 |
|
|
\[ {}\left (1+x \right ) y^{\prime }+y \left (y-x \right ) = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.158 |
|
|
\[ {}2 x y^{\prime }-y-2 x^{3} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.897 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime }-4 \,{\mathrm e}^{-y}+2 = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.796 |
|
|
\[ {}3 x y^{\prime }-3 x \ln \left (x \right ) y^{4}-y = 0 \] |
1 |
1 |
3 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
2.128 |
|
|
\[ {}x^{2} y^{\prime }+y-x = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.943 |
|
|
\[ {}x^{2} y^{\prime }-y+x^{2} {\mathrm e}^{x -\frac {1}{x}} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.091 |
|
|
\[ {}x^{2} y^{\prime }-\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.096 |
|
|
\[ {}x^{2} y^{\prime }+y^{2}+x y+x^{2} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.293 |
|
|
\[ {}x^{2} y^{\prime }-y^{2}-x y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.117 |
|
|
\[ {}x^{2} y^{\prime }-y^{2}-x y-x^{2} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.257 |
|
|
\[ {}x^{2} \left (y^{\prime }+y^{2}\right )+a \,x^{k}-b \left (b -1\right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.506 |
|
|
\[ {}x^{2} \left (y^{\prime }+y^{2}\right )+4 x y+2 = 0 \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
1.753 |
|
|
\[ {}x^{2} \left (y^{\prime }+y^{2}\right )+a x y+b = 0 \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
2.507 |
|
|
\[ {}x^{2} \left (y^{\prime }-y^{2}\right )-a \,x^{2} y+x a +2 = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.374 |
|
|
\[ {}x^{2} \left (y^{\prime }+a y^{2}\right )-b = 0 \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
2.336 |
|
|
\[ {}x^{2} \left (y^{\prime }+a y^{2}\right )+b \,x^{\alpha }+c = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.549 |
|
|
\[ {}x^{2} y^{\prime }+a y^{3}-a \,x^{2} y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
2.115 |
|
|
\[ {}x^{2} y^{\prime }+x y^{3}+a y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
2.149 |
|
|
\[ {}x^{2} y^{\prime }+a \,x^{2} y^{3}+b y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
3.003 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+x y-1 = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.001 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+x y-x \left (x^{2}+1\right ) = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.016 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+2 x y-2 x^{2} = 0 \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.036 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+\left (1+y^{2}\right ) \left (2 x y-1\right ) = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
66.421 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right )-x \left (x^{2}+1\right ) \cos \left (y\right )^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
4.674 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }-x y+a = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.601 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+2 x y-\cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.161 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+y^{2}-2 x y+1 = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.912 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }-y \left (y-x \right ) = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.102 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+a \left (y^{2}-2 x y+1\right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.945 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+a x y^{2}+x y = 0 \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.3 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }-2 x y \ln \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.702 |
|
|
\[ {}\left (x^{2}-4\right ) y^{\prime }+\left (2+x \right ) y^{2}-4 y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.078 |
|
|
\[ {}\left (x^{2}-5 x +6\right ) y^{\prime }+3 x y-8 y+x^{2} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.147 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right ) = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
5.833 |
|
|
\[ {}2 x^{2} y^{\prime }-2 y^{2}-x y+2 x \,a^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.762 |
|
|
\[ {}2 x^{2} y^{\prime }-2 y^{2}-3 x y+2 x \,a^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.631 |
|
|
\[ {}x \left (2 x -1\right ) y^{\prime }+y^{2}-\left (1+4 x \right ) y+4 x = 0 \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
3.405 |
|
|
\[ {}2 x \left (-1+x \right ) y^{\prime }+\left (-1+x \right ) y^{2}-x = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.408 |
|
|
\[ {}3 x^{2} y^{\prime }-7 y^{2}-3 x y-x^{2} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.614 |
|
|
\[ {}3 \left (x^{2}-4\right ) y^{\prime }+y^{2}-x y-3 = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.335 |
|
|
\[ {}\left (x a +b \right )^{2} y^{\prime }+\left (x a +b \right ) y^{3}+c y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
6.46 |
|
|
\[ {}x^{3} y^{\prime }-y^{2}-x^{4} = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
1.056 |
|
|
\[ {}x^{3} y^{\prime }-y^{2}-x^{2} y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.181 |
|
|
\[ {}x^{3} y^{\prime }-x^{4} y^{2}+x^{2} y+20 = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
1.99 |
|
|
\[ {}x^{3} y^{\prime }-x^{6} y^{2}-\left (2 x -3\right ) x^{2} y+3 = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.912 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime }+x^{2} y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.967 |
|
|
\[ {}x \left (x^{2}-1\right ) y^{\prime }-\left (2 x^{2}-1\right ) y+a \,x^{3} = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.795 |
|
|
\[ {}x \left (x^{2}-1\right ) y^{\prime }+\left (x^{2}-1\right ) y^{2}-x^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.658 |
|
|
\[ {}x^{2} \left (-1+x \right ) y^{\prime }-y^{2}-x \left (-2+x \right ) y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.302 |
|
|
\[ {}2 x \left (x^{2}-1\right ) y^{\prime }+2 \left (x^{2}-1\right ) y^{2}-\left (3 x^{2}-5\right ) y+x^{2}-3 = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.309 |
|
|
\[ {}3 x \left (x^{2}-1\right ) y^{\prime }+x y^{2}-\left (x^{2}+1\right ) y-3 x = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.339 |
|
|
\[ {}\left (x^{2} a +b x +c \right ) \left (-y+x y^{\prime }\right )-y^{2}+x^{2} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
2.892 |
|
|
\[ {}x^{4} \left (y^{\prime }+y^{2}\right )+a = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
2.418 |
|
|
\[ {}x \left (x^{3}-1\right ) y^{\prime }-2 x y^{2}+y+x^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.973 |
|
|
\[ {}\left (2 x^{4}-x \right ) y^{\prime }-2 \left (x^{3}-1\right ) y = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.318 |
|
|
\[ {}\left (x^{2} a +b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )+A = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, _Riccati] |
✓ |
✓ |
5.824 |
|
|
\[ {}x^{7} y^{\prime }+2 \left (x^{2}+1\right ) y^{3}+5 x^{3} y^{2} = 0 \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
61.957 |
|
|
\[ {}x^{n} y^{\prime }+y^{2}-\left (n -1\right ) x^{n -1} y+x^{2 n -2} = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
2.357 |
|
|
\[ {}x^{n} y^{\prime }-a y^{2}-b \,x^{2 n -2} = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
5.237 |
|
|
\[ {}x^{2 n +1} y^{\prime }-a y^{3}-b \,x^{3 n} = 0 \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
✓ |
1.615 |
|
|
\[ {}x^{m \left (n -1\right )+n} y^{\prime }-a y^{n}-b \,x^{n \left (1+m \right )} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.809 |
|
|
\[ {}\sqrt {x^{2}-1}\, y^{\prime }-\sqrt {y^{2}-1} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
15.137 |
|
|
\[ {}\sqrt {-x^{2}+1}\, y^{\prime }-y \sqrt {y^{2}-1} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.783 |
|
|
\[ {}\sqrt {a^{2}+x^{2}}\, y^{\prime }+y-\sqrt {a^{2}+x^{2}}+x = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.701 |
|
|
\[ {}x y^{\prime } \ln \left (x \right )+y-a x \left (1+\ln \left (x \right )\right ) = 0 \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.624 |
|
|
\[ {}x y^{\prime } \ln \left (x \right )-y^{2} \ln \left (x \right )-\left (2 \ln \left (x \right )^{2}+1\right ) y-\ln \left (x \right )^{3} = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.189 |
|
|
\[ {}\sin \left (x \right ) y^{\prime }-y^{2} \sin \left (x \right )^{2}+\left (\cos \left (x \right )-3 \sin \left (x \right )\right ) y+4 = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
15.024 |
|
|
\[ {}\cos \left (x \right ) y^{\prime }+y+\left (\sin \left (x \right )+1\right ) \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.306 |
|
|
\[ {}\cos \left (x \right ) y^{\prime }-y^{4}-y \sin \left (x \right ) = 0 \] |
1 |
3 |
3 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
47.638 |
|
|
\[ {}\sin \left (x \right ) \cos \left (x \right ) y^{\prime }-y-\sin \left (x \right )^{3} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
4.431 |
|
|
\[ {}\sin \left (2 x \right ) y^{\prime }+\sin \left (2 y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
27.957 |
|
|
\[ {}\left (a \sin \left (x \right )^{2}+b \right ) y^{\prime }+a y \sin \left (2 x \right )+A x \left (a \sin \left (x \right )^{2}+c \right ) = 0 \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.359 |
|
|
\[ {}2 f \left (x \right ) y^{\prime }+2 y^{2} f \left (x \right )-f^{\prime }\left (x \right ) y-2 f \left (x \right )^{2} = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
0.951 |
|
|
\[ {}f \left (x \right ) y^{\prime }+g \left (x \right ) s \left (y\right )+h \left (x \right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.99 |
|
|
\[ {}y y^{\prime }+y+x^{3} = 0 \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.529 |
|
|
\[ {}y y^{\prime }+a y+x = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
22.648 |
|
|
\[ {}y y^{\prime }+a y+\frac {\left (a^{2}-1\right ) x}{4}+b \,x^{n} = 0 \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.325 |
|
|
\[ {}y y^{\prime }+a y+b \,{\mathrm e}^{x}-2 a = 0 \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.085 |
|
|
\[ {}y y^{\prime }+y^{2}+4 \left (1+x \right ) x = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
2.217 |
|
|
\[ {}y y^{\prime }+a y^{2}-b \cos \left (x +c \right ) = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
3.21 |
|
|
\[ {}y y^{\prime }-\sqrt {a y^{2}+b} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.638 |
|
|
\[ {}y y^{\prime }+x y^{2}-4 x = 0 \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.973 |
|
|
\[ {}y y^{\prime }-x \,{\mathrm e}^{\frac {x}{y}} = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.521 |
|
|
\[ {}y y^{\prime }+f \left (x^{2}+y^{2}\right ) g \left (x \right )+x = 0 \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
2.21 |
|
|
\[ {}\left (y+1\right ) y^{\prime }-y-x = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
12.486 |
|
|
\[ {}\left (x +y-1\right ) y^{\prime }-y+2 x +3 = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.851 |
|
|
\[ {}\left (y+2 x -2\right ) y^{\prime }-y+x +1 = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.328 |
|
|
\[ {}\left (y-2 x +1\right ) y^{\prime }+y+x = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.267 |
|
|
\[ {}\left (-x^{2}+y\right ) y^{\prime }-x = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
1.368 |
|
|
\[ {}\left (-x^{2}+y\right ) y^{\prime }+4 x y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.253 |
|
|
\[ {}\left (y+g \left (x \right )\right ) y^{\prime }-f_{2} \left (x \right ) y^{2}-f_{1} \left (x \right ) y-f_{0} \left (x \right ) = 0 \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
9.226 |
|
|
\[ {}2 y y^{\prime }-x y^{2}-x^{3} = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.375 |
|
|
\[ {}\left (2 y+x +1\right ) y^{\prime }-2 y-x +1 = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.27 |
|
|
\[ {}\left (2 y+x +7\right ) y^{\prime }-y+2 x +4 = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.214 |
|
|
\[ {}\left (2 y-x \right ) y^{\prime }-y-2 x = 0 \] |
1 |
1 |
2 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.146 |
|
|
\[ {}\left (2 y-6 x \right ) y^{\prime }-y+3 x +2 = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.451 |
|
|
\[ {}\left (4 y+2 x +3\right ) y^{\prime }-2 y-x -1 = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.293 |
|
|
\[ {}\left (4 y-2 x -3\right ) y^{\prime }+2 y-x -1 = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.315 |
|
|
\[ {}\left (4 y-3 x -5\right ) y^{\prime }-3 y+7 x +2 = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.281 |
|
|
\[ {}\left (4 y+11 x -11\right ) y^{\prime }-25 y-8 x +62 = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.253 |
|
|
\[ {}\left (12 y-5 x -8\right ) y^{\prime }-5 y+2 x +3 = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.935 |
|
|
\[ {}a y y^{\prime }+b y^{2}+f \left (x \right ) = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.867 |
|
|
\[ {}\left (a y+b x +c \right ) y^{\prime }+\alpha y+\beta x +\gamma = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
13.463 |
|
|
\[ {}x y y^{\prime }+y^{2}+x^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.885 |
|
|
\[ {}x y y^{\prime }-y^{2}+a \,x^{3} \cos \left (x \right ) = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _Bernoulli] |
✓ |
✓ |
1.794 |
|
|
\[ {}x y y^{\prime }-y^{2}+x y+x^{3}-2 x^{2} = 0 \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.738 |
|
|
\[ {}\left (x y+a \right ) y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, _with_exponential_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.537 |
|
|
\[ {}x \left (y+4\right ) y^{\prime }-y^{2}-2 y-2 x = 0 \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
7.61 |
|
|
\[ {}x \left (y+a \right ) y^{\prime }+b y+c x = 0 \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.77 |
|
|
\[ {}\left (x \left (x +y\right )+a \right ) y^{\prime }-y \left (x +y\right )-b = 0 \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.793 |
|
|
\[ {}\left (x y-x^{2}\right ) y^{\prime }+y^{2}-3 x y-2 x^{2} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.605 |
|
|
\[ {}2 x y y^{\prime }-y^{2}+x a = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.388 |
|
|
\[ {}2 x y y^{\prime }-y^{2}+x^{2} a = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.811 |
|
|
\[ {}2 x y y^{\prime }+2 y^{2}+1 = 0 \] |
1 |
1 |
2 |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.362 |
|
|
\[ {}x \left (2 y+x -1\right ) y^{\prime }-y \left (2 x +y+1\right ) = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.121 |
|
|
\[ {}x \left (2 y-x -1\right ) y^{\prime }+y \left (2 x -y-1\right ) = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.125 |
|
|
\[ {}\left (2 x y+4 x^{3}\right ) y^{\prime }+y^{2}+112 x^{2} y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.468 |
|
|
\[ {}x \left (3 y+2 x \right ) y^{\prime }+3 \left (x +y\right )^{2} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.754 |
|
|
\[ {}\left (3 x +2\right ) \left (y-2 x -1\right ) y^{\prime }-y^{2}+x y-7 x^{2}-9 x -3 = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
4.573 |
|
|
\[ {}\left (6 x y+x^{2}+3\right ) y^{\prime }+3 y^{2}+2 x y+2 x = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.834 |
|
|
\[ {}\left (a x y+b \,x^{n}\right ) y^{\prime }+\alpha y^{3}+\beta y^{2} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
10.773 |
|
|
\[ {}\left (B x y+A \,x^{2}+x a +b y+c \right ) y^{\prime }-B g \left (x \right )^{2}+A x y+\alpha x +\beta y+\gamma = 0 \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
6.216 |
|
|
\[ {}\left (x^{2} y-1\right ) y^{\prime }+x y^{2}-1 = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.626 |
|
|
\[ {}\left (x^{2} y-1\right ) y^{\prime }-x y^{2}+1 = 0 \] |
1 |
0 |
3 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.338 |
|
|
\[ {}\left (x^{2} y-1\right ) y^{\prime }+8 x y^{2}-8 = 0 \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.757 |
|
|
\[ {}x \left (x y-2\right ) y^{\prime }+x^{2} y^{3}+x y^{2}-2 y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
1.533 |
|
|
\[ {}x \left (x y-3\right ) y^{\prime }+x y^{2}-y = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.989 |
|
|
\[ {}x^{2} \left (y-1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.733 |
|
|
\[ {}x \left (x y+x^{4}-1\right ) y^{\prime }-y \left (x y-x^{4}-1\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.035 |
|
|
\[ {}2 x^{2} y y^{\prime }+y^{2}-2 x^{3}-x^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
2.801 |
|
|
\[ {}2 x^{2} y y^{\prime }-y^{2}-x^{2} {\mathrm e}^{x -\frac {1}{x}} = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.458 |
|
|
\[ {}\left (2 x^{2} y+x \right ) y^{\prime }-x^{2} y^{3}+2 x y^{2}+y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
1.589 |
|
|
\[ {}\left (2 x^{2} y-x \right ) y^{\prime }-2 x y^{2}-y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.831 |
|
|
\[ {}\left (2 x^{2} y-x^{3}\right ) y^{\prime }+y^{3}-4 x y^{2}+2 x^{3} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
3.803 |
|
|
\[ {}2 x^{3}+y y^{\prime }+3 x^{2} y^{2}+7 = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
3.064 |
|
|
\[ {}2 x \left (x^{3} y+1\right ) y^{\prime }+\left (3 x^{3} y-1\right ) y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.406 |
|
|
\[ {}\left (x^{n \left (n +1\right )} y-1\right ) y^{\prime }+2 \left (n +1\right )^{2} x^{n -1} \left (x^{n^{2}} y^{2}-1\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.551 |
|
|
\[ {}\left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime }-a \sqrt {\left (1+y^{2}\right )^{3}} = 0 \] |
1 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[‘x=_G(y,y’)‘] |
✓ |
✗ |
100.584 |
|
|
\[ {}y y^{\prime } \sin \left (x \right )^{2}+y^{2} \cos \left (x \right ) \sin \left (x \right )-1 = 0 \] |
1 |
1 |
2 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[_exact, _Bernoulli] |
✓ |
✓ |
12.353 |
|
|
\[ {}f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2}+h \left (x \right ) = 0 \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.951 |
|
|
\[ {}\left (g_{1} \left (x \right ) y+g_{0} \left (x \right )\right ) y^{\prime }-f_{1} \left (x \right ) y-f_{2} \left (x \right ) y^{2}-f_{3} \left (x \right ) y^{3}-f_{0} \left (x \right ) = 0 \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
❇ |
N/A |
48.582 |
|
|
\[ {}\left (y^{2}-x \right ) y^{\prime }-y+x^{2} = 0 \] |
1 |
1 |
3 |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
12.393 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (y+2 x \right ) = 0 \] |
1 |
1 |
3 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
10.35 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }-y^{2} = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.717 |
|
|
\[ {}\left (y^{2}+x^{2}+a \right ) y^{\prime }+2 x y = 0 \] |
1 |
1 |
3 |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
10.403 |
|
|
\[ {}\left (y^{2}+x^{2}+a \right ) y^{\prime }+2 x y+x^{2}+b = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational] |
✓ |
✓ |
2.233 |
|
|
\[ {}\left (y^{2}+x^{2}+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
exactByInspection |
[_rational] |
✓ |
✓ |
1.581 |
|
|
\[ {}\left (-x^{2}+y^{2}\right ) y^{\prime }+2 x y = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.542 |
|
|
\[ {}\left (y^{2}+x^{4}\right ) y^{\prime }-4 x^{3} y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.52 |
|
|
\[ {}\left (y^{2}+4 \sin \left (x \right )\right ) y^{\prime }-\cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.526 |
|
|
\[ {}\left (y^{2}+2 y+x \right ) y^{\prime }+\left (x +y\right )^{2} y^{2}+y \left (y+1\right ) = 0 \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
2.048 |
|
|
\[ {}\left (x +y\right )^{2} y^{\prime }-a^{2} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.276 |
|
|
\[ {}\left (y^{2}+2 x y-x^{2}\right ) y^{\prime }-y^{2}+2 x y+x^{2} = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.559 |
|
|
\[ {}\left (y+3 x -1\right )^{2} y^{\prime }-\left (2 y-1\right ) \left (4 y+6 x -3\right ) = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
4.532 |
|
|
\[ {}3 \left (-x^{2}+y^{2}\right ) y^{\prime }+2 y^{3}-6 x \left (1+x \right ) y-3 \,{\mathrm e}^{x} = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
2.594 |
|
|
\[ {}\left (4 y^{2}+x^{2}\right ) y^{\prime }-x y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.81 |
|
|
\[ {}\left (4 y^{2}+2 x y+3 x^{2}\right ) y^{\prime }+y^{2}+6 x y+2 x^{2} = 0 \] |
1 |
1 |
3 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
2.693 |
|
|
\[ {}\left (2 y-3 x +1\right )^{2} y^{\prime }-\left (3 y-2 x -4\right )^{2} = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
4.156 |
|
|
\[ {}\left (2 y-4 x +1\right )^{2} y^{\prime }-\left (-2 x +y\right )^{2} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
1.72 |
|
|
\[ {}\left (6 y^{2}-3 x^{2} y+1\right ) y^{\prime }-3 x y^{2}+x = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.181 |
|
|
\[ {}\left (6 y-x \right )^{2} y^{\prime }-6 y^{2}+2 x y+a = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
1.95 |
|
|
\[ {}\left (a y^{2}+2 b x y+c \,x^{2}\right ) y^{\prime }+b y^{2}+2 c x y+d \,x^{2} = 0 \] |
1 |
1 |
3 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
4.631 |
|
|
\[ {}\left (b \left (\beta y+\alpha x \right )^{2}-\beta \left (x a +b y\right )\right ) y^{\prime }+a \left (\beta y+\alpha x \right )^{2}-\alpha \left (x a +b y\right ) = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
2.774 |
|
|
\[ {}\left (a y+b x +c \right )^{2} y^{\prime }+\left (\alpha y+\beta x +\gamma \right )^{2} = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
2.499 |
|
|
\[ {}x \left (y^{2}-3 x \right ) y^{\prime }+2 y^{3}-5 x y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
5.332 |
|
|
\[ {}x \left (y^{2}+x^{2}-a \right ) y^{\prime }-y \left (y^{2}+x^{2}+a \right ) = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
5.089 |
|
|
\[ {}x \left (y^{2}+x y-x^{2}\right ) y^{\prime }-y^{3}+x y^{2}+x^{2} y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.986 |
|
|
\[ {}x \left (y^{2}+x^{2} y+x^{2}\right ) y^{\prime }-2 y^{3}-2 x^{2} y^{2}+x^{4} = 0 \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
4.655 |
|
|
\[ {}2 x \left (y^{2}+5 x^{2}\right ) y^{\prime }+y^{3}-x^{2} y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.911 |
|
|
\[ {}3 x y^{2} y^{\prime }+y^{3}-2 x = 0 \] |
1 |
1 |
3 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
1.665 |
|
|
\[ {}\left (3 x y^{2}-x^{2}\right ) y^{\prime }+y^{3}-2 x y = 0 \] |
1 |
1 |
3 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
✓ |
2.623 |
|
|
\[ {}6 x y^{2} y^{\prime }+2 y^{3}+x = 0 \] |
1 |
1 |
3 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
1.436 |
|
|
\[ {}\left (6 x y^{2}+x^{2}\right ) y^{\prime }-y \left (3 y^{2}-x \right ) = 0 \] |
1 |
1 |
1 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.961 |
|
|
\[ {}\left (x^{2} y^{2}+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
4 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.464 |
|
|
\[ {}\left (x y-1\right )^{2} x y^{\prime }+\left (x^{2} y^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.829 |
|
|
\[ {}\left (10 x^{3} y^{2}+x^{2} y+2 x \right ) y^{\prime }+5 x^{2} y^{3}+x y^{2} = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.185 |
|
|
\[ {}\left (y^{3}-3 x \right ) y^{\prime }-3 y+x^{2} = 0 \] |
1 |
1 |
1 |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
1.838 |
|
|
\[ {}\left (y^{3}-x^{3}\right ) y^{\prime }-x^{2} y = 0 \] |
1 |
1 |
10 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.938 |
|
|
\[ {}\left (y^{2}+x^{2}+a \right ) y y^{\prime }+\left (y^{2}+x^{2}-a \right ) x = 0 \] |
1 |
1 |
4 |
exact |
[_exact, _rational] |
✓ |
✓ |
2.072 |
|
|
\[ {}2 y^{3} y^{\prime }+x y^{2} = 0 \] |
1 |
1 |
3 |
exact, separable, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.089 |
|
|
\[ {}\left (2 y^{3}+y\right ) y^{\prime }-2 x^{3}-x = 0 \] |
1 |
1 |
4 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.007 |
|
|
\[ {}\left (2 y^{3}+5 x^{2} y\right ) y^{\prime }+5 x y^{2}+x^{3} = 0 \] |
1 |
1 |
4 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
2.622 |
|
|
\[ {}\left (20 y^{3}-3 x y^{2}+6 x^{2} y+3 x^{3}\right ) y^{\prime }-y^{3}+6 x y^{2}+9 x^{2} y+4 x^{3} = 0 \] |
1 |
1 |
1 |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
4.909 |
|
|
\[ {}\left (\frac {y^{2}}{b}+\frac {x^{2}}{a}\right ) \left (y y^{\prime }+x \right )+\frac {\left (-b +a \right ) \left (y y^{\prime }-x \right )}{a +b} = 0 \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
2.642 |
|
|
\[ {}\left (2 a y^{3}+3 a x y^{2}-b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 b \,x^{2} y+2 b \,x^{3} = 0 \] |
1 |
0 |
3 |
unknown |
[_rational] |
✗ |
N/A |
2.221 |
|
|
\[ {}x y^{3} y^{\prime }+y^{4}-x \sin \left (x \right ) = 0 \] |
1 |
1 |
4 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
7.618 |
|
|
\[ {}\left (2 x y^{3}-x^{4}\right ) y^{\prime }-y^{4}+2 x^{3} y = 0 \] |
1 |
1 |
3 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.033 |
|
|
\[ {}\left (2 x y^{3}+y\right ) y^{\prime }+2 y^{2} = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.514 |
|
|
\[ {}\left (2 x y^{3}+x y+x^{2}\right ) y^{\prime }+y^{2}-x y = 0 \] |
1 |
1 |
1 |
exactByInspection |
[_rational] |
✓ |
✓ |
1.961 |
|
|
\[ {}\left (3 x y^{3}-4 x y+y\right ) y^{\prime }+y^{2} \left (y^{2}-2\right ) = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.52 |
|
|
\[ {}\left (7 x y^{3}+y-5 x \right ) y^{\prime }+y^{4}-5 y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.996 |
|
|
\[ {}\left (x^{2} y^{3}+x y\right ) y^{\prime }-1 = 0 \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✗ |
N/A |
1.332 |
|
|
\[ {}\left (2 x^{2} y^{3}+x^{2} y^{2}-2 x \right ) y^{\prime }-2 y-1 = 0 \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
3.747 |
|
|
\[ {}\left (10 x^{2} y^{3}-3 y^{2}-2\right ) y^{\prime }+5 y^{4} x +x = 0 \] |
1 |
1 |
1 |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.027 |
|
|
\[ {}\left (a x y^{3}+c \right ) x y^{\prime }+\left (b \,x^{3} y+c \right ) y = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
2.47 |
|
|
\[ {}\left (2 y^{3} x^{3}-x \right ) y^{\prime }+2 y^{3} x^{3}-y = 0 \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.992 |
|
|
\[ {}y \left (y^{3}-2 x^{3}\right ) y^{\prime }+\left (2 y^{3}-x^{3}\right ) x = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
13.04 |
|
|
\[ {}y \left (\left (b x +a y\right )^{3}+b \,x^{3}\right ) y^{\prime }+x \left (\left (b x +a y\right )^{3}+a y^{3}\right ) = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
6.096 |
|
|
\[ {}\left (y^{4} x +2 x^{2} y^{3}+2 y+x \right ) y^{\prime }+y^{5}+y = 0 \] |
1 |
0 |
3 |
unknown |
[_rational] |
✗ |
N/A |
2.097 |
|
|
\[ {}a \,x^{2} y^{n} y^{\prime }-2 x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.52 |
|
|
\[ {}y^{m} x^{n} \left (a x y^{\prime }+b y\right )+\alpha x y^{\prime }+\beta y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.395 |
|
|
\[ {}\left (f \left (x +y\right )+1\right ) y^{\prime }+f \left (x +y\right ) = 0 \] |
1 |
1 |
1 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _dAlembert] |
✓ |
✓ |
1.382 |
|
|
\[ {}\frac {y^{\prime } f_{\nu }\left (x \right ) \left (-y+y^{p +1}\right )}{y-1}-\frac {g_{\nu }\left (x \right ) \left (-y+y^{q +1}\right )}{y-1} = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.638 |
|
|
\[ {}\left (\sqrt {x y}-1\right ) x y^{\prime }-\left (\sqrt {x y}+1\right ) y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
5.008 |
|
|
\[ {}\left (2 x^{\frac {5}{2}} y^{\frac {3}{2}}+x^{2} y-x \right ) y^{\prime }-x^{\frac {3}{2}} y^{\frac {5}{2}}+x y^{2}-y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
191.197 |
|
|
\[ {}\left (1+\sqrt {x +y}\right ) y^{\prime }+1 = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.214 |
|
|
\[ {}\sqrt {y^{2}-1}\, y^{\prime }-\sqrt {x^{2}-1} = 0 \] |
1 |
1 |
1 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.625 |
|
|
\[ {}\left (\sqrt {1+y^{2}}+x a \right ) y^{\prime }+\sqrt {x^{2}+1}+a y = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
2.711 |
|
|
\[ {}\left (\sqrt {x^{2}+y^{2}}+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
5.527 |
|
|
\[ {}\left (y \sqrt {x^{2}+y^{2}}+\left (-x^{2}+y^{2}\right ) \sin \left (\alpha \right )-2 x y \cos \left (\alpha \right )\right ) y^{\prime }+x \sqrt {x^{2}+y^{2}}+2 x y \sin \left (\alpha \right )+\left (-x^{2}+y^{2}\right ) \cos \left (\alpha \right ) = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
9.542 |
|
|
\[ {}\left (x \sqrt {1+x^{2}+y^{2}}-y \left (x^{2}+y^{2}\right )\right ) y^{\prime }-y \sqrt {1+x^{2}+y^{2}}-x \left (x^{2}+y^{2}\right ) = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.592 |
|
|
\[ {}\left (\frac {\operatorname {e1} \left (x +a \right )}{\left (\left (x +a \right )^{2}+y^{2}\right )^{\frac {3}{2}}}+\frac {\operatorname {e2} \left (x -a \right )}{\left (\left (x -a \right )^{2}+y^{2}\right )^{\frac {3}{2}}}\right ) y^{\prime }-y \left (\frac {\operatorname {e1}}{\left (\left (x +a \right )^{2}+y^{2}\right )^{\frac {3}{2}}}+\frac {\operatorname {e2}}{\left (\left (x -a \right )^{2}+y^{2}\right )^{\frac {3}{2}}}\right ) = 0 \] |
1 |
1 |
0 |
exactWithIntegrationFactor |
unknown |
✓ |
✓ |
67.142 |
|
|
\[ {}\left (x \,{\mathrm e}^{y}+{\mathrm e}^{x}\right ) y^{\prime }+{\mathrm e}^{y}+{\mathrm e}^{x} y = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
2.319 |
|
|
\[ {}x \left (3 \,{\mathrm e}^{x y}+2 \,{\mathrm e}^{-x y}\right ) \left (x y^{\prime }+y\right )+1 = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
35.467 |
|
|
\[ {}\left (\ln \left (y\right )+x \right ) y^{\prime }-1 = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
✓ |
1.513 |
|
|
\[ {}\left (\ln \left (y\right )+2 x -1\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
2.277 |
|
|
\[ {}x \left (2 x^{2} y \ln \left (y\right )+1\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.725 |
|
|
\[ {}x \left (y \ln \left (x y\right )+y-x a \right ) y^{\prime }-y \left (a x \ln \left (x y\right )-y+x a \right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
2.546 |
|
|
\[ {}y^{\prime } \left (\sin \left (x \right )+1\right ) \sin \left (y\right )+\cos \left (x \right ) \left (\cos \left (y\right )-1\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
6.272 |
|
|
\[ {}\left (x \cos \left (y\right )+\sin \left (x \right )\right ) y^{\prime }+y \cos \left (x \right )+\sin \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
13.434 |
|
|
\[ {}x y^{\prime } \cot \left (\frac {y}{x}\right )+2 x \sin \left (\frac {y}{x}\right )-y \cot \left (\frac {y}{x}\right ) = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘]] |
✓ |
✓ |
2.95 |
|
|
\[ {}y^{\prime } \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right )^{2}-\sin \left (y\right ) = 0 \] |
1 |
0 |
2 |
unknown |
unknown |
✗ |
N/A |
41.768 |
|
|
\[ {}y^{\prime } \cos \left (y\right )+x \sin \left (y\right ) \cos \left (y\right )^{2}-\sin \left (y\right )^{3} = 0 \] |
1 |
0 |
2 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
48.569 |
|
|
\[ {}y^{\prime } \left (\cos \left (y\right )-\sin \left (\alpha \right ) \sin \left (x \right )\right ) \cos \left (y\right )+\left (\cos \left (x \right )-\sin \left (\alpha \right ) \sin \left (y\right )\right ) \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
exact |
unknown |
✓ |
✓ |
6.481 |
|
|
\[ {}x y^{\prime } \cos \left (y\right )+\sin \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.712 |
|
|
\[ {}\left (x \sin \left (y\right )-1\right ) y^{\prime }+\cos \left (y\right ) = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
13.398 |
|
|
\[ {}\left (x \cos \left (y\right )+\cos \left (x \right )\right ) y^{\prime }-y \sin \left (x \right )+\sin \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
8.404 |
|
|
\[ {}\left (x^{2} \cos \left (y\right )+2 y \sin \left (x \right )\right ) y^{\prime }+2 x \sin \left (y\right )+y^{2} \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
33.297 |
|
|
\[ {}x y^{\prime } \ln \left (x \right ) \sin \left (y\right )+\cos \left (y\right ) \left (1-x \cos \left (y\right )\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
33.409 |
|
|
\[ {}y^{\prime } \sin \left (y\right ) \cos \left (x \right )+\cos \left (y\right ) \sin \left (x \right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.329 |
|
|
\[ {}3 y^{\prime } \sin \left (x \right ) \sin \left (y\right )+5 \cos \left (x \right )^{4} y = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
32.413 |
|
|
\[ {}y^{\prime } \cos \left (a y\right )-b \left (1-c \cos \left (a y\right )\right ) \sqrt {\cos \left (a y\right )^{2}-1+c \cos \left (a y\right )} = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
67.808 |
|
|
\[ {}\left (x \sin \left (x y\right )+\cos \left (x +y\right )-\sin \left (y\right )\right ) y^{\prime }+y \sin \left (x y\right )+\cos \left (x +y\right )+\cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
37.063 |
|
|
\[ {}\left (x^{2} y \sin \left (x y\right )-4 x \right ) y^{\prime }+x y^{2} \sin \left (x y\right )-y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
33.174 |
|
|
\[ {}\left (-y+x y^{\prime }\right ) \cos \left (\frac {y}{x}\right )^{2}+x = 0 \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
4.676 |
|
|
\[ {}\left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime }-\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
4.265 |
|
|
\[ {}\left (y f \left (x^{2}+y^{2}\right )-x \right ) y^{\prime }+y+x f \left (x^{2}+y^{2}\right ) = 0 \] |
1 |
1 |
1 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
9.193 |
|
|
\[ {}f \left (x^{2}+a y^{2}\right ) \left (a y y^{\prime }+x \right )-y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
2.684 |
|
|
\[ {}f \left (x^{c} y\right ) \left (b x y^{\prime }-a \right )-x^{a} y^{b} \left (x y^{\prime }+c y\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
3.075 |
|
|
\[ {}{y^{\prime }}^{2}+a y+b \,x^{2} = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
6.334 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2}-a^{2} = 0 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.372 |
|
|
\[ {}{y^{\prime }}^{2}+y^{2}-f \left (x \right )^{2} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
2.067 |
|
|
\[ {}{y^{\prime }}^{2}-y^{3}+y^{2} = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.365 |
|
|
\[ {}{y^{\prime }}^{2}-4 y^{3}+a y+b = 0 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
17.666 |
|
|
\[ {}{y^{\prime }}^{2}+a^{2} y^{2} \left (\ln \left (y\right )^{2}-1\right ) = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.986 |
|
|
\[ {}{y^{\prime }}^{2}-2 y^{\prime }-y^{2} = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.912 |
|
|
\[ {}{y^{\prime }}^{2}+a y^{\prime }+b x = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.733 |
|
|
\[ {}{y^{\prime }}^{2}+a y^{\prime }+b y = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.322 |
|
|
\[ {}{y^{\prime }}^{2}+\left (-2+x \right ) y^{\prime }-y+1 = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.513 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x +a \right ) y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.466 |
|
|
\[ {}{y^{\prime }}^{2}-\left (1+x \right ) y^{\prime }+y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.458 |
|
|
\[ {}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.687 |
|
|
\[ {}{y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.692 |
|
|
\[ {}{y^{\prime }}^{2}+a x y^{\prime }-b \,x^{2}-c = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.967 |
|
|
\[ {}{y^{\prime }}^{2}+a x y^{\prime }+b y+c \,x^{2} = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
8.793 |
|
|
\[ {}{y^{\prime }}^{2}+\left (x a +b \right ) y^{\prime }-a y+c = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.502 |
|
|
\[ {}{y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
52.138 |
|
|
\[ {}{y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
23.797 |
|
|
\[ {}{y^{\prime }}^{2}+\left (y^{\prime }-y\right ) {\mathrm e}^{x} = 0 \] |
2 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
13.042 |
|
|
\[ {}{y^{\prime }}^{2}-2 y y^{\prime }-2 x = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.981 |
|
|
\[ {}{y^{\prime }}^{2}-\left (4 y+1\right ) y^{\prime }+\left (4 y+1\right ) y = 0 \] |
2 |
2 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.441 |
|
|
\[ {}{y^{\prime }}^{2}+a y y^{\prime }-b x -c = 0 \] |
2 |
4 |
1 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
91.843 |
|
|
\[ {}{y^{\prime }}^{2}+\left (b x +a y\right ) y^{\prime }+a b x y = 0 \] |
2 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.919 |
|
|
\[ {}{y^{\prime }}^{2}-x y y^{\prime }+y^{2} \ln \left (a y\right ) = 0 \] |
2 |
2 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
8.968 |
|
|
\[ {}{y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right )-y^{2} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
8.635 |
|
|
\[ {}{y^{\prime }}^{2}+2 f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2}+h \left (x \right ) = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
14.407 |
|
|
\[ {}{y^{\prime }}^{2}+y \left (y-x \right ) y^{\prime }-x y^{3} = 0 \] |
2 |
1 |
2 |
quadrature, separable |
[_separable] |
✓ |
✓ |
0.816 |
|
|
\[ {}{y^{\prime }}^{2}-2 x^{3} y^{2} y^{\prime }-4 x^{2} y^{3} = 0 \] |
2 |
1 |
6 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
35.307 |
|
|
\[ {}{y^{\prime }}^{2}-3 x y^{\frac {2}{3}} y^{\prime }+9 y^{\frac {5}{3}} = 0 \] |
2 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
14.772 |
|
|
\[ {}2 {y^{\prime }}^{2}+\left (-1+x \right ) y^{\prime }-y = 0 \] |
2 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.484 |
|
|
\[ {}2 {y^{\prime }}^{2}-2 x^{2} y^{\prime }+3 x y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
14.767 |
|
|
\[ {}3 {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
0.743 |
|
|
|
|||||||||
|
\[ {}3 {y^{\prime }}^{2}+4 x y^{\prime }-y+x^{2} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.991 |
|
|
\[ {}a {y^{\prime }}^{2}+b y^{\prime }-y = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.518 |
|
|
\[ {}a {y^{\prime }}^{2}+b \,x^{2} y^{\prime }+c x y = 0 \] |
2 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
44.898 |
|
|
\[ {}a {y^{\prime }}^{2}+y y^{\prime }-x = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
91.692 |
|
|
\[ {}a {y^{\prime }}^{2}-y y^{\prime }-x = 0 \] |
2 |
4 |
2 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
1.307 |
|
|
\[ {}x {y^{\prime }}^{2}-y = 0 \] |
2 |
3 |
3 |
dAlembert, first_order_nonlinear_p_but_separable |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.808 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y+x = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.03 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y^{\prime }-y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.774 |
|
|
\[ {}x {y^{\prime }}^{2}+4 y^{\prime }-2 y = 0 \] |
2 |
4 |
1 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
0.786 |
|
|
\[ {}x {y^{\prime }}^{2}+x y^{\prime }-y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.789 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }+a = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
✓ |
1.003 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }-x^{2} = 0 \] |
2 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
75.885 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }+x^{3} = 0 \] |
2 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
18.308 |
|
|
\[ {}x {y^{\prime }}^{2}+y y^{\prime }-y^{4} = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
5.524 |
|
|
\[ {}x {y^{\prime }}^{2}+\left (-3 x +y\right ) y^{\prime }+y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.282 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }+a = 0 \] |
2 |
3 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.625 |
|
|
\[ {}x {y^{\prime }}^{2}-y y^{\prime }+a y = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.109 |
|
|
\[ {}x {y^{\prime }}^{2}+2 y y^{\prime }-x = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.729 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+a = 0 \] |
2 |
2 |
3 |
dAlembert |
[[_homogeneous, ‘class G‘], _rational, _dAlembert] |
✓ |
✓ |
1.007 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.658 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+4 x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.582 |
|
|
\[ {}x {y^{\prime }}^{2}-2 y y^{\prime }+2 y+x = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.724 |
|
|
\[ {}x {y^{\prime }}^{2}+a y y^{\prime }+b x = 0 \] |
2 |
3 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.891 |
|
|
\[ {}\left (1+x \right ) {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.747 |
|
|
\[ {}\left (1+3 x \right ) {y^{\prime }}^{2}-3 \left (y+2\right ) y^{\prime }+9 = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.732 |
|
|
\[ {}\left (3 x +5\right ) {y^{\prime }}^{2}-\left (x +3 y\right ) y^{\prime }+y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.777 |
|
|
\[ {}a x {y^{\prime }}^{2}+\left (b x -a y+c \right ) y^{\prime }-b y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.829 |
|
|
\[ {}a x {y^{\prime }}^{2}-\left (a y+b x -a -b \right ) y^{\prime }+b y = 0 \] |
2 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert] |
✓ |
✓ |
0.886 |
|
|
\[ {}\left (\operatorname {a2} x +\operatorname {c2} \right ) {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {a0} x +\operatorname {b0} y+\operatorname {c0} = 0 \] |
2 |
4 |
2 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✓ |
128.933 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-y^{4}+y^{2} = 0 \] |
2 |
2 |
5 |
first_order_nonlinear_p_but_separable |
[_separable] |
✓ |
✓ |
2.795 |
|
|
\[ {}\left (x y^{\prime }+a \right )^{2}-2 a y+x^{2} = 0 \] |
2 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
4.225 |
|
|
\[ {}\left (x y^{\prime }+y+2 x \right )^{2}-4 x y-4 x^{2}-4 a = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
8.199 |
|
|
\[ {}y^{\prime }-1 = 0 \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.181 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+y \left (y+1\right )-x = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
8.697 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2} \left (-x^{2}+1\right )-x^{4} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
6.124 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-\left (2 x y+a \right ) y^{\prime }+y^{2} = 0 \] |
2 |
4 |
3 |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
1.856 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+3 x y y^{\prime }+2 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
1.616 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+3 x y y^{\prime }+3 y^{2} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
1.979 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+4 x y y^{\prime }-5 y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
1.934 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-4 x \left (y+2\right ) y^{\prime }+4 y \left (y+2\right ) = 0 \] |
2 |
2 |
5 |
separable |
[_separable] |
✓ |
✓ |
7.426 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+\left (x^{2} y-2 x y+x^{3}\right ) y^{\prime }+\left (y^{2}-x^{2} y\right ) \left (1-x \right ) = 0 \] |
2 |
1 |
2 |
linear, separable |
[_linear] |
✓ |
✓ |
1.335 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}-y \left (-2 x +y\right ) y^{\prime }+y^{2} = 0 \] |
2 |
3 |
6 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.429 |
|
|
\[ {}x^{2} {y^{\prime }}^{2}+\left (a \,x^{2} y^{3}+b \right ) y^{\prime }+a b y^{3} = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.109 |
|
|
\[ {}\left (x^{2}+1\right ) {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}-1 = 0 \] |
2 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.944 |
|
|
\[ {}\left (x^{2}-1\right ) {y^{\prime }}^{2}-1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.809 |
|
|
\[ {}\left (x^{2}-1\right ) {y^{\prime }}^{2}-y^{2}+1 = 0 \] |
2 |
2 |
4 |
first_order_nonlinear_p_but_separable |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.639 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+y^{2} = 0 \] |
2 |
1 |
2 |
separable |
[_separable] |
✓ |
✓ |
1.438 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}-2 x y y^{\prime }-x^{2} = 0 \] |
2 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
132.964 |
|
|
\[ {}\left (x^{2}+a \right ) {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}+b = 0 \] |
2 |
5 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
3.489 |
|
|
\[ {}\left (2 x^{2}+1\right ) {y^{\prime }}^{2}+\left (y^{2}+2 x y+x^{2}+2\right ) y^{\prime }+2 y^{2}+1 = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
26.482 |
|
|
\[ {}\left (a^{2}-1\right ) x^{2} {y^{\prime }}^{2}+2 x y y^{\prime }-y^{2}+a^{2} x^{2} = 0 \] |
2 |
8 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
213.142 |
|
|
\[ {}a \,x^{2} {y^{\prime }}^{2}-2 a x y y^{\prime }+y^{2}-a \left (a -1\right ) x^{2} = 0 \] |
2 |
8 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.766 |
|
|
\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+a = 0 \] |
2 |
2 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
6.274 |
|
|
\[ {}x \left (x^{2}-1\right ) {y^{\prime }}^{2}+2 \left (-x^{2}+1\right ) y y^{\prime }+x y^{2}-x = 0 \] |
2 |
0 |
3 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
8.06 |
|
|
\[ {}x^{4} {y^{\prime }}^{2}-x y^{\prime }-y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
4.196 |
|
|
\[ {}x^{2} \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}-1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.685 |
|
|
\[ {}{\mathrm e}^{-2 x} {y^{\prime }}^{2}-\left (y^{\prime }-1\right )^{2}+{\mathrm e}^{-2 y} = 0 \] |
2 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
12.216 |
|
|
\[ {}\left ({y^{\prime }}^{2}+y^{2}\right ) \cos \left (x \right )^{4}-a^{2} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
11.382 |
|
|
\[ {}\operatorname {d0} \left (x \right ) {y^{\prime }}^{2}+2 \operatorname {b0} \left (x \right ) y y^{\prime }+\operatorname {c0} \left (x \right ) y^{2}+2 \operatorname {d0} \left (x \right ) y^{\prime }+2 \operatorname {e0} \left (x \right ) y+\operatorname {f0} \left (x \right ) = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
77.711 |
|
|
\[ {}y {y^{\prime }}^{2}-1 = 0 \] |
2 |
2 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.422 |
|
|
\[ {}y {y^{\prime }}^{2}-{\mathrm e}^{2 x} = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
4.508 |
|
|
\[ {}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.859 |
|
|
\[ {}y {y^{\prime }}^{2}+2 x y^{\prime }-9 y = 0 \] |
2 |
3 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.352 |
|
|
\[ {}y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.612 |
|
|
\[ {}y {y^{\prime }}^{2}-4 x y^{\prime }+y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.174 |
|
|
\[ {}y {y^{\prime }}^{2}-4 a^{2} x y^{\prime }+a^{2} y = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.326 |
|
|
\[ {}y {y^{\prime }}^{2}+a x y^{\prime }+b y = 0 \] |
2 |
4 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
22.196 |
|
|
\[ {}y {y^{\prime }}^{2}+x^{3} y^{\prime }-x^{2} y = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.781 |
|
|
\[ {}y {y^{\prime }}^{2}-\left (y-x \right ) y^{\prime }-x = 0 \] |
2 |
2 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.495 |
|
|
\[ {}\left (x +y\right ) {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.971 |
|
|
\[ {}\left (-2 x +y\right ) {y^{\prime }}^{2}-2 \left (-1+x \right ) y^{\prime }+y-2 = 0 \] |
2 |
5 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.936 |
|
|
\[ {}2 y {y^{\prime }}^{2}-\left (4 x -5\right ) y^{\prime }+2 y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.817 |
|
|
\[ {}4 y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0 \] |
2 |
5 |
7 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.879 |
|
|
\[ {}9 y {y^{\prime }}^{2}+4 x^{3} y^{\prime }-4 x^{2} y = 0 \] |
2 |
2 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.589 |
|
|
\[ {}a y {y^{\prime }}^{2}+\left (2 x -b \right ) y^{\prime }-y = 0 \] |
2 |
5 |
5 |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
1.192 |
|
|
\[ {}\left (a y+b \right ) \left (1+{y^{\prime }}^{2}\right )-c = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
6.11 |
|
|
\[ {}\left (b_{2} y+a_{2} x +c_{2} \right ) {y^{\prime }}^{2}+\left (a_{1} x +b_{1} y+c_{1} \right ) y^{\prime }+a_{0} x +b_{0} y+c_{0} = 0 \] |
2 |
3 |
2 |
dAlembert |
[_rational, _dAlembert] |
✓ |
✗ |
9.348 |
|
|
\[ {}\left (a y-x^{2}\right ) {y^{\prime }}^{2}+2 x y {y^{\prime }}^{2}-y^{2} = 0 \] |
2 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
2.563 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
0.627 |
|
|
\[ {}x y {y^{\prime }}^{2}+\left (x^{22}-y^{2}+a \right ) y^{\prime }-x y = 0 \] |
2 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
24.905 |
|
|
\[ {}\left (2 x y-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+2 x y-y^{2} = 0 \] |
2 |
5 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.207 |
|
|
\[ {}\left (2 x y-x^{2}\right ) {y^{\prime }}^{2}-6 x y y^{\prime }-y^{2}+2 x y = 0 \] |
2 |
9 |
3 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
219.059 |
|
|
\[ {}a x y {y^{\prime }}^{2}-\left (a y^{2}+b \,x^{2}+c \right ) y^{\prime }+b x y = 0 \] |
2 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
100.549 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+y^{2}-a^{2} = 0 \] |
2 |
2 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.729 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y = 0 \] |
2 |
2 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
6.134 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-4 a y y^{\prime }+y^{2}-4 x a +4 a^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
5.074 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }+a y^{2}+b x +c = 0 \] |
2 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
5.374 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 y^{2}-x^{2}+a = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
5.46 |
|
|
\[ {}y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}+x^{2} a +\left (a -1\right ) b = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
7.128 |
|
|
\[ {}\left (y^{2}-a^{2}\right ) {y^{\prime }}^{2}+y^{2} = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.174 |
|
|
\[ {}\left (y^{2}-2 x a +a^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2} = 0 \] |
2 |
0 |
2 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
7.22 |
|
|
\[ {}\left (y^{2}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+\left (-a^{2}+1\right ) x^{2} = 0 \] |
2 |
10 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.904 |
|
|
\[ {}\left (y^{2}+\left (1-a \right ) x^{2}\right ) {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}+x^{2} = 0 \] |
2 |
4 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.012 |
|
|
\[ {}\left (y-x \right )^{2} \left (1+{y^{\prime }}^{2}\right )-a^{2} \left (y^{\prime }+1\right )^{2} = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
18.796 |
|
|
\[ {}3 y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+4 y^{2}-x^{2} = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.49 |
|
|
\[ {}\left (3 y-2\right ) {y^{\prime }}^{2}-4+4 y = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.652 |
|
|
\[ {}\left (-a^{2}+1\right ) y^{2} {y^{\prime }}^{2}-2 a^{2} x y y^{\prime }+y^{2}-a^{2} x^{2} = 0 \] |
2 |
7 |
4 |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
10.343 |
|
|
\[ {}\left (-b +a \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }+a y^{2}-b \,x^{2}-a b = 0 \] |
2 |
0 |
4 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
6.755 |
|
|
\[ {}\left (a y^{2}+b x +c \right ) {y^{\prime }}^{2}-b y y^{\prime }+d y^{2} = 0 \] |
2 |
0 |
3 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
13.713 |
|
|
\[ {}\left (a y-b x \right )^{2} \left (a^{2} {y^{\prime }}^{2}+b^{2}\right )-c^{2} \left (a y^{\prime }+b \right )^{2} = 0 \] |
2 |
6 |
4 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
30.862 |
|
|
\[ {}\left (\operatorname {b2} y+\operatorname {a2} x +\operatorname {c2} \right )^{2} {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {b0} y+\operatorname {a0} +\operatorname {c0} = 0 \] |
2 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
90.787 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}-\left (y^{3}+x^{3}-a \right ) y^{\prime }+x^{2} y = 0 \] |
2 |
0 |
9 |
unknown |
[_rational] |
✗ |
N/A |
15.87 |
|
|
\[ {}x y^{2} {y^{\prime }}^{2}-2 y^{3} y^{\prime }+2 x y^{2}-x^{3} = 0 \] |
2 |
1 |
4 |
separable, homogeneousTypeD2 |
[_separable] |
✓ |
✓ |
1.033 |
|
|
\[ {}x^{2} \left (x y^{2}-1\right ) {y^{\prime }}^{2}+2 x^{2} y^{2} \left (y-x \right ) y^{\prime }-y^{2} \left (x^{2} y-1\right ) = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
30.803 |
|
|
\[ {}\left (y^{4}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+y^{2} \left (y^{2}-a^{2}\right ) = 0 \] |
2 |
0 |
2 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
12.389 |
|
|
\[ {}\left (y^{4}+x^{2} y^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }-y^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
8.423 |
|
|
\[ {}9 y^{4} \left (x^{2}-1\right ) {y^{\prime }}^{2}-6 x y^{5} y^{\prime }-4 x^{2} = 0 \] |
2 |
0 |
9 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
7.985 |
|
|
\[ {}x^{2} \left (y^{4} x^{2}-1\right ) {y^{\prime }}^{2}+2 x^{3} y^{3} \left (-x^{2}+y^{2}\right ) y^{\prime }-y^{2} \left (x^{4} y^{2}-1\right ) = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
26.609 |
|
|
\[ {}\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a^{2} \sqrt {x^{2}+y^{2}}-y^{2} = 0 \] |
2 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
22.525 |
|
|
\[ {}\left (a \left (x^{2}+y^{2}\right )^{\frac {3}{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a \left (x^{2}+y^{2}\right )^{\frac {3}{2}}-y^{2} = 0 \] |
2 |
0 |
3 |
unknown |
[[_1st_order, _with_linear_symmetries]] |
✗ |
N/A |
39.219 |
|
|
\[ {}{y^{\prime }}^{2} \sin \left (y\right )+2 x y^{\prime } \cos \left (y\right )^{3}-\sin \left (y\right ) \cos \left (y\right )^{4} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
99.033 |
|
|
\[ {}{y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )+d = 0 \] |
2 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
11.155 |
|
|
\[ {}f \left (x^{2}+y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0 \] |
2 |
1 |
5 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
4.023 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) f \left (\frac {x}{\sqrt {x^{2}+y^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0 \] |
2 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘]] |
✓ |
✓ |
6.807 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) f \left (\frac {y}{\sqrt {x^{2}+y^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0 \] |
2 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘]] |
✓ |
✓ |
7.245 |
|
|
\[ {}{y^{\prime }}^{3}-\left (y-a \right )^{2} \left (y-b \right )^{2} = 0 \] |
3 |
3 |
5 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.117 |
|
|
\[ {}{y^{\prime }}^{3}-f \left (x \right ) \left (a y^{2}+b y+c \right )^{2} = 0 \] |
3 |
3 |
3 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.878 |
|
|
\[ {}{y^{\prime }}^{3}+y^{\prime }-y = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.805 |
|
|
\[ {}{y^{\prime }}^{3}+x y^{\prime }-y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.293 |
|
|
\[ {}{y^{\prime }}^{3}-\left (x +5\right ) y^{\prime }+y = 0 \] |
3 |
3 |
3 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.324 |
|
|
\[ {}{y^{\prime }}^{3}-a x y^{\prime }+x^{3} = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.045 |
|
|
\[ {}{y^{\prime }}^{3}-2 y y^{\prime }+y^{2} = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
2.474 |
|
|
\[ {}{y^{\prime }}^{2}-a x y y^{\prime }+2 a y^{2} = 0 \] |
2 |
2 |
3 |
separable |
[_separable] |
✓ |
✓ |
6.857 |
|
|
\[ {}{y^{\prime }}^{3}-\left (x^{2}+x y+y^{2}\right ) {y^{\prime }}^{2}+\left (x y^{3}+x^{2} y^{2}+x^{3} y\right ) y^{\prime }-y^{3} x^{3} = 0 \] |
3 |
1 |
3 |
quadrature, separable |
[_quadrature] |
✓ |
✓ |
0.309 |
|
|
\[ {}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0 \] |
3 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
15.946 |
|
|
\[ {}{y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y+a b x = 0 \] |
3 |
4 |
1 |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
118.026 |
|
|
\[ {}{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y = 0 \] |
3 |
5 |
4 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
163.223 |
|
|
\[ {}{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2} = 0 \] |
3 |
3 |
4 |
quadrature |
[_quadrature] |
✓ |
✓ |
1.626 |
|
|
\[ {}{y^{\prime }}^{2}-\left (y^{4}+x y^{2}+x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{6}+y^{4} x^{2}+x^{3} y^{2}\right ) y^{\prime }-x^{3} y^{6} = 0 \] |
2 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
21.569 |
|
|
\[ {}a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+c y^{\prime }-y-d = 0 \] |
3 |
3 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
65.237 |
|
|
\[ {}x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+a = 0 \] |
3 |
4 |
4 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.572 |
|
|
\[ {}4 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}+3 y-x = 0 \] |
3 |
6 |
5 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
10.048 |
|
|
\[ {}8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y = 0 \] |
3 |
6 |
5 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
19.878 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{3}+b x \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+y^{\prime }+b x = 0 \] |
3 |
1 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.499 |
|
|
\[ {}x^{3} {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+\left (3 x y^{2}+x^{6}\right ) y^{\prime }-y^{3}-2 x^{5} y = 0 \] |
3 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
179.494 |
|
|
\[ {}2 \left (x y^{\prime }+y\right )^{3}-y y^{\prime } = 0 \] |
3 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
17.451 |
|
|
\[ {}{y^{\prime }}^{3} \sin \left (x \right )-\left (y \sin \left (x \right )-\cos \left (x \right )^{2}\right ) {y^{\prime }}^{2}-\left (y \cos \left (x \right )^{2}+\sin \left (x \right )\right ) y^{\prime }+y \sin \left (x \right ) = 0 \] |
3 |
2 |
3 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.508 |
|
|
\[ {}2 y {y^{\prime }}^{3}-y {y^{\prime }}^{2}+2 x y^{\prime }-x = 0 \] |
3 |
4 |
3 |
dAlembert, quadrature |
[_quadrature] |
✓ |
✓ |
0.523 |
|
|
\[ {}y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
81.011 |
|
|
\[ {}16 y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
81.368 |
|
|
\[ {}x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-x^{2} y = 0 \] |
3 |
0 |
0 |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
146.769 |
|
|
\[ {}x^{7} y^{2} {y^{\prime }}^{3}-\left (3 x^{6} y^{3}-1\right ) {y^{\prime }}^{2}+3 x^{5} y^{4} y^{\prime }-x^{4} y^{5} = 0 \] |
3 |
1 |
7 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
126.033 |
|
|
\[ {}{y^{\prime }}^{4}-\left (y-a \right )^{3} \left (y-b \right )^{2} = 0 \] |
4 |
4 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.864 |
|
|
\[ {}{y^{\prime }}^{4}+3 \left (-1+x \right ) {y^{\prime }}^{2}-3 \left (2 y-1\right ) y^{\prime }+3 x = 0 \] |
4 |
3 |
4 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
1.138 |
|
|
\[ {}{y^{\prime }}^{4}-4 y \left (x y^{\prime }-2 y\right )^{2} = 0 \] |
4 |
0 |
3 |
unknown |
[[_homogeneous, ‘class G‘]] |
✗ |
N/A |
1.073 |
|
|
\[ {}{y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3} = 0 \] |
6 |
6 |
8 |
quadrature |
[_quadrature] |
✓ |
✓ |
73.431 |
|
|
\[ {}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0 \] |
6 |
6 |
6 |
quadrature |
[_quadrature] |
✓ |
✓ |
3.214 |
|
|
\[ {}{y^{\prime }}^{r}-a y^{s}-b \,x^{\frac {r s}{r -s}} = 0 \] |
0 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
1.43 |
|
|
\[ {}{y^{\prime }}^{n}-f \left (x \right )^{n} \left (y-a \right )^{n +1} \left (y-b \right )^{n -1} = 0 \] |
0 |
1 |
1 |
separable, first_order_nonlinear_p_but_separable |
[_separable] |
✓ |
✓ |
40.158 |
|
|
\[ {}{y^{\prime }}^{n}-f \left (x \right ) g \left (y\right ) = 0 \] |
0 |
1 |
1 |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
0.72 |
|
|
\[ {}a {y^{\prime }}^{m}+b {y^{\prime }}^{n}-y = 0 \] |
0 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.252 |
|
|
\[ {}x^{n -1} {y^{\prime }}^{n}-n x y^{\prime }+y = 0 \] |
0 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.272 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
2 |
2 |
1 |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
1.257 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}}+x {y^{\prime }}^{2}+y = 0 \] |
4 |
6 |
5 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
210.911 |
|
|
\[ {}x \left (\sqrt {1+{y^{\prime }}^{2}}+y^{\prime }\right )-y = 0 \] |
1 |
2 |
2 |
bernoulli, dAlembert, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
3.247 |
|
|
\[ {}a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
2 |
2 |
2 |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.607 |
|
|
\[ {}y \sqrt {1+{y^{\prime }}^{2}}-a y y^{\prime }-x a = 0 \] |
2 |
1 |
4 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
180.667 |
|
|
\[ {}a y \sqrt {1+{y^{\prime }}^{2}}-2 x y y^{\prime }+y^{2}-x^{2} = 0 \] |
2 |
2 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
19.386 |
|
|
\[ {}f \left (x^{2}+y^{2}\right ) \sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }+y = 0 \] |
2 |
0 |
1 |
unknown |
[[_1st_order, _with_linear_symmetries]] |
✗ |
N/A |
6.225 |
|
|
\[ {}a \left ({y^{\prime }}^{3}+1\right )^{\frac {1}{3}}+b x y^{\prime }-y = 0 \] |
3 |
4 |
3 |
dAlembert |
[_dAlembert] |
✓ |
✓ |
4.359 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a y+b = 0 \] |
0 |
2 |
1 |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.369 |
|
|
\[ {}\ln \left (y^{\prime }\right )+a \left (-y+x y^{\prime }\right ) = 0 \] |
0 |
2 |
2 |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.636 |
|
|
\[ {}y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-x y = 0 \] |
0 |
1 |
1 |
separable, homogeneousTypeD2 |
[_separable] |
✓ |
✓ |
1.886 |
|
|
\[ {}\sin \left (y^{\prime }\right )+y^{\prime }-x = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.199 |
|
|
\[ {}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.16 |
|
|
\[ {}{y^{\prime }}^{2} \sin \left (y^{\prime }\right )-y = 0 \] |
0 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.272 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \sin \left (-y+x y^{\prime }\right )^{2}-1 = 0 \] |
0 |
6 |
6 |
clairaut |
[_Clairaut] |
✓ |
✓ |
2.947 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+x a \right )+y^{\prime } = 0 \] |
0 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✗ |
1.307 |
|
|
\[ {}a \,x^{n} f \left (y^{\prime }\right )+x y^{\prime }-y = 0 \] |
0 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.537 |
|
|
\[ {}\left (-y+x y^{\prime }\right )^{n} f \left (y^{\prime }\right )+y g \left (y^{\prime }\right )+x h \left (y^{\prime }\right ) = 0 \] |
0 |
0 |
0 |
unknown |
[‘x=_G(y,y’)‘] |
❇ |
N/A |
2.117 |
|
|
\[ {}f \left (x {y^{\prime }}^{2}\right )+2 x y^{\prime }-y = 0 \] |
0 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.347 |
|
|
\[ {}f \left (x -\frac {3 {y^{\prime }}^{2}}{2}\right )+{y^{\prime }}^{3}-y = 0 \] |
0 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.781 |
|
|
\[ {}y^{\prime } f \left (x y y^{\prime }-y^{2}\right )-x^{2} y^{\prime }+x y = 0 \] |
0 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.613 |
|
|
\[ {}\phi \left (f \left (x , y, y^{\prime }\right ), g \left (x , y, y^{\prime }\right )\right ) = 0 \] |
0 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
1.004 |
|
|
\[ {}y^{\prime } = F \left (\frac {y}{x +a}\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime } = 2 x +F \left (-x^{2}+y\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
0.488 |
|
|
\[ {}y^{\prime } = -\frac {x a}{2}+F \left (y+\frac {x^{2} a}{4}+\frac {b x}{2}\right ) \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
0.688 |
|
|
\[ {}y^{\prime } = F \left (y \,{\mathrm e}^{-b x}\right ) {\mathrm e}^{b x} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime } = \frac {1+2 F \left (\frac {4 x^{2} y+1}{4 x^{2}}\right ) x}{2 x^{3}} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
0.969 |
|
|
\[ {}y^{\prime } = \frac {1+F \left (\frac {y a x +1}{a x}\right ) a \,x^{2}}{a \,x^{2}} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.977 |
|
|
\[ {}y^{\prime } = -\frac {\left (x^{2} a -2 F \left (y+\frac {a \,x^{4}}{8}\right )\right ) x}{2} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.083 |
|
|
\[ {}y^{\prime } = \frac {2 a}{y+2 F \left (y^{2}-4 x a \right ) a} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
0.882 |
|
|
\[ {}y^{\prime } = F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.955 |
|
|
\[ {}y^{\prime } = \frac {F \left (\frac {y}{\sqrt {x^{2}+1}}\right ) x}{\sqrt {x^{2}+1}} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.4 |
|
|
\[ {}y^{\prime } = \frac {\left (x^{\frac {3}{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.075 |
|
|
\[ {}y^{\prime } = \frac {x +F \left (-\left (x -y\right ) \left (x +y\right )\right )}{y} \] |
1 |
0 |
3 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.993 |
|
|
\[ {}y^{\prime } = \frac {F \left (-\frac {-1+y \ln \left (x \right )}{y}\right ) y^{2}}{x} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.248 |
|
|
\[ {}y^{\prime } = \frac {x}{-y+F \left (x^{2}+y^{2}\right )} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.041 |
|
|
\[ {}y^{\prime } = \frac {F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y} \] |
1 |
0 |
3 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.084 |
|
|
\[ {}y^{\prime } = \frac {6 x^{3}+5 \sqrt {x}+5 F \left (y-\frac {2 x^{3}}{5}-2 \sqrt {x}\right )}{5 x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.298 |
|
|
\[ {}y^{\prime } = \frac {F \left (y^{\frac {3}{2}}-\frac {3 \,{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{x}}{\sqrt {y}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.175 |
|
|
\[ {}y^{\prime } = \frac {F \left (-\frac {-y^{2}+b}{x^{2}}\right ) x}{y} \] |
1 |
0 |
3 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.985 |
|
|
\[ {}y^{\prime } = \frac {F \left (\frac {x y^{2}+1}{x}\right )}{y x^{2}} \] |
1 |
0 |
3 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.931 |
|
|
\[ {}y^{\prime } = \frac {-2 x^{2}+x +F \left (y+x^{2}-x \right )}{x} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
1.263 |
|
|
\[ {}y^{\prime } = \frac {2 a}{x^{2} \left (-y+2 F \left (\frac {x y^{2}-4 a}{x}\right ) a \right )} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
2.313 |
|
|
\[ {}y^{\prime } = \frac {y+F \left (\frac {y}{x}\right )}{-1+x} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
1.589 |
|
|
\[ {}y^{\prime } = \frac {-x +F \left (x^{2}+y^{2}\right )}{y} \] |
1 |
0 |
3 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.027 |
|
|
\[ {}y^{\prime } = \frac {F \left (-\frac {2 y \ln \left (x \right )-1}{y}\right ) y^{2}}{x} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.318 |
|
|
\[ {}y^{\prime } = \frac {F \left (-\left (x -y\right ) \left (x +y\right )\right ) x}{y} \] |
1 |
0 |
3 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.099 |
|
|
\[ {}y^{\prime } = \frac {y^{2} \left (2+F \left (\frac {x^{2}-y}{y x^{2}}\right ) x^{2}\right )}{x^{3}} \] |
1 |
0 |
2 |
unknown |
[NONE] |
✗ |
N/A |
1.191 |
|
|
|
|||||||||
|
\[ {}y^{\prime } = \frac {2 F \left (y+\ln \left (2 x +1\right )\right ) x +F \left (y+\ln \left (2 x +1\right )\right )-2}{2 x +1} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.044 |
|
|
\[ {}y^{\prime } = \frac {2 y^{3}}{1+2 F \left (\frac {1+4 x y^{2}}{y^{2}}\right ) y} \] |
1 |
0 |
2 |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
1.141 |
|
|
\[ {}y^{\prime } = -\frac {y^{2} \left (2 x -F \left (-\frac {x y-2}{2 y}\right )\right )}{4 x} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[NONE] |
✓ |
✓ |
2.022 |
|
|
\[ {}y^{\prime } = -\left (-{\mathrm e}^{-x^{2}}+x^{2} {\mathrm e}^{-x^{2}}-F \left (y-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}\right )\right ) x \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.217 |
|
|
\[ {}y^{\prime } = \frac {2 y+F \left (\frac {y}{x^{2}}\right ) x^{3}}{x} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.956 |
|
|
\[ {}y^{\prime } = \frac {\sqrt {y}}{\sqrt {y}+F \left (\frac {x -y}{\sqrt {y}}\right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.148 |
|
|
\[ {}y^{\prime } = \frac {-3 x^{2} y+F \left (x^{3} y\right )}{x^{3}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
0.961 |
|
|
\[ {}y^{\prime } = \frac {y+F \left (\frac {y}{x}\right ) x^{2}}{x} \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
0.601 |
|
|
\[ {}y^{\prime } = \frac {-2 x -y+F \left (x \left (x +y\right )\right )}{x} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.066 |
|
|
\[ {}y^{\prime } = \frac {\left (y \,{\mathrm e}^{-\frac {x^{2}}{4}} x +2 F \left (y \,{\mathrm e}^{-\frac {x^{2}}{4}}\right )\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.669 |
|
|
\[ {}y^{\prime } = \frac {x +y+F \left (-\frac {-y+x \ln \left (x \right )}{x}\right ) x^{2}}{x} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.165 |
|
|
\[ {}y^{\prime } = \frac {x \left (a -1\right ) \left (1+a \right )}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.178 |
|
|
\[ {}y^{\prime } = \frac {y}{x \left (-1+F \left (x y\right ) y\right )} \] |
1 |
1 |
3 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
1.014 |
|
|
\[ {}y^{\prime } = -\frac {-x^{2}+2 x^{3} y-F \left (\left (x y-1\right ) x \right )}{x^{4}} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.303 |
|
|
\[ {}y^{\prime } = \frac {F \left (\frac {\left (3+y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.406 |
|
|
\[ {}y^{\prime } = \frac {\left (y+1\right ) \left (\left (y-\ln \left (y+1\right )-\ln \left (x \right )\right ) x +1\right )}{y x} \] |
1 |
0 |
2 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.148 |
|
|
\[ {}y^{\prime } = \frac {6 y}{8 y^{4}+9 y^{3}+12 y^{2}+6 y-F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )} \] |
1 |
0 |
1 |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
1.25 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{2 F \left (-\left (x -y\right ) \left (x +y\right )\right )}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{2 F \left (-\left (x -y\right ) \left (x +y\right )\right )}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.622 |
|
|
\[ {}y^{\prime } = \frac {1}{y+\sqrt {x}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
1.198 |
|
|
\[ {}y^{\prime } = \frac {1}{y+2+\sqrt {1+3 x}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
8.208 |
|
|
\[ {}y^{\prime } = \frac {x^{2}}{y+x^{\frac {3}{2}}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
8.038 |
|
|
\[ {}y^{\prime } = \frac {x^{\frac {5}{3}}}{y+x^{\frac {4}{3}}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
2.276 |
|
|
\[ {}y^{\prime } = \frac {i x^{2} \left (i-2 \sqrt {-x^{3}+6 y}\right )}{2} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.931 |
|
|
\[ {}y^{\prime } = \frac {x}{y+\sqrt {x^{2}+1}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
2.449 |
|
|
\[ {}y^{\prime } = \frac {\left (-1+y \ln \left (x \right )\right )^{2}}{x} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
1.95 |
|
|
\[ {}y^{\prime } = \frac {x \left (-2+3 \sqrt {x^{2}+3 y}\right )}{3} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
2.971 |
|
|
\[ {}y^{\prime } = \frac {\left (2 y \ln \left (x \right )-1\right )^{2}}{x} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_Riccati] |
✓ |
✓ |
2.034 |
|
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{b x}}{y \,{\mathrm e}^{-b x}+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
5.324 |
|
|
\[ {}y^{\prime } = \frac {x^{2} \left (1+2 \sqrt {x^{3}-6 y}\right )}{2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
3.208 |
|
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{x}}{y \,{\mathrm e}^{-x}+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
6.884 |
|
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{\frac {2 x}{3}}}{y \,{\mathrm e}^{-\frac {2 x}{3}}+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
5.501 |
|
|
\[ {}y^{\prime } = \frac {1+2 x^{5} \sqrt {4 x^{2} y+1}}{2 x^{3}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.5 |
|
|
\[ {}y^{\prime } = \frac {x \left (x +2 \sqrt {x^{3}-6 y}\right )}{2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.243 |
|
|
\[ {}y^{\prime } = \left (-\ln \left (y\right )+x^{2}\right ) y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.331 |
|
|
\[ {}y^{\prime } = \frac {{\mathrm e}^{-x^{2}} x}{y \,{\mathrm e}^{x^{2}}+1} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.121 |
|
|
\[ {}y^{\prime } = -\left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right ) y \] |
1 |
0 |
1 |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
0.998 |
|
|
\[ {}y^{\prime } = \left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right )^{2} y \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
0.984 |
|
|
\[ {}y^{\prime } = \frac {y}{\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )+1} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.066 |
|
|
\[ {}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{4}}{2 x^{3}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.386 |
|
|
\[ {}y^{\prime } = \frac {\left (-y^{2}+4 x a \right )^{2}}{y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.089 |
|
|
\[ {}y^{\prime } = \frac {x \left (-2+3 x \sqrt {x^{2}+3 y}\right )}{3} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.1 |
|
|
\[ {}y^{\prime } = -\frac {x^{2} \left (x a -2 \sqrt {a \left (a \,x^{4}+8 y\right )}\right )}{2} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.37 |
|
|
\[ {}y^{\prime } = \left (-\ln \left (y\right )+x \right ) y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.007 |
|
|
\[ {}y^{\prime } = \frac {x^{3}+x^{2}+2 \sqrt {x^{3}-6 y}}{2 x +2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.084 |
|
|
\[ {}y^{\prime } = \frac {\left (a y^{2}+b \,x^{2}\right )^{2} x}{a^{\frac {5}{2}} y} \] |
1 |
0 |
2 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.277 |
|
|
\[ {}y^{\prime } = -\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
2.175 |
|
|
\[ {}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+x \sqrt {x^{2}-2 x +1+8 y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.399 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+x \sqrt {x^{2}+2 x a +a^{2}+4 y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.252 |
|
|
\[ {}y^{\prime } = \frac {\left (\ln \left (y\right )+x^{2}\right ) y}{x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.096 |
|
|
\[ {}y^{\prime } = \frac {2 a +x \sqrt {-y^{2}+4 x a}}{y} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.353 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+1+x \sqrt {x^{2}-4 x +4 y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.486 |
|
|
\[ {}y^{\prime } = -\frac {2 x^{2}+2 x -3 \sqrt {x^{2}+3 y}}{3 \left (1+x \right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.467 |
|
|
\[ {}y^{\prime } = \frac {y^{3} {\mathrm e}^{-\frac {4 x}{3}}}{y \,{\mathrm e}^{-\frac {2 x}{3}}+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
4.551 |
|
|
\[ {}y^{\prime } = \frac {\left (\ln \left (y\right )+x^{3}\right ) y}{x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.328 |
|
|
\[ {}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+x^{2} \sqrt {x^{2}-2 x +1+8 y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.495 |
|
|
\[ {}y^{\prime } = -\frac {x^{2}-1-4 \sqrt {x^{2}-2 x +1+8 y}}{4 \left (1+x \right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.266 |
|
|
\[ {}y^{\prime } = -\frac {x a}{2}-\frac {b}{2}+x \sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.892 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+x^{2} \sqrt {x^{2}+2 x a +a^{2}+4 y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.317 |
|
|
\[ {}y^{\prime } = -\frac {x a}{2}-\frac {b}{2}+x^{2} \sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
3.178 |
|
|
\[ {}y^{\prime } = \frac {x}{2}+\frac {1}{2}+x^{2} \sqrt {x^{2}+2 x +1-4 y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.891 |
|
|
\[ {}y^{\prime } = \frac {2 a +x^{2} \sqrt {-y^{2}+4 x a}}{y} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.323 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+1+x^{2} \sqrt {x^{2}-4 x +4 y} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.686 |
|
|
\[ {}y^{\prime } = -\frac {\left (\sqrt {a}\, x^{4}+\sqrt {a}\, x^{3}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
2.103 |
|
|
\[ {}y^{\prime } = \left (-\ln \left (y\right )+1+x^{2}+x^{3}\right ) y \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.563 |
|
|
\[ {}y^{\prime } = \frac {y^{3} {\mathrm e}^{-2 b x}}{y \,{\mathrm e}^{-b x}+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
1.837 |
|
|
\[ {}y^{\prime } = \frac {y^{3} {\mathrm e}^{-2 x}}{y \,{\mathrm e}^{-x}+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
4.694 |
|
|
\[ {}y^{\prime } = \frac {\left (-2 y^{\frac {3}{2}}+3 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{x}}{4 \sqrt {y}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
1.619 |
|
|
\[ {}y^{\prime } = \frac {i x \left (i-2 \sqrt {-x^{2}+4 \ln \left (a \right )+4 \ln \left (y\right )}\right ) y}{2} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
2.197 |
|
|
\[ {}y^{\prime } = \frac {\left (x y^{2}+1\right )^{2}}{y x^{4}} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.087 |
|
|
\[ {}y^{\prime } = \frac {x^{2} \left (3 x +\sqrt {-9 x^{4}+4 y^{3}}\right )}{y^{2}} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
3.261 |
|
|
\[ {}y^{\prime } = \frac {-\sin \left (2 y\right )+\cos \left (2 y\right ) x^{2}+x^{2}}{2 x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.942 |
|
|
\[ {}y^{\prime } = -\frac {x^{2}-x -2-2 \sqrt {x^{2}-4 x +4 y}}{2 \left (1+x \right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.381 |
|
|
\[ {}y^{\prime } = \frac {y+x^{3} a \,{\mathrm e}^{x}+a \,x^{4}+a \,x^{3}-x y^{2} {\mathrm e}^{x}-x^{2} y^{2}-x y^{2}}{x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
1.763 |
|
|
\[ {}y^{\prime } = \frac {x +1+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3} \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.501 |
|
|
\[ {}y^{\prime } = \frac {y+x^{3} a \ln \left (1+x \right )+a \,x^{4}+a \,x^{3}-x y^{2} \ln \left (1+x \right )-x^{2} y^{2}-x y^{2}}{x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
1.858 |
|
|
\[ {}y^{\prime } = \frac {x^{2} \left (x +1+2 x \sqrt {x^{3}-6 y}\right )}{2 x +2} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.516 |
|
|
\[ {}y^{\prime } = \frac {y+x^{3} \ln \left (x \right )+x^{4}+x^{3}+7 x y^{2} \ln \left (x \right )+7 x^{2} y^{2}+7 x y^{2}}{x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
1.745 |
|
|
\[ {}y^{\prime } = \frac {x^{2}+2 x +1+2 \sqrt {x^{2}+2 x +1-4 y}}{2 x +2} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.301 |
|
|
\[ {}y^{\prime } = \frac {y+x^{3} b \ln \left (\frac {1}{x}\right )+x^{4} b +b \,x^{3}+x a y^{2} \ln \left (\frac {1}{x}\right )+a \,x^{2} y^{2}+a x y^{2}}{x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
1.874 |
|
|
\[ {}y^{\prime } = \frac {2 a}{x \left (-x y+2 a x y^{2}-8 a^{2}\right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.07 |
|
|
\[ {}y^{\prime } = \frac {y \left (-1+\ln \left (\left (1+x \right ) x \right ) y x^{4}-\ln \left (\left (1+x \right ) x \right ) x^{3}\right )}{x} \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
3.632 |
|
|
\[ {}y^{\prime } = \frac {y+\sqrt {x^{2}+y^{2}}\, x^{2}}{x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.227 |
|
|
\[ {}y^{\prime } = \frac {y+\ln \left (\left (-1+x \right ) \left (1+x \right )\right ) x^{3}+7 \ln \left (\left (-1+x \right ) \left (1+x \right )\right ) x y^{2}}{x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
1.882 |
|
|
\[ {}y^{\prime } = \frac {y^{3} x \,{\mathrm e}^{2 x^{2}}}{y \,{\mathrm e}^{x^{2}}+1} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.201 |
|
|
\[ {}y^{\prime } = \frac {y-\ln \left (\frac {1+x}{-1+x}\right ) x^{3}+\ln \left (\frac {1+x}{-1+x}\right ) x y^{2}}{x} \] |
1 |
1 |
1 |
riccati, exactByInspection, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
2.185 |
|
|
\[ {}y^{\prime } = \frac {y+{\mathrm e}^{\frac {1+x}{-1+x}} x^{3}+{\mathrm e}^{\frac {1+x}{-1+x}} x y^{2}}{x} \] |
1 |
1 |
1 |
riccati, exactByInspection, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
3.069 |
|
|
\[ {}y^{\prime } = \frac {x y-y-{\mathrm e}^{1+x} x^{3}+{\mathrm e}^{1+x} x y^{2}}{\left (-1+x \right ) x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
2.608 |
|
|
\[ {}y^{\prime } = \frac {-x^{2}+1+4 x^{3} \sqrt {x^{2}-2 x +1+8 y}}{4 x +4} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.624 |
|
|
\[ {}y^{\prime } = \frac {-\sin \left (2 y\right )+\cos \left (2 y\right ) x^{3}+x^{3}}{2 x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
2.125 |
|
|
\[ {}y^{\prime } = \frac {y+x^{3} \sqrt {x^{2}+y^{2}}}{x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.397 |
|
|
\[ {}y^{\prime } = \left (1+y^{2} {\mathrm e}^{-2 b x}+y^{3} {\mathrm e}^{-3 b x}\right ) {\mathrm e}^{b x} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
2.971 |
|
|
\[ {}y^{\prime } = \frac {x +1+2 \sqrt {4 x^{2} y+1}\, x^{3}}{2 x^{3} \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.526 |
|
|
\[ {}y^{\prime } = \frac {y \ln \left (-1+x \right )+x^{4}+x^{3}+x^{2} y^{2}+x y^{2}}{\ln \left (-1+x \right ) x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
1.908 |
|
|
\[ {}y^{\prime } = \frac {y \ln \left (-1+x \right )+{\mathrm e}^{1+x} x^{3}+7 \,{\mathrm e}^{1+x} x y^{2}}{\ln \left (-1+x \right ) x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
2.854 |
|
|
\[ {}y^{\prime } = \left (1+y^{2} {\mathrm e}^{-\frac {4 x}{3}}+y^{3} {\mathrm e}^{-2 x}\right ) {\mathrm e}^{\frac {2 x}{3}} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
3.481 |
|
|
\[ {}y^{\prime } = \left (1+y^{2} {\mathrm e}^{-2 x}+y^{3} {\mathrm e}^{-3 x}\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
2.64 |
|
|
\[ {}y^{\prime } = \frac {x \left (-2 x -2+3 x^{2} \sqrt {x^{2}+3 y}\right )}{3+3 x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.567 |
|
|
\[ {}y^{\prime } = \frac {1}{x \left (x y^{2}+1+x \right ) y} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✗ |
N/A |
0.954 |
|
|
\[ {}y^{\prime } = \frac {2 x \,{\mathrm e}^{x}-2 x -\ln \left (x \right )-1+x^{4} \ln \left (x \right )+x^{4}-2 y x^{2} \ln \left (x \right )-2 x^{2} y+y^{2} \ln \left (x \right )+y^{2}}{{\mathrm e}^{x}-1} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
17.087 |
|
|
\[ {}y^{\prime } = \frac {-{\mathrm e}^{x} y+x y-x^{3} \ln \left (x \right )-x^{3}-x y^{2} \ln \left (x \right )-x y^{2}}{\left (-{\mathrm e}^{x}+x \right ) x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
2.129 |
|
|
\[ {}y^{\prime } = \frac {y \left (1-x +y x^{2} \ln \left (x \right )+x^{3} y-x \ln \left (x \right )-x^{2}\right )}{\left (-1+x \right ) x} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.721 |
|
|
\[ {}y^{\prime } = \frac {y \ln \left (x \right ) x -y+2 x^{5} b +2 x^{3} a y^{2}}{\left (x \ln \left (x \right )-1\right ) x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
2.013 |
|
|
\[ {}y^{\prime } = \frac {\left (\ln \left (y\right )+x +x^{3}+x^{4}\right ) y}{x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.543 |
|
|
\[ {}y^{\prime } = -\frac {\left (-\ln \left (y-1\right )+\ln \left (y+1\right )+2 \ln \left (x \right )\right ) x \left (y+1\right )^{2}}{8} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
4.471 |
|
|
\[ {}y^{\prime } = \frac {\left (-\ln \left (y-1\right )+\ln \left (y+1\right )+2 \ln \left (x \right )\right )^{2} x \left (y+1\right )^{2}}{16} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
4.617 |
|
|
\[ {}y^{\prime } = \frac {\left (-y^{2}+4 x a \right )^{3}}{\left (-y^{2}+4 x a -1\right ) y} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
1.386 |
|
|
\[ {}y^{\prime } = \frac {2 x a +2 a +x^{3} \sqrt {-y^{2}+4 x a}}{\left (1+x \right ) y} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
6.14 |
|
|
\[ {}y^{\prime } = \frac {-\ln \left (x \right )+{\mathrm e}^{\frac {1}{x}}+4 x^{2} y+2 x +2 x y^{2}+2 x^{3}}{\ln \left (x \right )-{\mathrm e}^{\frac {1}{x}}} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
10.543 |
|
|
\[ {}y^{\prime } = -\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-1\right ) y}{1+x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.255 |
|
|
\[ {}y^{\prime } = \frac {x^{2}+2 x +1+2 x^{3} \sqrt {x^{2}+2 x +1-4 y}}{2 x +2} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.58 |
|
|
\[ {}y^{\prime } = \frac {-y a b +b^{2}+a b +b^{2} x -b a \sqrt {x}-a^{2}}{a \left (-a y+b +a +b x -\sqrt {x}\, a \right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.737 |
|
|
\[ {}y^{\prime } = -\frac {y \left (-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}+y x^{2} \ln \left (x \right )+x^{3} y-x \ln \left (x \right )-x^{2}\right )}{\left (-\ln \left (\frac {1}{x}\right )+{\mathrm e}^{x}\right ) x} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.694 |
|
|
\[ {}y^{\prime } = \frac {-x^{2}+x +2+2 x^{3} \sqrt {x^{2}-4 x +4 y}}{2 x +2} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.619 |
|
|
\[ {}y^{\prime } = \frac {3 x^{4}+3 x^{3}+\sqrt {9 x^{4}-4 y^{3}}}{\left (1+x \right ) y^{2}} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
3.016 |
|
|
\[ {}y^{\prime } = -\frac {x^{2}+x +x a +a -2 \sqrt {x^{2}+2 x a +a^{2}+4 y}}{2 \left (1+x \right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
2.842 |
|
|
\[ {}y^{\prime } = \left (1+y^{2} {\mathrm e}^{2 x^{2}}+y^{3} {\mathrm e}^{3 x^{2}}\right ) {\mathrm e}^{-x^{2}} x \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
4.792 |
|
|
\[ {}y^{\prime } = \frac {y \left (-{\mathrm e}^{x}+\ln \left (2 x \right ) x^{2} y-\ln \left (2 x \right ) x \right ) {\mathrm e}^{-x}}{x} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.566 |
|
|
\[ {}y^{\prime } = \frac {x^{3} \left (3 x +3+\sqrt {9 x^{4}-4 y^{3}}\right )}{\left (1+x \right ) y^{2}} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.8 |
|
|
\[ {}y^{\prime } = \frac {\left (18 x^{\frac {3}{2}}+36 y^{2}-12 x^{3} y+x^{6}\right ) \sqrt {x}}{36} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
4.141 |
|
|
\[ {}y^{\prime } = -\frac {y^{3}}{\left (-1+2 y \ln \left (x \right )-y\right ) x} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
2.492 |
|
|
\[ {}y^{\prime } = \frac {2 a}{y+2 a y^{4}-16 a^{2} x y^{2}+32 a^{3} x^{2}} \] |
1 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
0.999 |
|
|
\[ {}y^{\prime } = -\frac {y^{3}}{\left (-1+y \ln \left (x \right )-y\right ) x} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
2.178 |
|
|
\[ {}y^{\prime } = \frac {-\ln \left (x \right )+2 \ln \left (2 x \right ) x y+\ln \left (2 x \right )+\ln \left (2 x \right ) y^{2}+\ln \left (2 x \right ) x^{2}}{\ln \left (x \right )} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
3.72 |
|
|
\[ {}y^{\prime } = -\frac {y a b -b c +b^{2} x +b a \sqrt {x}-a^{2}}{a \left (a y-c +b x +\sqrt {x}\, a \right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.285 |
|
|
\[ {}y^{\prime } = \frac {\left (2 x +2+y\right ) y}{\left (\ln \left (y\right )+2 x -1\right ) \left (1+x \right )} \] |
1 |
0 |
2 |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
1.342 |
|
|
\[ {}y^{\prime } = \frac {\left (x^{3}+3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x} \] |
1 |
1 |
1 |
exactByInspection |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.286 |
|
|
\[ {}y^{\prime } = \frac {y \left (x -y\right )}{x \left (x -y^{3}\right )} \] |
1 |
1 |
3 |
exactByInspection |
[_rational] |
✓ |
✓ |
0.972 |
|
|
\[ {}y^{\prime } = \frac {\left (2 y^{\frac {3}{2}}-3 \,{\mathrm e}^{x}\right )^{3} {\mathrm e}^{x}}{4 \left (2 y^{\frac {3}{2}}-3 \,{\mathrm e}^{x}+2\right ) \sqrt {y}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
2.519 |
|
|
\[ {}y^{\prime } = \frac {1+2 y}{x \left (-2+x y^{2}+2 x y^{3}\right )} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.129 |
|
|
\[ {}y^{\prime } = \frac {-x^{2}-x -x a -a +2 x^{3} \sqrt {x^{2}+2 x a +a^{2}+4 y}}{2 x +2} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
1.704 |
|
|
\[ {}y^{\prime } = \frac {2 x \sin \left (x \right )-\ln \left (2 x \right )+\ln \left (2 x \right ) x^{4}-2 \ln \left (2 x \right ) x^{2} y+\ln \left (2 x \right ) y^{2}}{\sin \left (x \right )} \] |
1 |
0 |
0 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
❇ |
N/A |
81.822 |
|
|
\[ {}y^{\prime } = \frac {\left (-\ln \left (y\right ) x -\ln \left (y\right )+x^{3}\right ) y}{1+x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.288 |
|
|
\[ {}y^{\prime } = \frac {\left (2 y \ln \left (x \right )-1\right )^{3}}{\left (-1+2 y \ln \left (x \right )-y\right ) x} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
2.694 |
|
|
\[ {}y^{\prime } = \frac {2 x^{2}+2 x +x^{4}-2 x^{2} y-1+y^{2}}{1+x} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
2.947 |
|
|
\[ {}y^{\prime } = \frac {x \left (-1+x -2 x y+2 x^{3}\right )}{x^{2}-y} \] |
1 |
0 |
1 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.118 |
|
|
\[ {}y^{\prime } = \frac {2 a}{-x^{2} y+2 a y^{4} x^{2}-16 a^{2} x y^{2}+32 a^{3}} \] |
1 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
2.482 |
|
|
\[ {}y^{\prime } = \frac {1+2 y}{x \left (-2+x y+2 x y^{2}\right )} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.319 |
|
|
\[ {}y^{\prime } = \frac {x +y^{4}-2 x^{2} y^{2}+x^{4}}{y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.194 |
|
|
\[ {}y^{\prime } = \frac {\left (a y^{2}+b \,x^{2}\right )^{3} x}{a^{\frac {5}{2}} \left (a y^{2}+b \,x^{2}+a \right ) y} \] |
1 |
0 |
1 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
2.455 |
|
|
\[ {}y^{\prime } = -\frac {\cos \left (y\right ) \left (x -\cos \left (y\right )+1\right )}{\left (x \sin \left (y\right )-1\right ) \left (1+x \right )} \] |
1 |
0 |
2 |
unknown |
unknown |
✗ |
N/A |
9.466 |
|
|
\[ {}y^{\prime } = -\frac {i \left (8 i x +16 y^{4}+8 x^{2} y^{2}+x^{4}\right )}{32 y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.939 |
|
|
\[ {}y^{\prime } = \frac {x}{-y+x^{4}+2 x^{2} y^{2}+y^{4}} \] |
1 |
0 |
3 |
unknown |
[_rational] |
✗ |
N/A |
1.28 |
|
|
\[ {}y^{\prime } = \frac {\left (-1+y \ln \left (x \right )\right )^{3}}{\left (-1+y \ln \left (x \right )-y\right ) x} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
2.63 |
|
|
\[ {}y^{\prime } = -\frac {i \left (i x +x^{4}+2 x^{2} y^{2}+y^{4}\right )}{y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.878 |
|
|
\[ {}y^{\prime } = -\frac {y \left (\tan \left (x \right )+\ln \left (2 x \right ) x -\ln \left (2 x \right ) x^{2} y\right )}{x \tan \left (x \right )} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
37.084 |
|
|
\[ {}y^{\prime } = \frac {y \left (x +y\right )}{x \left (x +y^{3}\right )} \] |
1 |
1 |
3 |
exactByInspection |
[_rational] |
✓ |
✓ |
1.083 |
|
|
\[ {}y^{\prime } = \frac {\left (x -y\right )^{2} \left (x +y\right )^{2} x}{y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
1.149 |
|
|
\[ {}y^{\prime } = \frac {\left (x^{2}+3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x} \] |
1 |
1 |
1 |
exactByInspection |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.331 |
|
|
\[ {}y^{\prime } = \frac {\left (\ln \left (y\right ) x +\ln \left (y\right )+x^{4}\right ) y}{x \left (1+x \right )} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
2.066 |
|
|
\[ {}y^{\prime } = \frac {\cos \left (y\right ) \left (\cos \left (y\right ) x^{3}-x -1\right )}{\left (x \sin \left (y\right )-1\right ) \left (1+x \right )} \] |
1 |
0 |
2 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
43.079 |
|
|
\[ {}y^{\prime } = \frac {\left (x +1+x^{4} \ln \left (y\right )\right ) y \ln \left (y\right )}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
1.426 |
|
|
\[ {}y^{\prime } = \frac {x y+x^{3}+x y^{2}+y^{3}}{x^{2}} \] |
1 |
1 |
1 |
abelFirstKind, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Abel] |
✓ |
✓ |
4.526 |
|
|
\[ {}y^{\prime } = \frac {y^{\frac {3}{2}}}{y^{\frac {3}{2}}+x^{2}-2 x y+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
1.806 |
|
|
\[ {}y^{\prime } = \frac {2 x^{3} y+x^{6}+x^{2} y^{2}+y^{3}}{x^{4}} \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
3.535 |
|
|
\[ {}y^{\prime } = \frac {-4 x y+x^{3}+2 x^{2}-4 x -8}{-8 y+2 x^{2}+4 x -8} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.186 |
|
|
\[ {}y^{\prime } = \frac {\left (2 x +2+x^{3} y\right ) y}{\left (\ln \left (y\right )+2 x -1\right ) \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
1.496 |
|
|
\[ {}y^{\prime } = -\frac {i \left (54 i x^{2}+81 y^{4}+18 x^{4} y^{2}+x^{8}\right ) x}{243 y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
2.324 |
|
|
\[ {}y^{\prime } = \frac {\left (x y^{2}+1\right )^{3}}{x^{4} \left (x y^{2}+1+x \right ) y} \] |
1 |
0 |
5 |
unknown |
[_rational] |
✗ |
N/A |
1.355 |
|
|
\[ {}y^{\prime } = \frac {-4 x y-x^{3}+4 x^{2}-4 x +8}{8 y+2 x^{2}-8 x +8} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.204 |
|
|
\[ {}y^{\prime } = -\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-x \right ) y}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.484 |
|
|
\[ {}y^{\prime } = \frac {\left (\ln \left (y\right ) x +\ln \left (y\right )+x \right ) y}{x \left (1+x \right )} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.83 |
|
|
\[ {}y^{\prime } = \frac {\left (-\ln \left (y\right ) x -\ln \left (y\right )+x^{4}\right ) y}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.325 |
|
|
\[ {}y^{\prime } = \frac {y \left (-1-\ln \left (\frac {\left (-1+x \right ) \left (1+x \right )}{x}\right )+\ln \left (\frac {\left (-1+x \right ) \left (1+x \right )}{x}\right ) x y\right )}{x} \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
12.734 |
|
|
\[ {}y^{\prime } = \frac {y \left (-\ln \left (x \right )-x \ln \left (\frac {\left (-1+x \right ) \left (1+x \right )}{x}\right )+\ln \left (\frac {\left (-1+x \right ) \left (1+x \right )}{x}\right ) x^{2} y\right )}{x \ln \left (x \right )} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
5.017 |
|
|
\[ {}y^{\prime } = \frac {-8 x y-x^{3}+2 x^{2}-8 x +32}{32 y+4 x^{2}-8 x +32} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.227 |
|
|
\[ {}y^{\prime } = \frac {y \left (y+1\right )}{x \left (-y-1+x y\right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.378 |
|
|
\[ {}y^{\prime } = -\frac {i \left (16 i x^{2}+16 y^{4}+8 x^{4} y^{2}+x^{8}\right ) x}{32 y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
2.345 |
|
|
\[ {}y^{\prime } = \frac {2 y^{6}}{y^{3}+2+16 x y^{2}+32 y^{4} x^{2}} \] |
1 |
0 |
3 |
unknown |
[_rational] |
✗ |
N/A |
1.35 |
|
|
\[ {}y^{\prime } = \frac {-4 y a x -a^{2} x^{3}-2 a b \,x^{2}-4 x a +8}{8 y+2 x^{2} a +4 b x +8} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.213 |
|
|
\[ {}y^{\prime } = \frac {\left (x +1+\ln \left (y\right ) x \right ) \ln \left (y\right ) y}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[‘x=_G(y,y’)‘] |
✗ |
N/A |
1.279 |
|
|
\[ {}y^{\prime } = \frac {x y+x +y^{2}}{\left (-1+x \right ) \left (x +y\right )} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.531 |
|
|
\[ {}y^{\prime } = \frac {-4 x y-x^{3}-2 x^{2} a -4 x +8}{8 y+2 x^{2}+4 x a +8} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.769 |
|
|
\[ {}y^{\prime } = \frac {x -y+\sqrt {y}}{x -y+\sqrt {y}+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
1.77 |
|
|
\[ {}y^{\prime } = \frac {y \left (-\ln \left (\frac {1}{x}\right )-\ln \left (\frac {x^{2}+1}{x}\right ) x +\ln \left (\frac {x^{2}+1}{x}\right ) x^{2} y\right )}{x \ln \left (\frac {1}{x}\right )} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
5.582 |
|
|
\[ {}y^{\prime } = \frac {y \left (y+1\right )}{x \left (-y-1+y^{4} x \right )} \] |
1 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.521 |
|
|
\[ {}y^{\prime } = \frac {-3 x^{2} y+1+y^{2} x^{6}+y^{3} x^{9}}{x^{3}} \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
3.98 |
|
|
\[ {}y^{\prime } = \frac {x^{3} y+x^{3}+x y^{2}+y^{3}}{\left (-1+x \right ) x^{3}} \] |
1 |
1 |
1 |
abelFirstKind, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘], _rational, _Abel] |
✓ |
✓ |
5.474 |
|
|
\[ {}y^{\prime } = \frac {x y+y+x \sqrt {x^{2}+y^{2}}}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
1.931 |
|
|
\[ {}y^{\prime } = \frac {\left (x^{4}+x^{3}+x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x} \] |
1 |
1 |
1 |
exactByInspection |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
1.605 |
|
|
\[ {}y^{\prime } = \frac {y \left (-\tanh \left (\frac {1}{x}\right )-\ln \left (\frac {x^{2}+1}{x}\right ) x +\ln \left (\frac {x^{2}+1}{x}\right ) x^{2} y\right )}{x \tanh \left (\frac {1}{x}\right )} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
36.352 |
|
|
\[ {}y^{\prime } = -\frac {y \left (\tanh \left (x \right )+\ln \left (2 x \right ) x -\ln \left (2 x \right ) x^{2} y\right )}{x \tanh \left (x \right )} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
14.089 |
|
|
\[ {}y^{\prime } = \frac {-\sinh \left (x \right )+\ln \left (x \right ) x^{2}+2 y \ln \left (x \right ) x +\ln \left (x \right )+y^{2} \ln \left (x \right )}{\sinh \left (x \right )} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
41.107 |
|
|
\[ {}y^{\prime } = -\frac {\ln \left (x \right )-\sinh \left (x \right ) x^{2}-2 \sinh \left (x \right ) x y-\sinh \left (x \right )-\sinh \left (x \right ) y^{2}}{\ln \left (x \right )} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
79.651 |
|
|
\[ {}y^{\prime } = \frac {y \ln \left (x \right )+\cosh \left (x \right ) x a y^{2}+\cosh \left (x \right ) x^{3} b}{x \ln \left (x \right )} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
4.746 |
|
|
\[ {}y^{\prime } = \frac {x \left (-x -1+x^{2}-2 x^{2} y+2 x^{4}\right )}{\left (x^{2}-y\right ) \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
2.211 |
|
|
\[ {}y^{\prime } = -\frac {y \left (\ln \left (-1+x \right )+\coth \left (1+x \right ) x -\coth \left (1+x \right ) x^{2} y\right )}{x \ln \left (-1+x \right )} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
46.734 |
|
|
\[ {}y^{\prime } = -\frac {\ln \left (-1+x \right )-\coth \left (1+x \right ) x^{2}-2 \coth \left (1+x \right ) x y-\coth \left (1+x \right )-\coth \left (1+x \right ) y^{2}}{\ln \left (-1+x \right )} \] |
1 |
1 |
0 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
93.694 |
|
|
\[ {}y^{\prime } = \frac {2 x \ln \left (\frac {1}{-1+x}\right )-\coth \left (\frac {1+x}{-1+x}\right )+\coth \left (\frac {1+x}{-1+x}\right ) y^{2}-2 \coth \left (\frac {1+x}{-1+x}\right ) x^{2} y+\coth \left (\frac {1+x}{-1+x}\right ) x^{4}}{\ln \left (\frac {1}{-1+x}\right )} \] |
1 |
0 |
0 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
❇ |
N/A |
93.109 |
|
|
\[ {}y^{\prime } = \frac {2 x^{2} \cosh \left (\frac {1}{-1+x}\right )-2 x \cosh \left (\frac {1}{-1+x}\right )-1+y^{2}-2 x^{2} y+x^{4}-x +x y^{2}-2 x^{3} y+x^{5}}{\left (-1+x \right ) \cosh \left (\frac {1}{-1+x}\right )} \] |
1 |
0 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
N/A |
87.348 |
|
|
\[ {}y^{\prime } = \frac {y \left (-\cosh \left (\frac {1}{1+x}\right ) x +\cosh \left (\frac {1}{1+x}\right )-x +x^{2} y-x^{2}+x^{3} y\right )}{x \left (-1+x \right ) \cosh \left (\frac {1}{1+x}\right )} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
18.688 |
|
|
\[ {}y^{\prime } = -\frac {y \left (x y+1\right )}{x \left (x y+1-y\right )} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.964 |
|
|
\[ {}y^{\prime } = \frac {y}{x \left (-1+y+x^{2} y^{3}+y^{4} x^{3}\right )} \] |
1 |
1 |
5 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
2.698 |
|
|
\[ {}y^{\prime } = \frac {x^{3}+3 x^{2} a +3 x \,a^{2}+a^{3}+x y^{2}+a y^{2}+y^{3}}{\left (x +a \right )^{3}} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, _Abel] |
✓ |
✓ |
8.093 |
|
|
\[ {}y^{\prime } = \frac {y^{3} x \,{\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+3 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+9 y} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
3.549 |
|
|
\[ {}y^{\prime } = \frac {y \left (-1-\cosh \left (\frac {1+x}{-1+x}\right ) x +\cosh \left (\frac {1+x}{-1+x}\right ) x^{2} y-\cosh \left (\frac {1+x}{-1+x}\right ) x^{2}+\cosh \left (\frac {1+x}{-1+x}\right ) x^{3} y\right )}{x} \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
34.423 |
|
|
\[ {}y^{\prime } = \frac {\left (1+x +y\right ) y}{\left (2 y^{3}+y+x \right ) \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
2.457 |
|
|
\[ {}y^{\prime } = \frac {y \left (-1-{\mathrm e}^{\frac {1+x}{-1+x}} x +x^{2} {\mathrm e}^{\frac {1+x}{-1+x}} y-x^{2} {\mathrm e}^{\frac {1+x}{-1+x}}+x^{3} {\mathrm e}^{\frac {1+x}{-1+x}} y\right )}{x} \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
5.467 |
|
|
\[ {}y^{\prime } = \frac {-b^{3}+6 b^{2} x -12 b \,x^{2}+8 x^{3}-4 b y^{2}+8 x y^{2}+8 y^{3}}{\left (2 x -b \right )^{3}} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, _Abel] |
✓ |
✓ |
8.224 |
|
|
\[ {}y^{\prime } = \frac {\left (y \,{\mathrm e}^{-\frac {x^{2}}{4}} x +2+2 y^{2} {\mathrm e}^{-\frac {x^{2}}{2}}+2 y^{3} {\mathrm e}^{-\frac {3 x^{2}}{4}}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[_Abel] |
✓ |
✓ |
29.868 |
|
|
\[ {}y^{\prime } = -\frac {-\frac {1}{x}-\textit {\_F1} \left (y+\frac {1}{x}\right )}{x} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
2.216 |
|
|
|
|||||||||
|
\[ {}y^{\prime } = \frac {\textit {\_F1} \left (y^{2}-2 \ln \left (x \right )\right )}{\sqrt {y^{2}}\, x} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
2.939 |
|
|
\[ {}y^{\prime } = \frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+\cos \left (2 y\right ) x^{4}+x^{4}}{2 x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
11.148 |
|
|
\[ {}y^{\prime } = \frac {x y+y+x^{4} \sqrt {x^{2}+y^{2}}}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.475 |
|
|
\[ {}y^{\prime } = \frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+x \cos \left (2 y\right )+x}{2 x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
7.264 |
|
|
\[ {}y^{\prime } = -\frac {1}{-x -\textit {\_F1} \left (y-\ln \left (x \right )\right ) y \,{\mathrm e}^{y}} \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
3.126 |
|
|
\[ {}y^{\prime } = \frac {\left (1+2 y\right ) \left (y+1\right )}{x \left (-2 y-2+x +2 x y\right )} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
5.21 |
|
|
\[ {}y^{\prime } = \frac {-125+300 x -240 x^{2}+64 x^{3}-80 y^{2}+64 x y^{2}+64 y^{3}}{\left (4 x -5\right )^{3}} \] |
1 |
1 |
1 |
abelFirstKind, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, _Abel] |
✓ |
✓ |
8.832 |
|
|
\[ {}y^{\prime } = \frac {x +y+y^{2}-2 y \ln \left (x \right ) x +x^{2} \ln \left (x \right )^{2}}{x} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
2.726 |
|
|
\[ {}y^{\prime } = \frac {x^{3} {\mathrm e}^{y}+x^{4}+{\mathrm e}^{y} y-{\mathrm e}^{y} \ln \left ({\mathrm e}^{y}+x \right )+x y-\ln \left ({\mathrm e}^{y}+x \right ) x +x}{x^{2}} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
3.885 |
|
|
\[ {}y^{\prime } = \frac {x^{2}}{2}+\sqrt {x^{3}-6 y}+x^{2} \sqrt {x^{3}-6 y}+x^{3} \sqrt {x^{3}-6 y} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
34.612 |
|
|
\[ {}y^{\prime } = \frac {\left (-\sqrt {a}\, x^{3}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
3.765 |
|
|
\[ {}y^{\prime } = \frac {y \left (-3 x^{3} y-3+y^{2} x^{7}\right )}{x \left (x^{3} y+1\right )} \] |
1 |
0 |
2 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
2.653 |
|
|
\[ {}y^{\prime } = \frac {\left (3+y\right )^{3} {\mathrm e}^{\frac {9 x^{2}}{2}} x \,{\mathrm e}^{\frac {3 x^{2}}{2}} {\mathrm e}^{-3 x^{2}}}{243 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+243 y} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
4.293 |
|
|
\[ {}y^{\prime } = \frac {\left (x -y\right )^{3} \left (x +y\right )^{3} x}{\left (-y^{2}+x^{2}-1\right ) y} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
2.537 |
|
|
\[ {}y^{\prime } = \frac {-2 \cos \left (y\right )+x^{3} \cos \left (2 y\right ) \ln \left (x \right )+x^{3} \ln \left (x \right )}{2 \sin \left (y\right ) \ln \left (x \right ) x} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
48.375 |
|
|
\[ {}y^{\prime } = \frac {y}{x \left (-1+x y+x y^{3}+y^{4} x \right )} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
4.684 |
|
|
\[ {}y^{\prime } = -\frac {2 x}{3}+\sqrt {x^{2}+3 y}+x^{2} \sqrt {x^{2}+3 y}+x^{3} \sqrt {x^{2}+3 y} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
2.898 |
|
|
\[ {}y^{\prime } = \frac {-2 \cos \left (y\right )+x^{2} \cos \left (2 y\right ) \ln \left (x \right )+\ln \left (x \right ) x^{2}}{2 \sin \left (y\right ) \ln \left (x \right ) x} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
47.844 |
|
|
\[ {}y^{\prime } = \frac {y \left (x y+1\right )}{x \left (-x y-1+y^{4} x^{3}\right )} \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
2.575 |
|
|
\[ {}y^{\prime } = \frac {\left (4 \,{\mathrm e}^{-x^{2}}-4 x^{2} {\mathrm e}^{-x^{2}}+4 y^{2}-4 x^{2} {\mathrm e}^{-x^{2}} y+x^{4} {\mathrm e}^{-2 x^{2}}\right ) x}{4} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
4.315 |
|
|
\[ {}y^{\prime } = \frac {y \left (x +y\right )}{x \left (x +y+y^{3}+y^{4}\right )} \] |
1 |
1 |
1 |
exactByInspection |
[_rational] |
✓ |
✓ |
2.342 |
|
|
\[ {}y^{\prime } = \frac {y \left (x^{3}+x^{2} y+y^{2}\right )}{x^{2} \left (-1+x \right ) \left (x +y\right )} \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
7.444 |
|
|
\[ {}y^{\prime } = \frac {\left (\left (x^{2}+1\right )^{\frac {3}{2}} x^{2}+\left (x^{2}+1\right )^{\frac {3}{2}}+y^{2} \left (x^{2}+1\right )^{\frac {3}{2}}+x^{2} y^{3}+y^{3}\right ) x}{\left (x^{2}+1\right )^{3}} \] |
1 |
0 |
1 |
abelFirstKind |
[_Abel] |
✗ |
N/A |
36.623 |
|
|
\[ {}y^{\prime } = \frac {\left (3 x y^{2}+x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
2.453 |
|
|
\[ {}y^{\prime } = -\frac {-y+x^{3} \sqrt {x^{2}+y^{2}}-x^{2} \sqrt {x^{2}+y^{2}}\, y}{x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.288 |
|
|
\[ {}y^{\prime } = \frac {\left (1+2 y\right ) \left (y+1\right )}{x \left (-2 y-2+x y^{3}+2 y^{4} x \right )} \] |
1 |
1 |
3 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
7.243 |
|
|
\[ {}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{3}+2 x^{5} \sqrt {4 x^{2} y+1}+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.35 |
|
|
\[ {}y^{\prime } = \frac {y \left (x -y\right )}{x \left (x -y-y^{3}-y^{4}\right )} \] |
1 |
1 |
1 |
exactByInspection |
[_rational] |
✓ |
✓ |
2.397 |
|
|
\[ {}y^{\prime } = \frac {2 a +\sqrt {-y^{2}+4 x a}+x^{2} \sqrt {-y^{2}+4 x a}+x^{3} \sqrt {-y^{2}+4 x a}}{y} \] |
1 |
0 |
1 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
25.428 |
|
|
\[ {}y^{\prime } = \frac {\left (1+x +y\right ) y}{\left (y^{4}+y^{3}+y^{2}+x \right ) \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
2.424 |
|
|
\[ {}y^{\prime } = -\frac {-y+x^{4} \sqrt {x^{2}+y^{2}}-x^{3} \sqrt {x^{2}+y^{2}}\, y}{x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.367 |
|
|
\[ {}y^{\prime } = \frac {\left (x^{4}+3 x y^{2}+3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.02 |
|
|
\[ {}y^{\prime } = -\frac {1}{-\left (y^{3}\right )^{\frac {2}{3}} x -\textit {\_F1} \left (y^{3}-3 \ln \left (x \right )\right ) \left (y^{3}\right )^{\frac {1}{3}} x} \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
2.438 |
|
|
\[ {}y^{\prime } = \frac {y \left (x -y\right ) \left (y+1\right )}{x \left (x y+x -y\right )} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
3.013 |
|
|
\[ {}y^{\prime } = -\frac {1}{-\ln \left (x \right ) \left (y^{3}\right )^{\frac {2}{3}}-\textit {\_F1} \left (y^{3}+3 \,\operatorname {expIntegral}_{1}\left (-\ln \left (x \right )\right )\right ) \ln \left (x \right ) \left (y^{3}\right )^{\frac {1}{3}}} \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
3.037 |
|
|
\[ {}y^{\prime } = \frac {30 x^{3}+25 \sqrt {x}+25 y^{2}-20 x^{3} y-100 y \sqrt {x}+4 x^{6}+40 x^{\frac {7}{2}}+100 x}{25 x} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
33.944 |
|
|
\[ {}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x^{2}\right ) {\mathrm e}^{\frac {y}{x}}}{x} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.023 |
|
|
\[ {}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x^{3}\right ) {\mathrm e}^{\frac {y}{x}}}{x} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
4.204 |
|
|
\[ {}y^{\prime } = \frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{\frac {3}{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{\frac {3}{2}} b \,x^{2}+a^{\frac {5}{2}} y^{4}}{a \,x^{2} y} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
4.005 |
|
|
\[ {}y^{\prime } = \frac {y+x^{2} \ln \left (x \right )^{3}+2 x^{2} \ln \left (x \right )^{2} y+x^{2} \ln \left (x \right ) y^{2}}{x \ln \left (x \right )} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
10.425 |
|
|
\[ {}y^{\prime } = \frac {y+x^{3} \ln \left (x \right )^{3}+2 x^{3} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right ) y^{2}}{x \ln \left (x \right )} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.578 |
|
|
\[ {}y^{\prime } = \frac {y \left (x +y\right ) \left (y+1\right )}{x \left (x y+x +y\right )} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
2.26 |
|
|
\[ {}y^{\prime } = \frac {3 x^{3}+\sqrt {-9 x^{4}+4 y^{3}}+x^{2} \sqrt {-9 x^{4}+4 y^{3}}+x^{3} \sqrt {-9 x^{4}+4 y^{3}}}{y^{2}} \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
6.203 |
|
|
\[ {}y^{\prime } = \frac {1}{-x +\left (\frac {1}{y}+1\right ) x +\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2}-\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2} \left (\frac {1}{y}+1\right )} \] |
1 |
0 |
2 |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.708 |
|
|
\[ {}y^{\prime } = \frac {x}{2}+\frac {1}{2}+\sqrt {x^{2}+2 x +1-4 y}+x^{2} \sqrt {x^{2}+2 x +1-4 y}+x^{3} \sqrt {x^{2}+2 x +1-4 y} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
3.155 |
|
|
\[ {}y^{\prime } = \frac {\cosh \left (x \right )}{\sinh \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sinh \left (x \right )\right )\right ) \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
4.316 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+1+\sqrt {x^{2}-4 x +4 y}+x^{2} \sqrt {x^{2}-4 x +4 y}+x^{3} \sqrt {x^{2}-4 x +4 y} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
3.218 |
|
|
\[ {}y^{\prime } = \frac {1}{\sin \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sin \left (x \right )\right )+\ln \left (\cos \left (x \right )+1\right )\right ) \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
5.882 |
|
|
\[ {}y^{\prime } = \frac {b^{3}+y^{2} b^{3}+2 y b^{2} a x +x^{2} b \,a^{2}+y^{3} b^{3}+3 y^{2} b^{2} a x +3 y b \,a^{2} x^{2}+a^{3} x^{3}}{b^{3}} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Abel] |
✓ |
✓ |
3.258 |
|
|
\[ {}y^{\prime } = \frac {\alpha ^{3}+y^{2} \alpha ^{3}+2 y \alpha ^{2} \beta x +\alpha \,\beta ^{2} x^{2}+y^{3} \alpha ^{3}+3 y^{2} \alpha ^{2} \beta x +3 y \alpha \,\beta ^{2} x^{2}+\beta ^{3} x^{3}}{\alpha ^{3}} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Abel] |
✓ |
✓ |
3.256 |
|
|
\[ {}y^{\prime } = \frac {14 x y+12+2 x +y^{3} x^{3}+6 x^{2} y^{2}}{x^{2} \left (x y+2+x \right )} \] |
1 |
1 |
2 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
3.396 |
|
|
\[ {}y^{\prime } = \frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+x^{2} \ln \left (x \right )^{2}+2 x^{2} \ln \left (y\right ) \ln \left (x \right )+x^{2} \ln \left (y\right )^{2}\right )}{x} \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
3.891 |
|
|
\[ {}y^{\prime } = \frac {y \left (\ln \left (y\right )-1+\ln \left (x \right )+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (y\right ) \ln \left (x \right )+x^{3} \ln \left (y\right )^{2}\right )}{x} \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
3.783 |
|
|
\[ {}y^{\prime } = -\frac {\left (-\frac {1}{x}-\textit {\_F1} \left (y^{2}-2 x \right )\right ) x}{\sqrt {y^{2}}} \] |
1 |
0 |
2 |
unknown |
[NONE] |
✗ |
N/A |
3.537 |
|
|
\[ {}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+\sqrt {x^{2}-2 x +1+8 y}+x^{2} \sqrt {x^{2}-2 x +1+8 y}+x^{3} \sqrt {x^{2}-2 x +1+8 y} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
3.575 |
|
|
\[ {}y^{\prime } = \frac {a^{3}+y^{2} a^{3}+2 y a^{2} b x +a \,b^{2} x^{2}+y^{3} a^{3}+3 y^{2} a^{2} b x +3 y a \,b^{2} x^{2}+b^{3} x^{3}}{a^{3}} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _Abel] |
✓ |
✓ |
3.294 |
|
|
\[ {}y^{\prime } = -\frac {-x -\textit {\_F1} \left (y^{2}-2 x \right )}{\sqrt {y^{2}}\, x} \] |
1 |
0 |
2 |
unknown |
[NONE] |
✗ |
N/A |
3.542 |
|
|
\[ {}y^{\prime } = \frac {-\sin \left (2 y\right )+x \cos \left (2 y\right )+\cos \left (2 y\right ) x^{3}+\cos \left (2 y\right ) x^{4}+x +x^{3}+x^{4}}{2 x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
7.437 |
|
|
\[ {}y^{\prime } = -\frac {\left (-\frac {y \,{\mathrm e}^{\frac {1}{x}}}{x}-\textit {\_F1} \left (y \,{\mathrm e}^{\frac {1}{x}}\right )\right ) {\mathrm e}^{-\frac {1}{x}}}{x} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
2.957 |
|
|
\[ {}y^{\prime } = \frac {y+x \sqrt {x^{2}+y^{2}}+x^{3} \sqrt {x^{2}+y^{2}}+x^{4} \sqrt {x^{2}+y^{2}}}{x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.476 |
|
|
\[ {}y^{\prime } = \frac {y \left ({\mathrm e}^{-\frac {x^{2}}{2}} x y+{\mathrm e}^{-\frac {x^{2}}{4}} x +2 y^{2} {\mathrm e}^{-\frac {3 x^{2}}{4}}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2 y \,{\mathrm e}^{-\frac {x^{2}}{4}}+2} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
5.991 |
|
|
\[ {}y^{\prime } = \left (\frac {\ln \left (y-1\right ) y}{\left (1-y\right ) \ln \left (x \right ) x}-\frac {\ln \left (y-1\right )}{\left (1-y\right ) \ln \left (x \right ) x}-f \left (x \right )\right ) \left (1-y\right ) \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.328 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+\sqrt {x^{2}+2 x a +a^{2}+4 y}+x^{2} \sqrt {x^{2}+2 x a +a^{2}+4 y}+x^{3} \sqrt {x^{2}+2 x a +a^{2}+4 y} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
N/A |
3.785 |
|
|
\[ {}y^{\prime } = -\frac {2 x}{3}+1+y^{2}+\frac {2 x^{2} y}{3}+\frac {x^{4}}{9}+y^{3}+x^{2} y^{2}+\frac {x^{4} y}{3}+\frac {x^{6}}{27} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
2.798 |
|
|
\[ {}y^{\prime } = 2 x +1+y^{2}-2 x^{2} y+x^{4}+y^{3}-3 x^{2} y^{2}+3 x^{4} y-x^{6} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
2.392 |
|
|
\[ {}y^{\prime } = \frac {-x +1-2 y+3 x^{2}-2 x^{2} y+2 x^{4}+x^{3}-2 x^{3} y+2 x^{5}}{x^{2}-y} \] |
1 |
0 |
1 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
2.652 |
|
|
\[ {}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x +x^{3}+x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
4.373 |
|
|
\[ {}y^{\prime } = \frac {2 x y^{2}+4 y \ln \left (2 x +1\right ) x +2 \ln \left (2 x +1\right )^{2} x +y^{2}-2+\ln \left (2 x +1\right )^{2}+2 y \ln \left (2 x +1\right )}{2 x +1} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
3.367 |
|
|
\[ {}y^{\prime } = \frac {-30 x^{3} y+12 x^{6}+70 x^{\frac {7}{2}}-30 x^{3}-25 y \sqrt {x}+50 x -25 \sqrt {x}-25}{5 \left (-5 y+2 x^{3}+10 \sqrt {x}-5\right ) x} \] |
1 |
1 |
2 |
exactByInspection |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
5.822 |
|
|
\[ {}y^{\prime } = \frac {1+2 y}{x \left (-2+x +x y^{2}+3 x y^{3}+2 x y+2 y^{4} x \right )} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
5.895 |
|
|
\[ {}y^{\prime } = \frac {\left (-256 x^{2} a +512+512 y^{2}+128 y a \,x^{4}+8 a^{2} x^{8}+512 y^{3}+192 x^{4} a y^{2}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512} \] |
1 |
1 |
1 |
abelFirstKind |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
6.243 |
|
|
\[ {}y^{\prime } = -\frac {-x y-y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
4.088 |
|
|
\[ {}y^{\prime } = -\frac {y^{2} \left (x^{2} y-2 x -2 x y+y\right )}{2 \left (-2+x y-2 y\right ) x} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
5.187 |
|
|
\[ {}y^{\prime } = \frac {-2 x y+2 x^{3}-2 x -y^{3}+3 x^{2} y^{2}-3 x^{4} y+x^{6}}{-y+x^{2}-1} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
2.837 |
|
|
\[ {}y^{\prime } = \frac {1+y^{4}-8 a x y^{2}+16 a^{2} x^{2}+y^{6}-12 y^{4} a x +48 y^{2} a^{2} x^{2}-64 a^{3} x^{3}}{y} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
2.915 |
|
|
\[ {}y^{\prime } = -\frac {-x y-y+\sqrt {x^{2}+y^{2}}\, x^{2}-x \sqrt {x^{2}+y^{2}}\, y}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.691 |
|
|
\[ {}y^{\prime } = -\frac {2 a}{-y-2 a -2 a y^{4}+16 a^{2} x y^{2}-32 a^{3} x^{2}-2 a y^{6}+24 y^{4} a^{2} x -96 y^{2} a^{3} x^{2}+128 a^{4} x^{3}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.668 |
|
|
\[ {}y^{\prime } = \frac {-18 x y-6 x^{3}-18 x +27 y^{3}+27 x^{2} y^{2}+9 x^{4} y+x^{6}}{27 y+9 x^{2}+27} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
2.931 |
|
|
\[ {}y^{\prime } = -\frac {\left (-108 x^{\frac {3}{2}}-216-216 y^{2}+72 x^{3} y-6 x^{6}-216 y^{3}+108 x^{3} y^{2}-18 x^{6} y+x^{9}\right ) \sqrt {x}}{216} \] |
1 |
1 |
1 |
abelFirstKind |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
37.031 |
|
|
\[ {}y^{\prime } = \frac {\left (a^{3}+y^{4} a^{3}+2 y^{2} a^{2} b \,x^{2}+a \,x^{4} b^{2}+y^{6} a^{3}+3 y^{4} a^{2} b \,x^{2}+3 y^{2} a \,b^{2} x^{4}+b^{3} x^{6}\right ) x}{a^{\frac {7}{2}} y} \] |
1 |
0 |
1 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
5.099 |
|
|
\[ {}y^{\prime } = -\frac {\left (-1-y^{4}+2 x^{2} y^{2}-x^{4}-y^{6}+3 y^{4} x^{2}-3 x^{4} y^{2}+x^{6}\right ) x}{y} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
2.701 |
|
|
\[ {}y^{\prime } = -\frac {i \left (32 i x +64+64 y^{4}+32 x^{2} y^{2}+4 x^{4}+64 y^{6}+48 y^{4} x^{2}+12 x^{4} y^{2}+x^{6}\right )}{128 y} \] |
1 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
8.035 |
|
|
\[ {}y^{\prime } = \frac {2 x^{2}-4 x^{3} y+1+x^{4} y^{2}+x^{6} y^{3}-3 y^{2} x^{5}+3 x^{4} y-x^{3}}{x^{4}} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[_rational, _Abel] |
✓ |
✓ |
10.175 |
|
|
\[ {}y^{\prime } = \frac {y a^{2} x +a +x \,a^{2}+y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 y a x +1}{a^{2} x^{2} \left (y a x +1+x a \right )} \] |
1 |
1 |
2 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
3.69 |
|
|
\[ {}y^{\prime } = \frac {6 x^{2} y-2 x +1-5 x^{3} y^{2}-2 x y+y^{3} x^{4}}{x^{2} \left (x^{2} y-x +1\right )} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
4.778 |
|
|
\[ {}y^{\prime } = -\frac {\left (-8-8 y^{3}+24 y^{\frac {3}{2}} {\mathrm e}^{x}-18 \,{\mathrm e}^{2 x}-8 y^{\frac {9}{2}}+36 y^{3} {\mathrm e}^{x}-54 y^{\frac {3}{2}} {\mathrm e}^{2 x}+27 \,{\mathrm e}^{3 x}\right ) {\mathrm e}^{x}}{8 \sqrt {y}} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
N/A |
5.087 |
|
|
\[ {}y^{\prime } = \frac {x}{-y+1+y^{4}+2 x^{2} y^{2}+x^{4}+y^{6}+3 y^{4} x^{2}+3 x^{4} y^{2}+x^{6}} \] |
1 |
0 |
7 |
unknown |
[_rational] |
✗ |
N/A |
2.939 |
|
|
\[ {}y^{\prime } = \frac {y^{2} \left (-2 y+2 x^{2}+2 x^{2} y+x^{4} y\right )}{x^{3} \left (x^{2}-y+x^{2} y\right )} \] |
1 |
0 |
2 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
3.371 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{-\frac {2}{-y^{2}+x^{2}-1}}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{-\frac {2}{-y^{2}+x^{2}-1}}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
6.133 |
|
|
\[ {}y^{\prime } = \frac {6 x +x^{3}+x^{3} y^{2}+4 x^{2} y+y^{3} x^{3}+6 x^{2} y^{2}+12 x y+8}{x^{3}} \] |
1 |
1 |
1 |
abelFirstKind |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
3.356 |
|
|
\[ {}y^{\prime } = -\frac {i \left (i x +1+x^{4}+2 x^{2} y^{2}+y^{4}+x^{6}+3 x^{4} y^{2}+3 y^{4} x^{2}+y^{6}\right )}{y} \] |
1 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
7.515 |
|
|
\[ {}y^{\prime } = \frac {\left (-256 a \,x^{2} y-32 a^{2} x^{6}-256 x^{2} a +512 y^{3}+192 x^{4} a y^{2}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512 y+64 a \,x^{4}+512} \] |
1 |
0 |
2 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
4.528 |
|
|
\[ {}y^{\prime } = \frac {x +1+y^{4}-2 x^{2} y^{2}+x^{4}+y^{6}-3 y^{4} x^{2}+3 x^{4} y^{2}-x^{6}}{y} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
2.694 |
|
|
\[ {}y^{\prime } = \frac {\left (-108 y x^{\frac {3}{2}}+18 x^{\frac {9}{2}}-108 x^{\frac {3}{2}}-216 y^{3}+108 x^{3} y^{2}-18 x^{6} y+x^{9}\right ) \sqrt {x}}{-216 y+36 x^{3}-216} \] |
1 |
0 |
2 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
31.148 |
|
|
\[ {}y^{\prime } = \frac {32 x^{5} y+8 x^{3}+32 x^{5}+64 x^{6} y^{3}+48 x^{4} y^{2}+12 x^{2} y+1}{16 x^{6} \left (4 x^{2} y+1+4 x^{2}\right )} \] |
1 |
0 |
2 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
4.204 |
|
|
\[ {}y^{\prime } = \frac {32 x^{5}+64 x^{6}+64 y^{2} x^{6}+32 x^{4} y+4 x^{2}+64 x^{6} y^{3}+48 x^{4} y^{2}+12 x^{2} y+1}{64 x^{8}} \] |
1 |
1 |
1 |
abelFirstKind |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
5.427 |
|
|
\[ {}y^{\prime } = \frac {2 a \left (-y^{2}+4 x a -1\right )}{-y^{3}+4 y a x -y-2 a y^{6}+24 y^{4} a^{2} x -96 y^{2} a^{3} x^{2}+128 a^{4} x^{3}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
4.06 |
|
|
\[ {}y^{\prime } = \frac {\left (y-a \ln \left (y\right ) x +x^{2}\right ) y}{\left (-y \ln \left (y\right )-y \ln \left (x \right )-y+x a \right ) x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[NONE] |
✓ |
✓ |
3.469 |
|
|
\[ {}y^{\prime } = \frac {-x y^{2}+x^{3}-x -y^{6}+3 y^{4} x^{2}-3 x^{4} y^{2}+x^{6}}{\left (-y^{2}+x^{2}-1\right ) y} \] |
1 |
0 |
4 |
unknown |
[_rational] |
✗ |
N/A |
3.136 |
|
|
\[ {}y^{\prime } = \frac {\sin \left (\frac {y}{x}\right ) \left (y+2 x^{2} \sin \left (\frac {y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )\right )}{2 \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right )} \] |
1 |
1 |
1 |
homogeneousTypeD2 |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
15.951 |
|
|
\[ {}y^{\prime } = \frac {\sin \left (\frac {y}{x}\right ) \left (y+2 x^{3} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )\right )}{2 \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right )} \] |
1 |
1 |
1 |
homogeneousTypeD2 |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
22.625 |
|
|
\[ {}y^{\prime } = \frac {x \,a^{2}+a^{3} x^{3}+a^{3} x^{3} y^{2}+2 a^{2} y x^{2}+x a +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 y a x +1}{a^{3} x^{3}} \] |
1 |
1 |
1 |
abelFirstKind |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
3.573 |
|
|
\[ {}y^{\prime } = \frac {x \left (1+x^{2}+y^{2}\right )}{-y^{3}-x^{2} y-y+y^{6}+3 y^{4} x^{2}+3 x^{4} y^{2}+x^{6}} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
3.5 |
|
|
\[ {}y^{\prime } = \frac {-2 \cos \left (x \right ) x +2 \sin \left (x \right ) x^{2}+2 x +2 y^{2}+4 y \cos \left (x \right ) x -4 x y+x^{2} \cos \left (2 x \right )+3 x^{2}-4 x^{2} \cos \left (x \right )}{2 x} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
9.873 |
|
|
\[ {}y^{\prime } = \frac {4 x \left (a -1\right ) \left (1+a \right )}{4 y+a^{2} y^{4}-2 a^{4} y^{2} x^{2}+4 y^{2} a^{2} x^{2}+a^{6} x^{4}-3 a^{4} x^{4}+3 a^{2} x^{4}-y^{4}-2 x^{2} y^{2}-x^{4}} \] |
1 |
0 |
3 |
unknown |
[_rational] |
✗ |
N/A |
4.1 |
|
|
\[ {}y^{\prime } = \frac {x^{3}+y^{4} x^{3}+2 x^{2} y^{2}+x +x^{3} y^{6}+3 y^{4} x^{2}+3 x y^{2}+1}{x^{5} y} \] |
1 |
0 |
8 |
unknown |
[_rational] |
✗ |
N/A |
3.374 |
|
|
\[ {}y^{\prime } = \frac {-2 x -y+1+x^{2} y^{2}+2 x^{3} y+x^{4}+y^{3} x^{3}+3 x^{4} y^{2}+3 x^{5} y+x^{6}}{x} \] |
1 |
1 |
1 |
abelFirstKind |
[_rational, _Abel] |
✓ |
✓ |
11.483 |
|
|
\[ {}y^{\prime } = -\left (-\frac {\ln \left (y\right )}{x}+\frac {\cos \left (x \right ) \ln \left (y\right )}{\sin \left (x \right )}-\textit {\_F1} \left (x \right )\right ) y \] |
1 |
1 |
1 |
exactByInspection |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
9.809 |
|
|
\[ {}y^{\prime } = \frac {2 a x}{-x^{3} y+2 a \,x^{3}+2 a y^{4} x^{3}-16 y^{2} a^{2} x^{2}+32 x \,a^{3}+2 a y^{6} x^{3}-24 y^{4} a^{2} x^{2}+96 y^{2} x \,a^{3}-128 a^{4}} \] |
1 |
1 |
0 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
7.266 |
|
|
\[ {}y^{\prime } = -\frac {-y^{3}-y+2 y^{2} \ln \left (x \right )-\ln \left (x \right )^{2} y^{3}-1+3 y \ln \left (x \right )-3 \ln \left (x \right )^{2} y^{2}+\ln \left (x \right )^{3} y^{3}}{y x} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
4.983 |
|
|
\[ {}y^{\prime } = \frac {2 a \left (x y^{2}-4 a +x \right )}{-y^{3} x^{3}+4 a \,x^{2} y-x^{3} y+2 a y^{6} x^{3}-24 y^{4} a^{2} x^{2}+96 y^{2} x \,a^{3}-128 a^{4}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
7.559 |
|
|
\[ {}y^{\prime } = -\frac {-y^{3}-y+4 y^{2} \ln \left (x \right )-4 \ln \left (x \right )^{2} y^{3}-1+6 y \ln \left (x \right )-12 \ln \left (x \right )^{2} y^{2}+8 \ln \left (x \right )^{3} y^{3}}{y x} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
4.929 |
|
|
\[ {}y^{\prime } = \frac {y \left (\ln \left (y\right ) x +\ln \left (y\right )-x -1+x \ln \left (x \right )+\ln \left (x \right )+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
12.056 |
|
|
\[ {}y^{\prime } = \frac {y \left (x \ln \left (x \right )+\ln \left (x \right )+\ln \left (y\right ) x +\ln \left (y\right )-x -1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}\right )}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
4.967 |
|
|
\[ {}y^{\prime } = \frac {2 y^{8}}{y^{5}+2 y^{6}+2 y^{2}+16 y^{4} x +32 y^{6} x^{2}+2+24 x y^{2}+96 y^{4} x^{2}+128 x^{3} y^{6}} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
3.741 |
|
|
\[ {}y^{\prime } = \frac {y^{\frac {3}{2}} \left (x -y+\sqrt {y}\right )}{y^{\frac {3}{2}} x -y^{\frac {5}{2}}+y^{2}+x^{3}-3 x^{2} y+3 x y^{2}-y^{3}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
44.891 |
|
|
\[ {}y^{\prime } = \frac {2 y^{6} \left (1+4 x y^{2}+y^{2}\right )}{y^{3}+4 y^{5} x +y^{5}+2+24 x y^{2}+96 y^{4} x^{2}+128 x^{3} y^{6}} \] |
1 |
0 |
0 |
unknown |
[_rational] |
❇ |
N/A |
3.885 |
|
|
\[ {}y^{\prime } = -\left (-\frac {\ln \left (y\right )}{x}+\frac {\ln \left (y\right )}{x \ln \left (x \right )}-\textit {\_F1} \left (x \right )\right ) y \] |
1 |
1 |
1 |
exactByInspection |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
3.809 |
|
|
\[ {}y^{\prime } = \frac {y^{2}}{y^{2}+y^{\frac {3}{2}}+\sqrt {y}\, x^{2}-2 y^{\frac {3}{2}} x +y^{\frac {5}{2}}+x^{3}-3 x^{2} y+3 x y^{2}-y^{3}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
4.431 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{-2 \left (x -y\right ) \left (x +y\right )}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{-2 \left (x -y\right ) \left (x +y\right )}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.3 |
|
|
\[ {}y^{\prime } = -\frac {\left (-\frac {\ln \left (y\right )^{2}}{2 x}-\textit {\_F1} \left (x \right )\right ) y}{\ln \left (y\right )} \] |
1 |
1 |
2 |
exactByInspection |
[NONE] |
✓ |
✓ |
3.614 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{2 \left (x -y\right )^{2} \left (x +y\right )^{2}}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{2 \left (x -y\right )^{2} \left (x +y\right )^{2}}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
5.895 |
|
|
\[ {}y^{\prime } = \frac {-8 x^{2} y^{3}+16 x y^{2}+16 x y^{3}-8+12 x y-6 x^{2} y^{2}+y^{3} x^{3}}{16 \left (-2+x y-2 y\right ) x} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
5.727 |
|
|
\[ {}y^{\prime } = -\frac {\left (-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8-8 y^{2}+8 x^{2} {\mathrm e}^{-x^{2}} y-2 x^{4} {\mathrm e}^{-2 x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{8} \] |
1 |
1 |
1 |
abelFirstKind |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
16.706 |
|
|
\[ {}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x \right ) {\mathrm e}^{\frac {y}{x}}}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
4.066 |
|
|
\[ {}y^{\prime } = -\frac {16 x y^{3}-8 y^{3}-8 y+8 x y^{2}-2 x^{2} y^{3}-8+12 x y-6 x^{2} y^{2}+y^{3} x^{3}}{32 y x} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
5.741 |
|
|
\[ {}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x \left (1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
3.941 |
|
|
\[ {}y^{\prime } = \frac {-3 x^{2} y-2 x^{3}-2 x -x y^{2}-y+y^{3} x^{3}+3 x^{4} y^{2}+3 x^{5} y+x^{6}}{x \left (x y+x^{2}+1\right )} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
8.714 |
|
|
\[ {}y^{\prime } = \frac {\left (27 y^{3}+27 \,{\mathrm e}^{3 x^{2}} y+18 \,{\mathrm e}^{3 x^{2}} y^{2}+3 y^{3} {\mathrm e}^{3 x^{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y+9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y^{2}+{\mathrm e}^{\frac {9 x^{2}}{2}} y^{3}\right ) {\mathrm e}^{3 x^{2}} x \,{\mathrm e}^{-\frac {9 x^{2}}{2}}}{243 y} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
35.108 |
|
|
\[ {}y^{\prime } = -\frac {-x^{2}-x y-x^{3}-x y^{2}+2 y x^{2} \ln \left (x \right )-x^{3} \ln \left (x \right )^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x^{2}} \] |
1 |
1 |
1 |
abelFirstKind |
[_Abel] |
✓ |
✓ |
11.313 |
|
|
\[ {}y^{\prime } = \frac {x}{2}+1+y^{2}+\frac {x^{2} y}{4}-x y-\frac {x^{4}}{8}+\frac {x^{3}}{8}+\frac {x^{2}}{4}+y^{3}-\frac {3 x^{2} y^{2}}{4}-\frac {3 x y^{2}}{2}+\frac {3 x^{4} y}{16}+\frac {3 x^{3} y}{4}-\frac {x^{6}}{64}-\frac {3 x^{5}}{32} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
3.691 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+1+y^{2}+\frac {7 x^{2} y}{2}-2 x y+\frac {13 x^{4}}{16}-\frac {3 x^{3}}{2}+x^{2}+y^{3}+\frac {3 x^{2} y^{2}}{4}-3 x y^{2}+\frac {3 x^{4} y}{16}-\frac {3 x^{3} y}{2}+\frac {x^{6}}{64}-\frac {3 x^{5}}{16} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
7.264 |
|
|
\[ {}y^{\prime } = -\frac {x}{4}+1+y^{2}+\frac {7 x^{2} y}{16}-\frac {x y}{2}+\frac {5 x^{4}}{128}-\frac {5 x^{3}}{64}+\frac {x^{2}}{16}+y^{3}+\frac {3 x^{2} y^{2}}{8}-\frac {3 x y^{2}}{4}+\frac {3 x^{4} y}{64}-\frac {3 x^{3} y}{16}+\frac {x^{6}}{512}-\frac {3 x^{5}}{256} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
3.816 |
|
|
\[ {}y^{\prime } = \frac {-2 y-2 \ln \left (2 x +1\right )-2+2 x y^{3}+y^{3}+6 y^{2} \ln \left (2 x +1\right ) x +3 y^{2} \ln \left (2 x +1\right )+6 y \ln \left (2 x +1\right )^{2} x +3 y \ln \left (2 x +1\right )^{2}+2 \ln \left (2 x +1\right )^{3} x +\ln \left (2 x +1\right )^{3}}{\left (2 x +1\right ) \left (y+\ln \left (2 x +1\right )+1\right )} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
35.34 |
|
|
\[ {}y^{\prime } = \frac {-x^{2}+x +1+y^{2}+5 x^{2} y-2 x y+4 x^{4}-3 x^{3}+y^{3}+3 x^{2} y^{2}-3 x y^{2}+3 x^{4} y-6 x^{3} y+x^{6}-3 x^{5}}{x} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
7.046 |
|
|
\[ {}y^{\prime } = \frac {-32 x y+16 x^{3}+16 x^{2}-32 x -64 y^{3}+48 x^{2} y^{2}+96 x y^{2}-12 x^{4} y-48 x^{3} y-48 x^{2} y+x^{6}+6 x^{5}+12 x^{4}}{-64 y+16 x^{2}+32 x -64} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
5.046 |
|
|
\[ {}y^{\prime } = \frac {y \ln \left (x \right ) x +\ln \left (x \right ) x^{2}-2 x y-x^{2}-y^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x \left (-y+x \ln \left (x \right )-x \right )} \] |
1 |
0 |
2 |
unknown |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
4.381 |
|
|
\[ {}y^{\prime } = \frac {-32 x y-72 x^{3}+32 x^{2}-32 x +64 y^{3}+48 x^{2} y^{2}-192 x y^{2}+12 x^{4} y-96 x^{3} y+192 x^{2} y+x^{6}-12 x^{5}+48 x^{4}}{64 y+16 x^{2}-64 x +64} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
4.492 |
|
|
\[ {}y^{\prime } = -\frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{\frac {2 \left (x -y\right )^{3} \left (x +y\right )^{3}}{-y^{2}+x^{2}-1}}}{-y^{2}-2 x y-x^{2}+{\mathrm e}^{\frac {2 \left (x -y\right )^{3} \left (x +y\right )^{3}}{-y^{2}+x^{2}-1}}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
13.535 |
|
|
\[ {}y^{\prime } = \frac {-128 x y-24 x^{3}+32 x^{2}-128 x +512 y^{3}+192 x^{2} y^{2}-384 x y^{2}+24 x^{4} y-96 x^{3} y+96 x^{2} y+x^{6}-6 x^{5}+12 x^{4}}{512 y+64 x^{2}-128 x +512} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
4.328 |
|
|
\[ {}y^{\prime } = \frac {-32 y a x -8 a^{2} x^{3}-16 a b \,x^{2}-32 x a +64 y^{3}+48 a \,x^{2} y^{2}+96 y^{2} b x +12 y a^{2} x^{4}+48 y a \,x^{3} b +48 y b^{2} x^{2}+a^{3} x^{6}+6 a^{2} x^{5} b +12 a \,x^{4} b^{2}+8 b^{3} x^{3}}{64 y+16 x^{2} a +32 b x +64} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
4.367 |
|
|
\[ {}y^{\prime } = \frac {-32 x y-8 x^{3}-16 x^{2} a -32 x +64 y^{3}+48 x^{2} y^{2}+96 a x y^{2}+12 x^{4} y+48 y a \,x^{3}+48 a^{2} y x^{2}+x^{6}+6 x^{5} a +12 a^{2} x^{4}+8 a^{3} x^{3}}{64 y+16 x^{2}+32 x a +64} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
3.667 |
|
|
\[ {}y^{\prime } = \frac {\left (-8 \,{\mathrm e}^{-x^{2}} y+4 x^{2} {\mathrm e}^{-2 x^{2}}-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}} y-4 x^{4} {\mathrm e}^{-2 x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{-8 y+4 x^{2} {\mathrm e}^{-x^{2}}-8} \] |
1 |
0 |
2 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
42.027 |
|
|
\[ {}y^{\prime } = \frac {2 x^{2} \cos \left (x \right )+2 \sin \left (x \right ) x^{3}-2 x \sin \left (x \right )+2 x +2 x^{2} y^{2}-4 y \sin \left (x \right ) x +4 y \cos \left (x \right ) x^{2}+4 x y+3-\cos \left (2 x \right )-2 \sin \left (2 x \right ) x -4 \sin \left (x \right )+x^{2} \cos \left (2 x \right )+x^{2}+4 \cos \left (x \right ) x}{2 x^{3}} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
22.888 |
|
|
\[ {}y^{\prime } = -\frac {216 y}{-216 y^{4}-252 y^{3}-396 y^{2}-216 y+36 x^{2}-72 x y+60 y^{5}-36 x y^{3}-72 x y^{2}-24 y^{4} x +4 y^{8}+12 y^{7}+33 y^{6}} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
4.053 |
|
|
\[ {}y^{\prime } = \frac {x^{2} y+x^{4}+2 x^{3}-3 x^{2}+x y+x +y^{3}+3 x^{2} y^{2}-3 x y^{2}+3 x^{4} y-6 x^{3} y+x^{6}-3 x^{5}}{x \left (y+x^{2}-x +1\right )} \] |
1 |
1 |
2 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
5.431 |
|
|
\[ {}y^{\prime } = -\frac {x a}{2}+1+y^{2}+\frac {a \,x^{2} y}{2}+b x y+\frac {a^{2} x^{4}}{16}+\frac {a b \,x^{3}}{4}+\frac {b^{2} x^{2}}{4}+y^{3}+\frac {3 a \,x^{2} y^{2}}{4}+\frac {3 y^{2} b x}{2}+\frac {3 y a^{2} x^{4}}{16}+\frac {3 y a \,x^{3} b}{4}+\frac {3 y b^{2} x^{2}}{4}+\frac {a^{3} x^{6}}{64}+\frac {3 a^{2} x^{5} b}{32}+\frac {3 a \,x^{4} b^{2}}{16}+\frac {b^{3} x^{3}}{8} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
3.684 |
|
|
\[ {}y^{\prime } = -\frac {x}{2}+1+y^{2}+\frac {x^{2} y}{2}+y a x +\frac {x^{4}}{16}+\frac {a \,x^{3}}{4}+\frac {a^{2} x^{2}}{4}+y^{3}+\frac {3 x^{2} y^{2}}{4}+\frac {3 a x y^{2}}{2}+\frac {3 x^{4} y}{16}+\frac {3 y a \,x^{3}}{4}+\frac {3 a^{2} y x^{2}}{4}+\frac {x^{6}}{64}+\frac {3 x^{5} a}{32}+\frac {3 a^{2} x^{4}}{16}+\frac {a^{3} x^{3}}{8} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
3.694 |
|
|
\[ {}y^{\prime } = -\frac {-y+\sqrt {x^{2}+y^{2}}\, x^{2}-x \sqrt {x^{2}+y^{2}}\, y+x^{4} \sqrt {x^{2}+y^{2}}-x^{3} \sqrt {x^{2}+y^{2}}\, y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x} \] |
1 |
0 |
1 |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
N/A |
5.291 |
|
|
\[ {}y^{\prime } = \frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (y\right ) \ln \left (x \right )+x^{3} \ln \left (y\right )^{2}+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x} \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
5.59 |
|
|
\[ {}y^{\prime } = \frac {150 x^{3}+125 \sqrt {x}+125+125 y^{2}-100 x^{3} y-500 y \sqrt {x}+20 x^{6}+200 x^{\frac {7}{2}}+500 x +125 y^{3}-150 x^{3} y^{2}-750 y^{2} \sqrt {x}+60 x^{6} y+600 y x^{\frac {7}{2}}+1500 x y-8 x^{9}-120 x^{\frac {13}{2}}-600 x^{4}-1000 x^{\frac {3}{2}}}{125 x} \] |
1 |
1 |
1 |
abelFirstKind |
[_rational, _Abel] |
✓ |
✓ |
14.825 |
|
|
\[ {}y^{\prime } = \frac {-150 x^{3} y+60 x^{6}+350 x^{\frac {7}{2}}-150 x^{3}-125 y \sqrt {x}+250 x -125 \sqrt {x}-125 y^{3}+150 x^{3} y^{2}+750 y^{2} \sqrt {x}-60 x^{6} y-600 y x^{\frac {7}{2}}-1500 x y+8 x^{9}+120 x^{\frac {13}{2}}+600 x^{4}+1000 x^{\frac {3}{2}}}{25 \left (-5 y+2 x^{3}+10 \sqrt {x}-5\right ) x} \] |
1 |
0 |
2 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
N/A |
50.033 |
|
|
\[ {}y^{\prime } = \frac {y \left (-1-x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2}-x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} \ln \left (x \right )+x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y+2 x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y \ln \left (x \right )+x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y \ln \left (x \right )^{2}\right )}{\left (1+\ln \left (x \right )\right ) x} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
11.429 |
|
|
\[ {}y^{\prime } = \frac {y \left (-1-x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}}-x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} \ln \left (x \right )+x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y+2 x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y \ln \left (x \right )+x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y \ln \left (x \right )^{2}\right )}{\left (1+\ln \left (x \right )\right ) x} \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
10.642 |
|
|
\[ {}y^{\prime } = \frac {2 x +4 y \ln \left (2 x +1\right ) x +6 y^{2} \ln \left (2 x +1\right ) x +6 y \ln \left (2 x +1\right )^{2} x +2 \ln \left (2 x +1\right )^{3} x +2 x y^{3}+2 \ln \left (2 x +1\right )^{2} x +2 x y^{2}-1+3 y^{2} \ln \left (2 x +1\right )+3 y \ln \left (2 x +1\right )^{2}+y^{2}+y^{3}+2 y \ln \left (2 x +1\right )+\ln \left (2 x +1\right )^{2}+\ln \left (2 x +1\right )^{3}}{2 x +1} \] |
1 |
1 |
1 |
abelFirstKind |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
5.588 |
|
|
\[ {}y^{\prime } = \frac {-y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{3} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x} \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
38.3 |
|
|
\[ {}y^{\prime } = \frac {-y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{2} \sin \left (\frac {y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x} \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
32.956 |
|
|
\[ {}y^{\prime } = \frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{2+2 y^{4}-4 x^{2} y^{2}+2 x^{4}+2 y^{6}-6 y^{4} x^{2}+6 x^{4} y^{2}-2 x^{6}}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{2+2 y^{4}-4 x^{2} y^{2}+2 x^{4}+2 y^{6}-6 y^{4} x^{2}+6 x^{4} y^{2}-2 x^{6}}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
7.367 |
|
|
\[ {}y^{\prime } = \frac {4 x \left (a -1\right ) \left (1+a \right ) \left (-y^{2}+a^{2} x^{2}-x^{2}-2\right )}{-4 y^{3}+4 a^{2} y x^{2}-4 x^{2} y-8 y-a^{2} y^{6}+3 a^{4} y^{4} x^{2}-6 y^{4} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}-4 a^{2} x^{6}+y^{6}+3 y^{4} x^{2}+3 x^{4} y^{2}+x^{6}} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
7.815 |
|
|
\[ {}y^{\prime } = \frac {-4 \cos \left (x \right ) x +4 \sin \left (x \right ) x^{2}+4 x +4+4 y^{2}+8 y \cos \left (x \right ) x -8 x y+2 x^{2} \cos \left (2 x \right )+6 x^{2}-8 x^{2} \cos \left (x \right )+4 y^{3}+12 y^{2} \cos \left (x \right ) x -12 x y^{2}+6 y x^{2} \cos \left (2 x \right )+18 x^{2} y-24 y \cos \left (x \right ) x^{2}+x^{3} \cos \left (3 x \right )+15 x^{3} \cos \left (x \right )-6 x^{3} \cos \left (2 x \right )-10 x^{3}}{4 x} \] |
1 |
1 |
1 |
abelFirstKind |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
44.799 |
|
|
\[ {}y^{\prime } = -\frac {8 x \left (a -1\right ) \left (1+a \right )}{8+2 x^{4}+2 y^{4}+3 x^{4} y^{2}-8 y+x^{6}-8 y^{2} a^{2} x^{2}-2 a^{2} y^{4}-6 y^{4} a^{2} x^{2}+y^{6}-8 a^{2}+4 x^{2} y^{2}+3 y^{4} x^{2}-2 a^{6} x^{4}+6 a^{4} x^{4}-6 a^{2} x^{4}-a^{2} y^{6}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}+4 a^{4} y^{2} x^{2}+3 a^{4} y^{4} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}-4 a^{2} x^{6}} \] |
1 |
0 |
7 |
unknown |
[_rational] |
✗ |
N/A |
9.055 |
|
|
\[ {}y^{\prime } = \frac {-y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x +2 \sin \left (\frac {y}{x}\right ) x^{3} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{4} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x} \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
54.065 |
|
|
\[ {}y^{\prime } = -\frac {1296 y}{216-432 x y+1080 x y^{3}-612 y^{5}+216 x y^{2}-126 y^{10}-8 y^{12}-36 y^{11}-1944 y^{4}-2376 y^{2}+216 x^{3}-1296 y+216 x^{2}-846 y^{7}-1728 y^{3}-648 x^{2} y-570 y^{8}-315 y^{9}+1152 y^{4} x -882 y^{6}-648 x^{2} y^{2}-216 y^{4} x^{2}-324 x^{2} y^{3}+72 y^{8} x +216 y^{7} x +594 x y^{6}+1080 y^{5} x} \] |
1 |
0 |
1 |
unknown |
[_rational] |
✗ |
N/A |
5.546 |
|
|
\[ {}y^{\prime } = -\frac {x \left (-513-432 x -216 y^{2} x^{6}-864 x^{4}-540 y^{2}-972 x^{4} y^{2}-576 x^{5}-756 x^{3}-378 y-1134 x^{2}-456 x^{6}-216 y^{3}-144 x^{7}-594 x^{2} y-96 x^{8}-1296 x^{2} y^{2}-216 x^{4} y+432 x^{3} y^{2}+432 y^{2} x^{7}-648 x^{2} y^{3}+64 x^{9}-288 y x^{8}+288 y x^{7}-216 x^{6} y^{3}-288 x^{6} y-648 y^{3} x^{4}+1008 x^{5} y+864 y^{2} x^{5}+720 x^{3} y\right )}{216 \left (x^{2}+1\right )^{4}} \] |
1 |
1 |
1 |
abelFirstKind |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
8.59 |
|
|
\[ {}y^{\prime } = \frac {-\sin \left (\frac {y}{x}\right ) y x -y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x +y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x +y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{4} \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x \left (1+x \right )} \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
71.731 |
|
|
\[ {}y^{\prime } = \frac {y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x +y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x +y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )-\sin \left (\frac {y}{x}\right ) y x -y \sin \left (\frac {y}{x}\right )+2 \sin \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x}{2 \cos \left (\frac {y}{x}\right ) \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right ) \left (1+x \right )} \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class D‘]] |
✓ |
✓ |
47.311 |
|
|
\[ {}y^{\prime } = -\frac {216 y \left (-2 y^{4}-3 y^{3}-6 y^{2}-6 y+6 x +6\right )}{-1296 x y-648 x y^{3}+4428 y^{5}-1944 x y^{2}-126 y^{10}-8 y^{12}-36 y^{11}+2808 y^{4}-1296 y^{2}+216 x^{3}-1296 y+594 y^{7}+1728 y^{3}-648 x^{2} y-18 y^{8}-315 y^{9}-432 y^{4} x +2484 y^{6}-648 x^{2} y^{2}-216 y^{4} x^{2}-324 x^{2} y^{3}+72 y^{8} x +216 y^{7} x +594 x y^{6}+1080 y^{5} x} \] |
1 |
0 |
2 |
unknown |
[_rational] |
✗ |
N/A |
6.566 |
|
|
\[ {}y^{\prime } = \frac {\left (x y+1\right )^{3}}{x^{5}} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
30.454 |
|
|
\[ {}y^{\prime } = \frac {x \left (-x^{2}+2 x^{2} y-2 x^{4}+1\right )}{-x^{2}+y} \] |
1 |
0 |
1 |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
3.581 |
|
|
\[ {}y^{\prime } = y \left (y^{2}+y \,{\mathrm e}^{b x}+{\mathrm e}^{2 b x}\right ) {\mathrm e}^{-2 b x} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
19.244 |
|
|
\[ {}y^{\prime } = y^{3}-3 x^{2} y^{2}+3 x^{4} y-x^{6}+2 x \] |
1 |
2 |
2 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
4.408 |
|
|
\[ {}y^{\prime } = y^{3}+x^{2} y^{2}+\frac {x^{4} y}{3}+\frac {x^{6}}{27}-\frac {2 x}{3} \] |
1 |
2 |
2 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Abel] |
✓ |
✓ |
4.05 |
|
|
\[ {}y^{\prime } = \frac {y \left (y^{2} x^{7}+x^{4} y+x -3\right )}{x} \] |
1 |
0 |
1 |
abelFirstKind |
[_rational, _Abel] |
✗ |
N/A |
11.227 |
|
|
\[ {}y^{\prime } = y \left (y^{2}+{\mathrm e}^{-x^{2}} y+{\mathrm e}^{-2 x^{2}}\right ) {\mathrm e}^{2 x^{2}} x \] |
1 |
0 |
1 |
abelFirstKind |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Abel] |
✗ |
N/A |
10.075 |
|
|
\[ {}y^{\prime } = \frac {y \left (y^{2}+x y+x^{2}+x \right )}{x^{2}} \] |
1 |
1 |
1 |
abelFirstKind, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Abel] |
✓ |
✓ |
5.063 |
|
|
\[ {}y^{\prime } = \frac {y^{3}-3 x y^{2}+3 x^{2} y-x^{3}+x}{x} \] |
1 |
1 |
2 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, _Abel] |
✓ |
✓ |
1.503 |
|
|
\[ {}y^{\prime } = \frac {y^{3} x^{3}+6 x^{2} y^{2}+12 x y+8+2 x}{x^{3}} \] |
1 |
2 |
2 |
abelFirstKind, exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
3.043 |
|
|
\[ {}y^{\prime } = \frac {y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 y a x +1+x \,a^{2}}{x^{3} a^{3}} \] |
1 |
2 |
2 |
abelFirstKind, exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
3.145 |
|
|
\[ {}y^{\prime } = \frac {y \,{\mathrm e}^{-\frac {x^{2}}{2}} \left (2 y^{2}+2 y \,{\mathrm e}^{\frac {x^{2}}{4}}+2 \,{\mathrm e}^{\frac {x^{2}}{2}}+x \,{\mathrm e}^{\frac {x^{2}}{2}}\right )}{2} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[_Abel] |
✓ |
✓ |
68.025 |
|
|
\[ {}y^{\prime } = \frac {y^{3}-3 x y^{2}+3 x^{2} y-x^{3}+x^{2}}{\left (-1+x \right ) \left (1+x \right )} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
5.314 |
|
|
\[ {}y^{\prime } = \frac {y \left (x^{2} y^{2}+y x \,{\mathrm e}^{x}+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x} \left (-1+x \right )}{x} \] |
1 |
0 |
1 |
abelFirstKind |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Abel] |
✗ |
N/A |
8.988 |
|
|
\[ {}y^{\prime } = \frac {\left (x y+1\right ) \left (x^{2} y^{2}+x^{2} y+2 x y+1+x +x^{2}\right )}{x^{5}} \] |
1 |
1 |
1 |
abelFirstKind, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
3.867 |
|
|
\[ {}y^{\prime } = \frac {y^{3}-3 x y^{2} \ln \left (x \right )+3 x^{2} \ln \left (x \right )^{2} y-x^{3} \ln \left (x \right )^{3}+x^{2}+x y}{x^{2}} \] |
1 |
1 |
2 |
abelFirstKind |
[_Abel] |
✓ |
✓ |
3.238 |
|
|
\[ {}y^{\prime } = -F \left (x \right ) \left (-x^{2} a +y^{2}\right )+\frac {y}{x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
3.55 |
|
|
\[ {}y^{\prime } = -F \left (x \right ) \left (-x^{2}-2 x y+y^{2}\right )+\frac {y}{x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
2.852 |
|
|
\[ {}y^{\prime } = -F \left (x \right ) \left (-a y^{2}-b \,x^{2}\right )+\frac {y}{x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
2.964 |
|
|
\[ {}y^{\prime } = -F \left (x \right ) \left (-y^{2}+2 x^{2} y+1-x^{4}\right )+2 x \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
3.02 |
|
|
\[ {}y^{\prime } = -F \left (x \right ) \left (x^{2}+2 x y-y^{2}\right )+\frac {y}{x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
2.735 |
|
|
\[ {}y^{\prime } = -F \left (x \right ) \left (-7 x y^{2}-x^{3}\right )+\frac {y}{x} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
✓ |
2.915 |
|
|
\[ {}y^{\prime } = -F \left (x \right ) \left (-y^{2}-2 y \ln \left (x \right )-\ln \left (x \right )^{2}\right )+\frac {y}{\ln \left (x \right ) x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.082 |
|
|
\[ {}y^{\prime } = -x^{3} \left (-y^{2}-2 y \ln \left (x \right )-\ln \left (x \right )^{2}\right )+\frac {y}{\ln \left (x \right ) x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.876 |
|
|
\[ {}y^{\prime } = \left (y-{\mathrm e}^{x}\right )^{2}+{\mathrm e}^{x} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
2.679 |
|
|
\[ {}y^{\prime } = \frac {\left (y-\operatorname {Si}\left (x \right )\right )^{2}+\sin \left (x \right )}{x} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
3.821 |
|
|
\[ {}y^{\prime } = \left (y+\cos \left (x \right )\right )^{2}+\sin \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
3.0 |
|
|
\[ {}y^{\prime } = \frac {\left (y-\ln \left (x \right )-\operatorname {Ci}\left (x \right )\right )^{2}+\cos \left (x \right )}{x} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
4.853 |
|
|
\[ {}y^{\prime } = \frac {\left (y-x +\ln \left (1+x \right )\right )^{2}+x}{1+x} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
2.306 |
|
|
\[ {}y^{\prime } = \frac {2 x^{2} y+x^{3}+y \ln \left (x \right ) x -y^{2}-x y}{x^{2} \left (x +\ln \left (x \right )\right )} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.15 |
|
|
\[ {}y^{\prime \prime } = 0 \] |
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, second_order_ode_can_be_made_integrable |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.749 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.899 |
|
|
\[ {}y^{\prime \prime }+y-\sin \left (n x \right ) = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.756 |
|
|
|
|||||||||
|
\[ {}y^{\prime \prime }+y-a \cos \left (b x \right ) = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.768 |
|
|
\[ {}y^{\prime \prime }+y-\sin \left (x a \right ) \sin \left (b x \right ) = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.983 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.702 |
|
|
\[ {}y^{\prime \prime }-2 y-4 x^{2} {\mathrm e}^{x^{2}} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.674 |
|
|
\[ {}y^{\prime \prime }+a^{2} y-\cot \left (x a \right ) = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.583 |
|
|
\[ {}y^{\prime \prime }+l y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.927 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.651 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.783 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2}+a \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.765 |
|
|
\[ {}y^{\prime \prime }-\left (a^{2} x^{2}+a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.831 |
|
|
\[ {}y^{\prime \prime }-c \,x^{a} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.343 |
|
|
\[ {}y^{\prime \prime }-\left (a^{2} x^{2 n}-1\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[_Titchmarsh] |
✓ |
✓ |
84.458 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{2 c}+b \,x^{c -1}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.78 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{2 x}-v^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode_form_A |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.193 |
|
|
\[ {}y^{\prime \prime }+a \,{\mathrm e}^{b x} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode_form_A |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.22 |
|
|
\[ {}y^{\prime \prime }-\left (4 a^{2} b^{2} x^{2} {\mathrm e}^{2 b \,x^{2}}-1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.934 |
|
|
\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{x}+c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.497 |
|
|
\[ {}y^{\prime \prime }+\left (a \cosh \left (x \right )^{2}+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.75 |
|
|
\[ {}y^{\prime \prime }+\left (a \cos \left (2 x \right )+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_ellipsoidal] |
✗ |
N/A |
0.596 |
|
|
\[ {}y^{\prime \prime }+\left (a \cos \left (x \right )^{2}+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_ellipsoidal] |
✗ |
N/A |
0.744 |
|
|
\[ {}y^{\prime \prime }-\left (1+2 \tan \left (x \right )^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.596 |
|
|
\[ {}y^{\prime \prime }-\left (\frac {m \left (m -1\right )}{\cos \left (x \right )^{2}}+\frac {n \left (n -1\right )}{\sin \left (x \right )^{2}}+a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
5.163 |
|
|
\[ {}y^{\prime \prime }-\left (n \left (n +1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+B \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.652 |
|
|
\[ {}y^{\prime \prime }-\left (n \left (n +1\right ) k^{2} \operatorname {JacobiSN}\left (x , k\right )^{2}+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.675 |
|
|
\[ {}y^{\prime \prime }-\left (\frac {p^{\prime \prime \prime \prime }\left (x \right )}{30}+\frac {7 p^{\prime \prime }\left (x \right )}{3}+a p \left (x \right )+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.05 |
|
|
\[ {}y^{\prime \prime }-\left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.169 |
|
|
\[ {}y^{\prime \prime }+\left (P \left (x \right )+l \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.125 |
|
|
\[ {}y^{\prime \prime }-f \left (x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.112 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+a \,{\mathrm e}^{-2 x} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode_form_A, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.664 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+{\mathrm e}^{2 x} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode_form_A, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.569 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.394 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b y-f \left (x \right ) = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.007 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.789 |
|
|
\[ {}y^{\prime \prime }+2 a y^{\prime }+f \left (x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.215 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.614 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-y = 0 \] |
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]] |
✓ |
✓ |
1.151 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+\left (n +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.588 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-n y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.56 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Hermite] |
✓ |
✓ |
0.908 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-a y = 0 \] |
1 |
0 |
1 |
unknown |
[_Hermite] |
✗ |
N/A |
0.576 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.598 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.602 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.626 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (3 x^{2}+2 n -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.689 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-1\right ) y-{\mathrm e}^{x} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.744 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.583 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-3\right ) y-{\mathrm e}^{x^{2}} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.314 |
|
|
\[ {}y^{\prime \prime }+a x y^{\prime }+b y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.721 |
|
|
\[ {}y^{\prime \prime }+2 a x y^{\prime }+a^{2} y x^{2} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.2 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.908 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
3.056 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.217 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-\left (1+x \right )^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.937 |
|
|
\[ {}y^{\prime \prime }-x^{2} \left (1+x \right ) y^{\prime }+x \left (x^{4}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.093 |
|
|
\[ {}y^{\prime \prime }+x^{4} y^{\prime }-x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.919 |
|
|
\[ {}y^{\prime \prime }+a \,x^{q -1} y^{\prime }+b \,x^{q -2} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.89 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } \sqrt {x}+\left (\frac {1}{4 \sqrt {x}}+\frac {x}{4}-9\right ) y-x \,{\mathrm e}^{-\frac {x^{\frac {3}{2}}}{3}} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.799 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.681 |
|
|
\[ {}y^{\prime \prime }-\left (2 \,{\mathrm e}^{x}+1\right ) y^{\prime }+{\mathrm e}^{2 x} y-{\mathrm e}^{3 x} = 0 \] |
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]] |
✓ |
✓ |
4.507 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+\tan \left (x \right )+b y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
5.553 |
|
|
\[ {}y^{\prime \prime }+2 n y^{\prime } \cot \left (x \right )+\left (-a^{2}+n^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.845 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } \tan \left (x \right )+y \cos \left (x \right )^{2} = 0 \] |
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]] |
✓ |
✓ |
10.078 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } \tan \left (x \right )-y \cos \left (x \right )^{2} = 0 \] |
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]] |
✓ |
✓ |
0.968 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } \cot \left (x \right )+v \left (v +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.166 |
|
|
\[ {}y^{\prime \prime }-y^{\prime } \cot \left (x \right )+y \sin \left (x \right )^{2} = 0 \] |
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]] |
✓ |
✓ |
10.019 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime } \tan \left (x \right )+b y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.436 |
|
|
\[ {}y^{\prime \prime }+2 a y^{\prime } \cot \left (x a \right )+\left (-a^{2}+b^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.75 |
|
|
\[ {}y^{\prime \prime }+a p^{\prime \prime }\left (x \right ) y^{\prime }+\left (a +b p \left (x \right )-4 n a p \left (x \right )^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.259 |
|
|
\[ {}y^{\prime \prime }+\frac {\left (11 \operatorname {WeierstrassP}\left (x , a , b\right ) \operatorname {WeierstrassPPrime}\left (x , a , b\right )-6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}+\frac {a}{2}\right ) y^{\prime }}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}}+\frac {\left (\operatorname {WeierstrassPPrime}\left (x , a , b\right )^{2}-\operatorname {WeierstrassP}\left (x , a , b\right )^{2} \operatorname {WeierstrassPPrime}\left (x , a , b\right )-\operatorname {WeierstrassP}\left (x , a , b\right ) \left (6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}-\frac {a}{2}\right )\right ) y}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
86.672 |
|
|
\[ {}y^{\prime \prime }+f \left (x \right ) y^{\prime }+g \left (x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.28 |
|
|
\[ {}y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right )+a \right ) y-g \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
0.244 |
|
|
\[ {}y^{\prime \prime }+\left (a f \left (x \right )+b \right ) y^{\prime }+\left (c f \left (x \right )+d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.681 |
|
|
\[ {}y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (\frac {f \left (x \right )^{2}}{4}+\frac {f^{\prime }\left (x \right )}{2}+a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.575 |
|
|
\[ {}y^{\prime \prime }-\frac {a f^{\prime }\left (x \right ) y^{\prime }}{f \left (x \right )}+b f \left (x \right )^{2 a} y = 0 \] |
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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.908 |
|
|
\[ {}y^{\prime \prime }-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+2 a \right ) y^{\prime }+\left (\frac {a f^{\prime }\left (x \right )}{f \left (x \right )}+a^{2}-b^{2} f \left (x \right )^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.642 |
|
|
\[ {}y^{\prime \prime }+\frac {f \left (x \right ) f^{\prime \prime \prime }\left (x \right ) y^{\prime }}{f \left (x \right )^{2}+b^{2}}-\frac {a^{2} {f^{\prime }\left (x \right )}^{2} y}{f \left (x \right )^{2}+b^{2}} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.052 |
|
|
\[ {}y^{\prime \prime }-\left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right ) y^{\prime }+\left (\frac {f^{\prime }\left (x \right ) \left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right )}{f \left (x \right )}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {v^{2} {g^{\prime }\left (x \right )}^{2}}{g \left (x \right )^{2}}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.604 |
|
|
\[ {}y^{\prime \prime }-\left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right ) y^{\prime }+\left (\frac {h^{\prime }\left (x \right ) \left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right )}{h \left (x \right )}-\frac {h^{\prime \prime }\left (x \right )}{h \left (x \right )}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.644 |
|
|
\[ {}4 y^{\prime \prime }+9 x y = 0 \] |
1 |
1 |
1 |
second_order_airy, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.422 |
|
|
\[ {}4 y^{\prime \prime }-\left (x^{2}+a \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.875 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.287 |
|
|
\[ {}a y^{\prime \prime }-\left (a b +c +x \right ) y^{\prime }+\left (b \left (x +c \right )+d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.168 |
|
|
\[ {}a^{2} y^{\prime \prime }+a \left (a^{2}-2 b \,{\mathrm e}^{-x a}\right ) y^{\prime }+b^{2} {\mathrm e}^{-2 x a} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.059 |
|
|
\[ {}x \left (y^{\prime \prime }+y\right )-\cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.277 |
|
|
\[ {}x y^{\prime \prime }+\left (x +a \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.911 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
1.478 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.458 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+l x y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.707 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+\left (x +a \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.947 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.494 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }-y a \,x^{3} = 0 \] |
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)]‘]] |
✓ |
✓ |
1.975 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+x^{3} \left ({\mathrm e}^{x^{2}}-v^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.431 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-x y-{\mathrm e}^{x} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.601 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+y a x = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+a \,x^{2} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.593 |
|
|
\[ {}x y^{\prime \prime }-2 y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.497 |
|
|
\[ {}x y^{\prime \prime }+v y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.578 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b x y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.937 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b \,x^{\operatorname {a1}} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.444 |
|
|
\[ {}x y^{\prime \prime }+\left (x +b \right ) y^{\prime }+a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.125 |
|
|
\[ {}x y^{\prime \prime }+\left (x +a +b \right ) y^{\prime }+a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.261 |
|
|
\[ {}x y^{\prime \prime }-x y^{\prime }-y-x \left (1+x \right ) {\mathrm e}^{x} = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.437 |
|
|
\[ {}x y^{\prime \prime }-x y^{\prime }-a y = 0 \] |
1 |
0 |
1 |
unknown |
[_Laguerre] |
✗ |
N/A |
0.635 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[_Laguerre] |
✓ |
✓ |
1.018 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }-2 \left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.914 |
|
|
\[ {}x y^{\prime \prime }+\left (-x +b \right ) y^{\prime }-a y = 0 \] |
1 |
0 |
1 |
unknown |
[_Laguerre] |
✗ |
N/A |
1.073 |
|
|
\[ {}x y^{\prime \prime }-2 \left (-1+x \right ) y^{\prime }-y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.741 |
|
|
\[ {}x y^{\prime \prime }-\left (3 x -2\right ) y^{\prime }-\left (2 x -3\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.956 |
|
|
\[ {}x y^{\prime \prime }+\left (x a +b +n \right ) y^{\prime }+n a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.461 |
|
|
\[ {}x y^{\prime \prime }-\left (a +b \right ) \left (1+x \right ) y^{\prime }+a b x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.168 |
|
|
\[ {}x y^{\prime \prime }+\left (x \left (a +b \right )+m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.511 |
|
|
\[ {}x y^{\prime \prime }-2 \left (x a +b \right ) y^{\prime }+\left (x \,a^{2}+2 a b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.359 |
|
|
\[ {}x y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.961 |
|
|
\[ {}x y^{\prime \prime }-\left (x^{2}-x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.277 |
|
|
\[ {}x y^{\prime \prime }-\left (x^{2}-x -2\right ) y^{\prime }-x \left (x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.05 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x^{2} a +1\right ) y^{\prime }+b \,x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.61 |
|
|
\[ {}x y^{\prime \prime }-2 \left (x^{2}-a \right ) y^{\prime }+2 n x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.467 |
|
|
\[ {}x y^{\prime \prime }+\left (4 x^{2}-1\right ) y^{\prime }-4 x^{3} y-4 x^{5} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.376 |
|
|
\[ {}x y^{\prime \prime }+\left (2 a \,x^{3}-1\right ) y^{\prime }+\left (a^{2} x^{3}+a \right ) x^{2} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.217 |
|
|
\[ {}x y^{\prime \prime }+\left (2 a x \ln \left (x \right )+1\right ) y^{\prime }+\left (a^{2} x \ln \left (x \right )^{2}+a \ln \left (x \right )+a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.072 |
|
|
\[ {}x y^{\prime \prime }+\left (f \left (x \right ) x +2\right ) y^{\prime }+f \left (x \right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.907 |
|
|
\[ {}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.245 |
|
|
\[ {}2 x y^{\prime \prime }+y^{\prime }+a y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.266 |
|
|
\[ {}2 x y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.946 |
|
|
\[ {}2 x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+a y = 0 \] |
1 |
0 |
1 |
unknown |
[_Laguerre] |
✗ |
N/A |
0.937 |
|
|
\[ {}\left (2 x -1\right ) y^{\prime \prime }-\left (3 x -4\right ) y^{\prime }+\left (x -3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.496 |
|
|
\[ {}4 x y^{\prime \prime }-\left (x +a \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.889 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }-y = 0 \] |
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)]‘]] |
✓ |
✓ |
1.116 |
|
|
\[ {}4 x y^{\prime \prime }+4 y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.951 |
|
|
\[ {}4 x y^{\prime \prime }+4 y-\left (2+x \right ) y+l y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.974 |
|
|
\[ {}4 x y^{\prime \prime }+4 m y^{\prime }-\left (x -2 m -4 n \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.152 |
|
|
\[ {}16 x y^{\prime \prime }+8 y^{\prime }-\left (x +a \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.112 |
|
|
\[ {}a x y^{\prime \prime }+b y^{\prime }+c y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.673 |
|
|
\[ {}a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+3 b y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.4 |
|
|
\[ {}5 \left (x a +b \right ) y^{\prime \prime }+8 a y^{\prime }+c \left (x a +b \right )^{\frac {1}{5}} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.859 |
|
|
\[ {}2 a x y^{\prime \prime }+\left (b x +a \right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.209 |
|
|
\[ {}2 a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.21 |
|
|
\[ {}\left (\operatorname {a2} x +\operatorname {b2} \right ) y^{\prime \prime }+\left (\operatorname {a1} x +\operatorname {b1} \right ) y^{\prime }+\left (\operatorname {a0} x +\operatorname {b0} \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
9.317 |
|
|
\[ {}x^{2} y^{\prime \prime }-6 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.43 |
|
|
\[ {}x^{2} y^{\prime \prime }-12 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.429 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.625 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x a +b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.53 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.99 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2} a +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.076 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-6\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.29 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a -v \left (v -1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.909 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.443 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{k}-b \left (b -1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.313 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}-x \,{\mathrm e}^{x} \left (2+x \ln \left (x \right )\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.433 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y^{\prime }-x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.746 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+a b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.299 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y-x^{2} a = 0 \] |
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, _nonhomogeneous]] |
✓ |
✓ |
4.91 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+a y = 0 \] |
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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.013 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x +a \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.573 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-v^{2}+x^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[_Bessel] |
✓ |
✓ |
0.852 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-v^{2}+x^{2}\right ) y-f \left (x \right ) = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.898 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (l \,x^{2}-v^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.954 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x +a \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.319 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y-3 x^{3} = 0 \] |
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, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.866 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+\left (a \,x^{m}+b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.355 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.836 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (x a -b^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.649 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (x^{2} a +b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.925 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (l \,x^{2}+x a -n \left (n +1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
29.175 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 \left (-1+x \right ) y^{\prime }+a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.578 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 \left (x +a \right ) y^{\prime }-b \left (b -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.797 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y-x^{5} \ln \left (x \right ) = 0 \] |
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_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.727 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-4 y-x \sin \left (x \right )-\left (x^{2} a +12 a +4\right ) \cos \left (x \right ) = 0 \] |
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_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
8.401 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.854 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y-\frac {x^{2}}{\cos \left (x \right )} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.411 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y-\frac {x^{3}}{\cos \left (x \right )} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.013 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (a^{2} x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.944 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (-v^{2}+x^{2}+1\right ) y-f \left (x \right ) = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.021 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.895 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y-5 x = 0 \] |
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 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.747 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }-5 y-\ln \left (x \right ) x^{2} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.193 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y-x^{4}+x^{2} = 0 \] |
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, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.567 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }-\left (2 x^{3}-4\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.637 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+8 y-\sin \left (x \right ) x^{3} = 0 \] |
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 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.214 |
|
|
\[ {}x^{2} y^{\prime \prime }+a x y^{\prime }+b y = 0 \] |
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 |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.714 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.231 |
|
|
\[ {}x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{m}+c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.385 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (x a +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.969 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.79 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.02 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (x -9\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.983 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (3 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.962 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x +3\right ) y^{\prime }-y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.93 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-1+x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.181 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}-2 x \right ) y^{\prime }-\left (x +a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.006 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}-2 x \right ) y^{\prime }-\left (3 x +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.055 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +4\right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.819 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-v \left (v -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.845 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2 x +1\right ) y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.879 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] |
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]] |
✓ |
✓ |
1.089 |
|
|
\[ {}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.892 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a +2 b \right ) x^{2} y^{\prime }+\left (\left (a +b \right ) b \,x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.175 |
|
|
\[ {}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+f \left (x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.274 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (2 x a +b \right ) x y^{\prime }+\left (a b x +c \,x^{2}+d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.618 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x a +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
3.365 |
|
|
|
|||||||||
|
\[ {}x^{2} y^{\prime \prime }+x^{3} y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.312 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}+2\right ) y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.545 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.682 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (4 x^{4}+2 x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.19 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a +b \right ) x y^{\prime }+f \left (x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.688 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{3}+1\right ) x y^{\prime }-y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.043 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-x^{4}+\left (2 n +2 a +1\right ) x^{2}+a \left (-1\right )^{n}-a^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
17.615 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2 n}+\operatorname {b1} \,x^{n}+\operatorname {c1} \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.336 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{\operatorname {a1}}+b \right ) x y^{\prime }+\left (A \,x^{2 \operatorname {a1}}+B \,x^{\operatorname {a1}}+C \,x^{\operatorname {b1}}+\operatorname {DD} \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.8 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 x^{2} \tan \left (x \right )-x \right ) y^{\prime }-\left (x \tan \left (x \right )+a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
3.028 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (2 x^{2} \cot \left (x \right )+x \right ) y^{\prime }+\left (x \cot \left (x \right )+a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
3.036 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right ) x +f \left (x \right )^{2}-f \left (x \right )+x^{2} a +b x +c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.887 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} f \left (x \right ) y^{\prime }+\left (x^{2} \left (f^{\prime }\left (x \right )+f \left (x \right )^{2}+a \right )-v \left (v -1\right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.888 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x -2 f \left (x \right ) x^{2}\right ) y^{\prime }+\left (x^{2} \left (1+f \left (x \right )^{2}-f^{\prime }\left (x \right )\right )-f \left (x \right ) x -v^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.897 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.028 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-9 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.615 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.71 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
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]] |
✓ |
✓ |
2.075 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-v \left (v -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.931 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
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]] |
✓ |
✓ |
1.867 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.014 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+4 x y^{\prime }+2 y-2 \cos \left (x \right )+2 x = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.898 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+a x y^{\prime }+\left (a -2\right ) y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.84 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-v \left (v +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
0.739 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-n \left (n +1\right ) y+\frac {\partial }{\partial x}\operatorname {LegendreP}\left (n , x\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.793 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime }+2 = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.165 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime }+a y = 0 \] |
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)]‘]] |
✓ |
✓ |
3.128 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime }+f \left (x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.514 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.871 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime }-a = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.484 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime }-l y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
1.037 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime }-v \left (v +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
1.086 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }-\left (v +2\right ) \left (v -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
375.082 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-\left (1+3 x \right ) y^{\prime }-\left (x^{2}-x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.693 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.661 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (v +n +1\right ) \left (v -n \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
1.046 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (v -n +1\right ) \left (v +n \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
1.001 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (v -1\right ) x y^{\prime }-2 v y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.973 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 a x y^{\prime }+a \left (a -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
1.091 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.645 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.996 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+8 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.738 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.974 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (3 x +2\right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.415 |
|
|
\[ {}\left (x^{2}+x -2\right ) y^{\prime \prime }+\left (x^{2}-x \right ) y^{\prime }-\left (6 x^{2}+7 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.934 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }+a y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.707 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }-v \left (v +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Jacobi] |
✗ |
N/A |
0.715 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }+\left (\left (1+a \right ) x +b \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.303 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[_Jacobi] |
✗ |
N/A |
1.017 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }+\left (\left (1+a \right ) x +b \right ) y^{\prime }-l y = 0 \] |
1 |
0 |
1 |
unknown |
[_Jacobi] |
✗ |
N/A |
1.032 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }+\left (\left (\operatorname {a1} +\operatorname {b1} +1\right ) x -\operatorname {d1} \right ) y^{\prime }+\operatorname {a1} \operatorname {b1} \operatorname {d1} = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
17.26 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+2 \left (n +1+\left (n +1-2 l \right ) x -l \,x^{2}\right ) y^{\prime }+\left (2 l \left (p -n -1\right ) x +2 p l +m \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.912 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+\left (x^{2}+x -1\right ) y^{\prime }-\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.96 |
|
|
\[ {}x \left (x +3\right ) y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+y-\left (20 x +30\right ) \left (x^{2}+3 x \right )^{\frac {7}{3}} = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
10.214 |
|
|
\[ {}\left (x^{2}+3 x +4\right ) y^{\prime \prime }+\left (x^{2}+x +1\right ) y^{\prime }-\left (2 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.81 |
|
|
\[ {}\left (-1+x \right ) \left (-2+x \right ) y^{\prime \prime }-\left (2 x -3\right ) y^{\prime }+y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
163.593 |
|
|
\[ {}\left (-2+x \right )^{2} y^{\prime \prime }-\left (-2+x \right ) y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.362 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-\left (2 x^{2}+l -5 x \right ) y^{\prime }-\left (4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.983 |
|
|
\[ {}2 x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (x a +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Jacobi] |
✗ |
N/A |
0.688 |
|
|
\[ {}2 x \left (-1+x \right ) y^{\prime \prime }+\left (\left (2 v +5\right ) x -2 v -3\right ) y^{\prime }+\left (v +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Jacobi] |
✗ |
N/A |
0.889 |
|
|
\[ {}\left (2 x^{2}+6 x +4\right ) y^{\prime \prime }+\left (10 x^{2}+21 x +8\right ) y^{\prime }+\left (12 x^{2}+17 x +8\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.766 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.299 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (4 a^{2} x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.466 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-\left (-4 k x +4 m^{2}+x^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.271 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-v^{2}+x \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.366 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-x^{2}+2 \left (1-m +2 l \right ) x -m^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.898 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (4 x^{2}+1\right ) y-4 \sqrt {x^{3}}\, {\mathrm e}^{x} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.011 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (x^{2} a +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.624 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+f \left (x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.18 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+5 x y^{\prime }-y-\ln \left (x \right ) = 0 \] |
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]] |
✓ |
✓ |
4.365 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }-\left (4 x^{2}+12 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.931 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (2 x -1\right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.582 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+6\right ) \left (x^{2}-4\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.589 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{2} \ln \left (x \right ) y^{\prime }+\left (x^{2} \ln \left (x \right )^{2}+2 x -8\right ) y-4 x^{2} \sqrt {{\mathrm e}^{x} x^{-x}} = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.788 |
|
|
\[ {}\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }-12 y-3 x -1 = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.525 |
|
|
\[ {}x \left (4 x -1\right ) y^{\prime \prime }+\left (\left (4 a +2\right ) x -a \right ) y^{\prime }+a \left (a -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.006 |
|
|
\[ {}\left (3 x -1\right )^{2} y^{\prime \prime }+3 \left (3 x -1\right ) y^{\prime }-9 y-\ln \left (3 x -1\right )^{2} = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.682 |
|
|
\[ {}9 x \left (-1+x \right ) y^{\prime \prime }+3 \left (2 x -1\right ) y^{\prime }-20 y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.776 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+\left (3+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.473 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }-\left (5+4 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.667 |
|
|
\[ {}\left (27 x^{2}+4\right ) y^{\prime \prime }+27 x y^{\prime }-3 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.43 |
|
|
\[ {}48 x \left (-1+x \right ) y^{\prime \prime }+\left (152 x -40\right ) y^{\prime }+53 y = 0 \] |
1 |
0 |
1 |
unknown |
[_Jacobi] |
✗ |
N/A |
2.178 |
|
|
\[ {}50 x \left (-1+x \right ) y^{\prime \prime }+25 \left (2 x -1\right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[_Jacobi, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.267 |
|
|
\[ {}144 x \left (-1+x \right ) y^{\prime \prime }+\left (120 x -48\right ) y^{\prime }+y = 0 \] |
1 |
0 |
1 |
unknown |
[_Jacobi] |
✗ |
N/A |
0.937 |
|
|
\[ {}144 x \left (-1+x \right ) y^{\prime \prime }+\left (168 x -96\right ) y^{\prime }+y = 0 \] |
1 |
0 |
1 |
unknown |
[_Jacobi] |
✗ |
N/A |
0.765 |
|
|
\[ {}a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
66.312 |
|
|
\[ {}\operatorname {a2} \,x^{2} y^{\prime \prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x \right ) y^{\prime }+\left (\operatorname {a0} \,x^{2}+\operatorname {b0} x +\operatorname {c0} \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.572 |
|
|
\[ {}\left (x^{2} a +1\right ) y^{\prime \prime }+a x y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
187.713 |
|
|
\[ {}\left (a^{2} x^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.988 |
|
|
\[ {}\left (a^{2} x^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime }-2 a^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[_Gegenbauer] |
✓ |
✓ |
1.786 |
|
|
\[ {}\left (x^{2} a +b x \right ) y^{\prime \prime }+2 b y^{\prime }-2 a y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.225 |
|
|
\[ {}\operatorname {A2} \left (x a +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (x a +b \right ) y^{\prime }+\operatorname {A0} \left (x a +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.467 |
|
|
\[ {}\left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (d x +f \right ) y^{\prime }+g y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
4.107 |
|
|
\[ {}x^{3} y^{\prime \prime }+x y^{\prime }-\left (2 x +3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.037 |
|
|
\[ {}x^{3} y^{\prime \prime }+2 x y^{\prime }-y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.298 |
|
|
\[ {}x^{3} y^{\prime \prime }+x^{2} y^{\prime }+\left (x^{2} a +b x +a \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.327 |
|
|
\[ {}x^{3} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.124 |
|
|
\[ {}x^{3} y^{\prime \prime }-x^{2} y^{\prime }+x y-\ln \left (x \right )^{3} = 0 \] |
1 |
1 |
1 |
kovacic, 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]] |
✓ |
✓ |
2.979 |
|
|
\[ {}x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.477 |
|
|
\[ {}x^{3} y^{\prime \prime }+3 x^{2} y^{\prime }+x y-1 = 0 \] |
1 |
1 |
1 |
kovacic, 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]] |
✓ |
✓ |
2.872 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 x^{2}+1\right ) y^{\prime }-v \left (v +1\right ) x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.915 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+2 \left (x^{2}-1\right ) y^{\prime }-2 x y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.48 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 \left (n +1\right ) x^{2}+2 n +1\right ) y^{\prime }-\left (v -n \right ) \left (v +n +1\right ) x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.099 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }-\left (2 \left (n -1\right ) x^{2}+2 n -1\right ) y^{\prime }+\left (v +n \right ) \left (-v +n -1\right ) x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.049 |
|
|
\[ {}x \left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime }+y a \,x^{3} = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.132 |
|
|
\[ {}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_elliptic, _class_II]] |
✗ |
N/A |
82.461 |
|
|
\[ {}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (3 x^{2}-1\right ) y^{\prime }+x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_elliptic, _class_I]] |
✗ |
N/A |
0.746 |
|
|
\[ {}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (x^{2} a +b \right ) y^{\prime }+c x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.424 |
|
|
\[ {}x \left (x^{2}+2\right ) y^{\prime \prime }-y^{\prime }-6 x y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.017 |
|
|
\[ {}x \left (x^{2}-2\right ) y^{\prime \prime }-\left (x^{3}+3 x^{2}-2 x -2\right ) y^{\prime }+\left (x^{2}+4 x +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.775 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+\left (2 x +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.434 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+2 x \left (3 x +2\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 \left (-2+x \right ) y^{\prime }}{x \left (-1+x \right )}+\frac {2 \left (1+x \right ) y}{x^{2} \left (-1+x \right )} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.983 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (5 x -4\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (9 x -6\right ) y}{x^{2} \left (-1+x \right )} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.243 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\left (a +b +1\right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (-1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.389 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{1+x}-\frac {y}{x \left (1+x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.036 |
|
|
\[ {}y^{\prime \prime } = \frac {2 y^{\prime }}{x \left (-2+x \right )}-\frac {y}{x^{2} \left (-2+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
78.828 |
|
|
\[ {}y^{\prime \prime } = \frac {2 y}{x \left (-1+x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.786 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (-1+x \right ) \left (x -a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{x \left (-1+x \right ) \left (x -a \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
3.026 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (-b +x \right ) \left (x -c \right )}-\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (-b +x \right ) \left (x -c \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
5.145 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (x -4\right ) y^{\prime }}{2 x \left (-2+x \right )}-\frac {\left (x -3\right ) y}{2 x^{2} \left (-2+x \right )} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.705 |
|
|
\[ {}y^{\prime \prime } = \frac {y^{\prime }}{1+x}-\frac {\left (1+3 x \right ) y}{4 x^{2} \left (1+x \right )} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.703 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}+\frac {v \left (v +1\right ) y}{4 x^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.895 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\left (1+a \right ) x -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (-1+x \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.076 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (x a +b \right ) y}{4 x \left (-1+x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.802 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (1-3 x \right ) y}{\left (-1+x \right ) \left (2 x -1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
8.41 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (3 x +a +2 b \right ) y^{\prime }}{2 \left (x +a \right ) \left (x +b \right )}-\frac {\left (-b +a \right ) y}{4 \left (x +a \right )^{2} \left (x +b \right )} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.638 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (6 x -1\right ) y^{\prime }}{3 x \left (-2+x \right )}+\frac {y}{3 x^{2} \left (-2+x \right )} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.767 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (a \left (b +2\right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (x a +1\right ) x^{2}}-\frac {\left (a b x -c d \right ) y}{\left (x a +1\right ) x^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.325 |
|
|
\[ {}y^{\prime \prime } = \frac {2 \left (x a +2 b \right ) y^{\prime }}{x \left (x a +b \right )}-\frac {\left (2 x a +6 b \right ) y}{\left (x a +b \right ) x^{2}} \] |
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]] |
✓ |
✓ |
0.803 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 x a +b \right ) y^{\prime }}{x \left (x a +b \right )}-\frac {\left (a v x -b \right ) y}{\left (x a +b \right ) x^{2}}+A x \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
84.643 |
|
|
\[ {}y^{\prime \prime } = -\frac {a y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.435 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}} \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.482 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.317 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x^{3}}+\frac {2 y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.438 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (a +b \right ) y^{\prime }}{x^{2}}-\frac {\left (x \left (a +b \right )+a b \right ) y}{x^{4}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.729 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {y}{x^{4}} \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.289 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}} \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.363 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.46 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {a^{2} y}{x^{4}} \] |
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)]‘]] |
✓ |
✓ |
1.519 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.283 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 \left (x +a \right ) y^{\prime }}{x^{2}}-\frac {b y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.389 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.269 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {2 y}{x^{4}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.865 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (x^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}+\frac {x y}{x^{3}+1} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.352 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.977 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (x^{2} a +a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.147 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (x^{2}-2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (x^{2}-2\right ) y}{x^{2} \left (x^{2}-1\right )} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.934 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2} \left (x^{2}-1\right )} \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.228 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {v \left (v +1\right ) y}{x^{2}} \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
11.417 |
|
|
\[ {}y^{\prime \prime } = \frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (a \left (1+a \right )-a \,x^{2} \left (a +3\right )\right ) y}{x^{2} \left (x^{2}-1\right )} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
54.459 |
|
|
\[ {}x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 x^{2} a +n \left (n +1\right ) \left (x^{2}-1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
38.754 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (x^{2} a +a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {b y}{x^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.461 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (a -1\right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (a -1\right )-v \left (v +1\right )\right )-a \left (1+a \right )\right ) y}{x^{2} \left (x^{2}-1\right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
5.592 |
|
|
\[ {}y^{\prime \prime } = -\frac {a y}{\left (x^{2}+1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[_Halm] |
✓ |
✓ |
0.843 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {y}{\left (x^{2}+1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, 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, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.445 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (n +1\right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.914 |
|
|
\[ {}y^{\prime \prime } = -\frac {a x y^{\prime }}{x^{2}+1}-\frac {b y}{\left (x^{2}+1\right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.74 |
|
|
\[ {}y^{\prime \prime } = -\frac {a y}{\left (x^{2}-1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.053 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {a^{2} y}{\left (x^{2}-1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.851 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.027 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (\left (x^{2}-1\right ) \left (x^{2} a +b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.645 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.523 |
|
|
\[ {}y^{\prime \prime } = \frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}-\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.355 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.651 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 x^{2}+a \right ) y^{\prime }}{x \left (x^{2}+a \right )}-\frac {b y}{x^{2} \left (x^{2}+a \right )} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.229 |
|
|
\[ {}y^{\prime \prime } = -\frac {b^{2} y}{\left (a^{2}+x^{2}\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.93 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 \left (x^{2}-1\right ) y^{\prime }}{x \left (-1+x \right )^{2}}-\frac {\left (-2 x^{2}+2 x +2\right ) y}{x^{2} \left (-1+x \right )^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.513 |
|
|
\[ {}y^{\prime \prime } = \frac {12 y}{\left (1+x \right )^{2} \left (x^{2}+2 x +3\right )} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.257 |
|
|
\[ {}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.246 |
|
|
\[ {}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}}+c \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
54.631 |
|
|
\[ {}y^{\prime \prime } = \frac {c y}{\left (x -a \right )^{2} \left (-b +x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.08 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) \left (x -a \right )^{2} \left (-b +x \right )+\left (1-\alpha -\beta \right ) \left (-b +x \right )^{2} \left (x -a \right )\right ) y^{\prime }}{\left (x -a \right )^{2} \left (-b +x \right )^{2}}-\frac {\alpha \beta \left (-b +a \right )^{2} y}{\left (x -a \right )^{2} \left (-b +x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.648 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}} \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.276 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (x^{2} a +a -3\right ) y}{4 \left (x^{2}+1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[_Halm] |
✓ |
✓ |
3.441 |
|
|
\[ {}y^{\prime \prime } = \frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.484 |
|
|
\[ {}y^{\prime \prime } = \frac {3 y}{4 \left (x^{2}+x +1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.257 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (v \left (v +1\right ) \left (-1+x \right )-x \,a^{2}\right ) y}{4 x^{2} \left (-1+x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.017 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (-v \left (v +1\right ) \left (-1+x \right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.108 |
|
|
\[ {}y^{\prime \prime } = -\frac {3 y}{16 x^{2} \left (-1+x \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.914 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (7 x^{2} a +5\right ) y^{\prime }}{x \left (x^{2} a +1\right )}-\frac {\left (15 x^{2} a +5\right ) y}{x^{2} \left (x^{2} a +1\right )} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.807 |
|
|
\[ {}y^{\prime \prime } = -\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.845 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (b \,x^{2}+c x +d \right ) y}{a \,x^{2} \left (-1+x \right )^{2}} \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
38.66 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {c y}{x^{2} \left (x a +b \right )^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.206 |
|
|
\[ {}y^{\prime \prime } = -\frac {y}{\left (x a +b \right )^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.93 |
|
|
\[ {}y^{\prime \prime } = -\frac {A y}{\left (x^{2} a +b x +c \right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.532 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x^{4}}+\frac {y}{x^{5}} \] |
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]] |
✓ |
✓ |
1.842 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.394 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (1+3 x \right ) y^{\prime }}{\left (-1+x \right ) \left (1+x \right )}-\frac {36 \left (1+x \right )^{2} y}{\left (-1+x \right )^{2} \left (3 x +5\right )^{2}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.965 |
|
|
\[ {}y^{\prime \prime } = \frac {y^{\prime }}{x}-\frac {a y}{x^{6}} \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.412 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (3 x^{2}+a \right ) y^{\prime }}{x^{3}}-\frac {b y}{x^{6}} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.582 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (1+a \right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
4.409 |
|
|
\[ {}y^{\prime \prime } = -\left (\frac {1-\operatorname {a1} -\operatorname {b1}}{x -\operatorname {c1}}+\frac {1-\operatorname {a2} -\operatorname {b2}}{x -\operatorname {c2}}+\frac {1-\operatorname {a3} -\operatorname {b3}}{x -\operatorname {c3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {a1} \operatorname {b1} \left (\operatorname {c1} -\operatorname {c3} \right ) \left (\operatorname {c1} -\operatorname {c2} \right )}{x -\operatorname {c1}}+\frac {\operatorname {a2} \operatorname {b2} \left (\operatorname {c2} -\operatorname {c1} \right ) \left (\operatorname {c2} -\operatorname {c3} \right )}{x -\operatorname {c2}}+\frac {\operatorname {a3} \operatorname {b3} \left (\operatorname {c3} -\operatorname {c2} \right ) \left (\operatorname {c3} -\operatorname {c1} \right )}{x -\operatorname {c3}}\right ) y}{\left (x -\operatorname {c1} \right ) \left (x -\operatorname {c2} \right ) \left (x -\operatorname {c3} \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
42.941 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (-2 x^{2}+1\right ) y}{4 x^{6}} \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.464 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}} \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.635 |
|
|
\[ {}y^{\prime \prime } = -\frac {27 x y}{16 \left (x^{3}-1\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
80.596 |
|
|
\[ {}y^{\prime \prime } = -\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
13.112 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (x^{2} \left (\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right )+\left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )+\left (x^{2}-\operatorname {a3} \right ) \left (x^{2}-\operatorname {a1} \right )\right )-\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}-\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
12.957 |
|
|
\[ {}y^{\prime \prime } = -a \,x^{2 a -1} x^{-2 a} y^{\prime }-b^{2} x^{-2 a} y \] |
1 |
1 |
1 |
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)]‘]] |
✓ |
✓ |
0.555 |
|
|
|
|||||||||
|
\[ {}y^{\prime \prime } = -\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}-\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.974 |
|
|
\[ {}y^{\prime \prime } = \frac {y}{1+{\mathrm e}^{x}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
N/A |
0.138 |
|
|
\[ {}y^{\prime \prime } = \frac {y^{\prime }}{x \ln \left (x \right )}+\ln \left (x \right )^{2} y \] |
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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.506 |
|
|
\[ {}y^{\prime \prime } = \frac {y^{\prime }}{x \left (-1+\ln \left (x \right )\right )}-\frac {y}{x^{2} \left (-1+\ln \left (x \right )\right )} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.015 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.822 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}-\left (-a^{2}+n^{2}\right ) y \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.396 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 n +1\right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\left (v +n +1\right ) \left (v -n \right ) y \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.869 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\sin \left (x \right )^{2}-\cos \left (x \right )\right ) y^{\prime }}{\sin \left (x \right )}-y \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.183 |
|
|
\[ {}y^{\prime \prime } = -\frac {x \sin \left (x \right ) y^{\prime }}{\cos \left (x \right ) x -\sin \left (x \right )}+\frac {\sin \left (x \right ) y}{\cos \left (x \right ) x -\sin \left (x \right )} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
30.27 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\sin \left (x \right ) x^{2}-2 \cos \left (x \right ) x \right ) y^{\prime }}{x^{2} \cos \left (x \right )}-\frac {\left (2 \cos \left (x \right )-x \sin \left (x \right )\right ) y}{x^{2} \cos \left (x \right )} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.051 |
|
|
\[ {}\cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.911 |
|
|
\[ {}y^{\prime \prime } = -\frac {a \left (n -1\right ) \sin \left (2 x a \right ) y^{\prime }}{\cos \left (x a \right )^{2}}-\frac {n \,a^{2} \left (\left (n -1\right ) \sin \left (x a \right )^{2}+\cos \left (x a \right )^{2}\right ) y}{\cos \left (x a \right )^{2}} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.421 |
|
|
\[ {}y^{\prime \prime } = \frac {2 y}{\sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.715 |
|
|
\[ {}y^{\prime \prime } = -\frac {a y}{\sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.625 |
|
|
\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-\left (a \sin \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.832 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.194 |
|
|
\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
5.682 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (-\left (a^{2} b^{2}-\left (1+a \right )^{2}\right ) \sin \left (x \right )^{2}-a \left (1+a \right ) b \sin \left (2 x \right )-a \left (a -1\right )\right ) y}{\sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.829 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.32 |
|
|
\[ {}y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}+\frac {y}{\sin \left (x \right )^{2}} \] |
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], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.096 |
|
|
\[ {}y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (v \left (v +1\right ) \sin \left (x \right )^{2}-n^{2}\right ) y}{\sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.603 |
|
|
\[ {}y^{\prime \prime } = \frac {\cos \left (2 x \right ) y^{\prime }}{\sin \left (2 x \right )}-2 y \] |
1 |
1 |
1 |
second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
10.22 |
|
|
\[ {}y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (-17 \sin \left (x \right )^{2}-1\right ) y}{4 \sin \left (x \right )^{2}} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.096 |
|
|
\[ {}y^{\prime \prime } = -\frac {\sin \left (x \right ) y^{\prime }}{\cos \left (x \right )}-\frac {\left (2 x^{2}+x^{2} \sin \left (x \right )^{2}-24 \cos \left (x \right )^{2}\right ) y}{4 x^{2} \cos \left (x \right )^{2}}+\sqrt {\cos \left (x \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
6.938 |
|
|
\[ {}y^{\prime \prime } = -\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}-\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
3.235 |
|
|
\[ {}y^{\prime \prime } = -\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.712 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.677 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (3 \sin \left (x \right )^{2}+1\right ) y^{\prime }}{\cos \left (x \right ) \sin \left (x \right )}+\frac {\sin \left (x \right )^{2} y}{\cos \left (x \right )^{2}} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.738 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (-a \cos \left (x \right )^{2} \sin \left (x \right )^{2}-m \left (m -1\right ) \sin \left (x \right )^{2}-n \left (n -1\right ) \cos \left (x \right )^{2}\right ) y}{\cos \left (x \right )^{2} \sin \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.011 |
|
|
\[ {}y^{\prime \prime } = \frac {\phi ^{\prime }\left (x \right ) y^{\prime }}{\phi \left (x \right )-\phi \left (a \right )}-\frac {\left (-n \left (n +1\right ) \left (\phi \left (x \right )-\phi \left (a \right )\right )^{2}+D^{\left (2\right )}\left (\phi \right )\left (a \right )\right ) y}{\phi \left (x \right )-\phi \left (a \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.618 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\phi \left (x^{3}\right )-\phi \left (x \right ) \phi ^{\prime }\left (x \right )-\phi ^{\prime \prime }\left (x \right )\right ) y^{\prime }}{\phi ^{\prime }\left (x \right )+\phi \left (x \right )^{2}}-\frac {\left ({\phi ^{\prime }\left (x \right )}^{2}-\phi \left (x \right )^{2} \phi ^{\prime }\left (x \right )-\phi \left (x \right ) \phi ^{\prime \prime }\left (x \right )\right ) y}{\phi ^{\prime }\left (x \right )+\phi \left (x \right )^{2}} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.445 |
|
|
\[ {}y^{\prime \prime } = \frac {2 \,\operatorname {JacobiSN}\left (x , k\right ) \operatorname {JacobiCN}\left (x , k\right ) \operatorname {JacobiDN}\left (x , k\right ) y^{\prime }-2 \left (1-2 \left (k^{2}+1\right ) \operatorname {JacobiSN}\left (a , k\right )^{2}+3 k^{2} \operatorname {JacobiSN}\left (a , k\right )^{4}\right ) y}{\operatorname {JacobiSN}\left (x , k\right )^{2}-\operatorname {JacobiSN}\left (a , k\right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
82.612 |
|
|
\[ {}y^{\prime \prime } = -\frac {x y^{\prime }}{f \left (x \right )}+\frac {y}{f \left (x \right )} \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.244 |
|
|
\[ {}y^{\prime \prime } = -\frac {f^{\prime }\left (x \right ) y^{\prime }}{2 f \left (x \right )}-\frac {g \left (x \right ) y}{f \left (x \right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.392 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 f \left (x \right ) {g^{\prime }\left (x \right )}^{2} g \left (x \right )-\left (g \left (x \right )^{2}-1\right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )\right ) y^{\prime }}{f \left (x \right ) g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}-\frac {\left (\left (g \left (x \right )^{2}-1\right ) \left (f^{\prime }\left (x \right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )-f \left (x \right ) f^{\prime \prime }\left (x \right ) g^{\prime }\left (x \right )\right )-\left (2 f^{\prime }\left (x \right ) g \left (x \right )+v \left (v +1\right ) f \left (x \right ) g^{\prime }\left (x \right )\right ) f \left (x \right ) {g^{\prime }\left (x \right )}^{2}\right ) y}{f \left (x \right )^{2} g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )} \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.621 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (-1+x \right ) y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.633 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (-x -1\right ) y}{x^{4}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.62 |
|
|
\[ {}y^{\prime \prime } = -\frac {b^{2} y}{\left (-a^{2}+x^{2}\right )^{2}} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.463 |
|
|
\[ {}y^{\prime \prime \prime }-\lambda y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.118 |
|
|
\[ {}y^{\prime \prime \prime }+y a \,x^{3}-b x = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.082 |
|
|
\[ {}y^{\prime \prime \prime }-a \,x^{b} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.082 |
|
|
\[ {}y^{\prime \prime \prime }+3 y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.355 |
|
|
\[ {}y^{\prime \prime \prime }-a^{2} y^{\prime }-{\mathrm e}^{2 x a} \sin \left (x \right )^{2} = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.267 |
|
|
\[ {}y^{\prime \prime \prime }+2 a x y^{\prime }+a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.122 |
|
|
\[ {}y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+\left (a +b -1\right ) x y^{\prime }-y a b = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.134 |
|
|
\[ {}y^{\prime \prime \prime }+x^{2 c -2} y^{\prime }+\left (c -1\right ) x^{2 c -3} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.083 |
|
|
\[ {}y^{\prime \prime \prime }-3 \left (2 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }+b y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.088 |
|
|
\[ {}y^{\prime \prime \prime }+\left (-n^{2}+1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }+\frac {\left (\left (-n^{2}+1\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right )-a \right ) y}{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.095 |
|
|
\[ {}y^{\prime \prime \prime }-\left (4 n \left (n +1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }-2 n \left (n +1\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime \prime }+\left (A \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }+B \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime \prime }-\left (3 k^{2} \operatorname {JacobiSN}\left (z , x\right )^{2}+a \right ) y^{\prime }+\left (b +c \operatorname {JacobiSN}\left (z , x\right )^{2}-3 k^{2} \operatorname {JacobiSN}\left (z , x\right ) \operatorname {JacobiCN}\left (z , x\right ) \operatorname {JacobiDN}\left (z , x\right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.093 |
|
|
\[ {}y^{\prime \prime \prime }-\left (6 k^{2} \sin \left (x \right )^{2}+a \right ) y^{\prime }+b y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.088 |
|
|
\[ {}y^{\prime \prime \prime }+2 f \left (x \right ) y^{\prime }+f^{\prime }\left (x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.086 |
|
|
\[ {}y^{\prime \prime \prime }-2 y^{\prime \prime }-3 y^{\prime }+10 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.272 |
|
|
\[ {}y^{\prime \prime \prime }-2 y^{\prime \prime }-a^{2} y^{\prime }+2 a^{2} y-\sinh \left (x \right ) = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.895 |
|
|
\[ {}y^{\prime \prime \prime }-3 a y^{\prime \prime }+3 a^{2} y^{\prime }-a^{3} y-{\mathrm e}^{x a} = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.161 |
|
|
\[ {}y^{\prime \prime \prime }+\operatorname {a2} y^{\prime \prime }+\operatorname {a1} y^{\prime }+\operatorname {a0} y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.241 |
|
|
\[ {}y^{\prime \prime \prime }-6 x y^{\prime \prime }+2 \left (4 x^{2}+2 a -1\right ) y^{\prime }-8 y a x = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.19 |
|
|
\[ {}y^{\prime \prime \prime }+3 a x y^{\prime \prime }+3 a^{2} x^{2} y^{\prime }+a^{3} x^{3} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.191 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime } \sin \left (x \right )-2 y^{\prime } \cos \left (x \right )+y \sin \left (x \right )-\ln \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _fully, _exact, _linear]] |
✗ |
N/A |
0.092 |
|
|
\[ {}y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime }+y^{\prime }+f \left (x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime \prime }+f \left (x \right ) \left (x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.087 |
|
|
\[ {}y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime }+g \left (x \right ) y^{\prime }+\left (f \left (x \right ) g \left (x \right )+g^{\prime }\left (x \right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.093 |
|
|
\[ {}y^{\prime \prime \prime }+3 f \left (x \right ) y^{\prime \prime }+\left (f^{\prime }\left (x \right )+2 f \left (x \right )^{2}+4 g \left (x \right )\right ) y^{\prime }+\left (4 f \left (x \right ) g \left (x \right )+2 g^{\prime }\left (x \right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.099 |
|
|
\[ {}4 y^{\prime \prime \prime }-8 y^{\prime \prime }-11 y^{\prime }-3 y+18 \,{\mathrm e}^{x} = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.579 |
|
|
\[ {}27 y^{\prime \prime \prime }-36 n^{2} \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }-2 n \left (n +3\right ) \left (4 n -3\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.092 |
|
|
\[ {}x y^{\prime \prime \prime }+3 y^{\prime \prime }+x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.228 |
|
|
\[ {}x y^{\prime \prime \prime }+3 y^{\prime \prime }-a \,x^{2} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.25 |
|
|
\[ {}x y^{\prime \prime \prime }+\left (a +b \right ) y^{\prime \prime }-x y^{\prime }-a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.234 |
|
|
\[ {}x y^{\prime \prime \prime }-\left (x +2 v \right ) y^{\prime \prime }-\left (x -2 v -1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.394 |
|
|
\[ {}x y^{\prime \prime \prime }+\left (x^{2}-3\right ) y^{\prime \prime }+4 x y^{\prime }+2 y-f \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _fully, _exact, _linear]] |
✗ |
N/A |
0.091 |
|
|
\[ {}2 x y^{\prime \prime \prime }+3 y^{\prime \prime }+y a x -b = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.088 |
|
|
\[ {}2 x y^{\prime \prime \prime }-4 \left (x +\nu -1\right ) y^{\prime \prime }+\left (2 x +6 \nu -5\right ) y^{\prime }+\left (1-2 \nu \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.362 |
|
|
\[ {}2 x y^{\prime \prime \prime }+3 \left (2 x a +k \right ) y^{\prime \prime }+6 \left (a k +b x \right ) y^{\prime }+\left (3 b k +2 c x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.424 |
|
|
\[ {}\left (-2+x \right ) x y^{\prime \prime \prime }-\left (-2+x \right ) x y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _exact, _linear, _homogeneous]] |
✗ |
N/A |
0.26 |
|
|
\[ {}\left (2 x -1\right ) y^{\prime \prime \prime }-8 x y^{\prime }+8 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.283 |
|
|
\[ {}\left (2 x -1\right ) y^{\prime \prime \prime }+\left (x +4\right ) y^{\prime \prime }+2 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
5.994 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-6 y^{\prime }+a \,x^{2} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.251 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }+\left (1+x \right ) y^{\prime \prime }-y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.05 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-x y^{\prime \prime }+\left (x^{2}+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.652 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }+3 x y^{\prime \prime }+\left (4 a^{2} x^{2 a}+1-4 \nu ^{2} a^{2}\right ) y^{\prime } = 4 a^{3} x^{2 a -1} y \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.098 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-3 \left (x -m \right ) x y^{\prime \prime }+\left (2 x^{2}+4 \left (n -m \right ) x +m \left (2 m -1\right )\right ) y^{\prime }-2 n \left (2 x -2 m +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.428 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }+4 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }+3 x y-f \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.098 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }+5 x y^{\prime \prime }+4 y^{\prime }-\ln \left (x \right ) = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.429 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }+6 x y^{\prime \prime }+6 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.44 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }+6 x y^{\prime \prime }+6 y^{\prime }+a \,x^{2} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.265 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-3 \left (p +q \right ) x y^{\prime \prime }+3 p \left (3 q +1\right ) y^{\prime }-x^{2} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.262 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-2 \left (n +1\right ) x y^{\prime \prime }+\left (x^{2} a +6 n \right ) y^{\prime }-2 y a x = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.267 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-\left (x^{2}-2 x \right ) y^{\prime \prime }-\left (x^{2}+\nu ^{2}-\frac {1}{4}\right ) y^{\prime }+\left (x^{2}-2 x +\nu ^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.452 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-\left (x +\nu \right ) x y^{\prime \prime }+\nu \left (2 x +1\right ) y^{\prime }-\nu \left (1+x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.34 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-2 \left (x^{2}-x \right ) y^{\prime \prime }+\left (x^{2}-2 x +\frac {1}{4}-\nu ^{2}\right ) y^{\prime }+\left (\nu ^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.378 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }-\left (x^{4}-6 x \right ) y^{\prime \prime }-\left (2 x^{3}-6\right ) y^{\prime }+2 x^{2} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.286 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime \prime }+8 x y^{\prime \prime }+10 y^{\prime }-3+\frac {1}{x^{2}}-2 \ln \left (x \right ) = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.729 |
|
|
\[ {}\left (x^{2}+2\right ) y^{\prime \prime \prime }-2 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }-2 x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.059 |
|
|
\[ {}2 x \left (-1+x \right ) y^{\prime \prime \prime }+3 \left (2 x -1\right ) y^{\prime \prime }+\left (2 x a +b \right ) y^{\prime }+a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.436 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+\left (-\nu ^{2}+1\right ) x y^{\prime }+\left (a \,x^{3}+\nu ^{2}-1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.261 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+\left (4 x^{3}+\left (-4 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (4 \nu ^{2}-1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.255 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+\left (a \,x^{2 \nu }+1-\nu ^{2}\right ) x y^{\prime }+\left (b \,x^{3 \nu }+a \left (\nu -1\right ) x^{2 \nu }+\nu ^{2}-1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.1 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+3 x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y-6 x^{3} \left (-1+x \right ) \ln \left (x \right )+x^{3} \left (x +8\right ) = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.369 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+3 x^{2} y^{\prime \prime }+\left (-a^{2}+1\right ) x y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.444 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }-4 x^{2} y^{\prime \prime }+\left (x^{2}+8\right ) x y^{\prime }-2 \left (x^{2}+4\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.251 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+6 x^{2} y^{\prime \prime }+\left (a \,x^{3}-12\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.246 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+3 \left (1-a \right ) x^{2} y^{\prime \prime }+\left (4 b^{2} c^{2} x^{2 c +1}+1-4 \nu ^{2} c^{2}+3 a \left (a -1\right ) x \right ) y^{\prime }+\left (4 b^{2} c^{2} \left (c -a \right ) x^{2 c}+a \left (4 \nu ^{2} c^{2}-a^{2}\right )\right ) y = 0 \] |
1 |
0 |
0 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.107 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+x^{2} \left (x +3\right ) y^{\prime \prime }+5 \left (-6+x \right ) x y^{\prime }+\left (4 x +30\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.277 |
|
|
\[ {}x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+\ln \left (x \right )+2 x y^{\prime }-y-2 x^{3} = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
15.638 |
|
|
\[ {}\left (x^{2}+1\right ) x y^{\prime \prime \prime }+3 \left (2 x^{2}+1\right ) y^{\prime \prime }-12 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.233 |
|
|
\[ {}\left (x +3\right ) x^{2} y^{\prime \prime \prime }-3 x \left (2+x \right ) y^{\prime \prime }+6 \left (1+x \right ) y^{\prime }-6 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.356 |
|
|
\[ {}2 \left (x -\operatorname {a1} \right ) \left (x -\operatorname {a2} \right ) \left (x -\operatorname {a3} \right ) y^{\prime \prime \prime }+\left (9 x^{2}-6 \left (\operatorname {a1} +\operatorname {a2} +\operatorname {a3} \right ) x +3 \operatorname {a1} \operatorname {a2} +3 \operatorname {a1} \operatorname {a3} +3 \operatorname {a2} \operatorname {a3} \right ) y^{\prime \prime }-2 \left (\left (n^{2}+n -3\right ) x +b \right ) y^{\prime }-n \left (n +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.168 |
|
|
\[ {}\left (1+x \right ) x^{3} y^{\prime \prime \prime }-\left (4 x +2\right ) x^{2} y^{\prime \prime }+\left (10 x +4\right ) x y^{\prime }-4 \left (1+3 x \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.52 |
|
|
\[ {}4 x^{4} y^{\prime \prime \prime }-4 x^{3} y^{\prime \prime }+4 x^{2} y^{\prime }-1 = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_3rd_order, _missing_y]] |
✓ |
✓ |
0.365 |
|
|
\[ {}\left (x^{2}+1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-4 \left (3 x^{2}+1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.267 |
|
|
\[ {}x^{6} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.054 |
|
|
\[ {}x^{6} y^{\prime \prime \prime }+6 x^{5} y^{\prime \prime }+a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.056 |
|
|
\[ {}x^{2} \left (x^{4}+2 x^{2}+2 x +1\right ) y^{\prime \prime \prime }-\left (2 x^{6}+3 x^{4}-6 x^{2}-6 x -1\right ) y^{\prime \prime }+\left (x^{6}-6 x^{3}-15 x^{2}-12 x -2\right ) y^{\prime }+\left (x^{4}+4 x^{3}+8 x^{2}+6 x +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
7.201 |
|
|
\[ {}\left (x -a \right )^{3} \left (-b +x \right )^{3} y^{\prime \prime \prime }-c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
78.088 |
|
|
\[ {}y^{\prime \prime \prime } \sin \left (x \right )+\left (2 \cos \left (x \right )+1\right ) y^{\prime \prime }-y^{\prime } \sin \left (x \right )-\cos \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _missing_y]] |
✗ |
N/A |
65.457 |
|
|
\[ {}\left (\sin \left (x \right )+x \right ) y^{\prime \prime \prime }+3 \left (\cos \left (x \right )+1\right ) y^{\prime \prime }-3 y^{\prime } \sin \left (x \right )-y \cos \left (x \right )+\sin \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _fully, _exact, _linear]] |
✗ |
N/A |
0.099 |
|
|
\[ {}y^{\prime \prime \prime } \sin \left (x \right )^{2}+3 y^{\prime \prime } \sin \left (x \right ) \cos \left (x \right )+\left (\cos \left (2 x \right )+4 \nu \left (\nu +1\right ) \sin \left (x \right )^{2}\right ) y^{\prime }+2 \nu \left (\nu +1\right ) y \sin \left (2 x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.105 |
|
|
\[ {}f^{\prime }\left (x \right ) y^{\prime \prime }+f \left (x \right ) y^{\prime \prime \prime }+g^{\prime }\left (x \right ) y^{\prime }+g \left (x \right ) y^{\prime \prime }+h^{\prime }\left (x \right ) y+h \left (x \right ) y^{\prime }+A \left (x \right ) \left (f \left (x \right ) y^{\prime \prime }+g \left (x \right ) y^{\prime }+h \left (x \right ) y\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.118 |
|
|
\[ {}y^{\prime \prime \prime }+x y^{\prime }+n y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.131 |
|
|
\[ {}y^{\prime \prime \prime }-x y^{\prime }-n y = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.129 |
|
|
\[ {}y^{\prime \prime \prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _quadrature]] |
✓ |
✓ |
0.151 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y-f = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
1.329 |
|
|
\[ {}y^{\prime \prime \prime \prime }+\lambda y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.088 |
|
|
\[ {}y^{\prime \prime \prime \prime }-12 y^{\prime \prime }+12 y-16 x^{4} {\mathrm e}^{x^{2}} = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
76.197 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 a^{2} y^{\prime \prime }+a^{4} y-\cosh \left (x a \right ) = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.877 |
|
|
\[ {}y^{\prime \prime \prime \prime }+\left (\lambda +1\right ) a^{2} y^{\prime \prime }+\lambda \,a^{4} y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.453 |
|
|
\[ {}y^{\prime \prime \prime \prime }+a \left (b x -1\right ) y^{\prime \prime }+a b y^{\prime }+\lambda y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
❇ |
N/A |
0.184 |
|
|
\[ {}y^{\prime \prime \prime \prime }+\left (x^{2} a +b \lambda +c \right ) y^{\prime \prime }+\left (x^{2} a +\beta \lambda +\gamma \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
❇ |
N/A |
0.211 |
|
|
\[ {}y^{\prime \prime \prime \prime }+a \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime \prime }+b \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }+\left (c \left (6 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )^{2}-\frac {\operatorname {g2}}{2}\right )+d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
❇ |
N/A |
0.103 |
|
|
\[ {}y^{\prime \prime \prime \prime }-\left (12 k^{2} \operatorname {JacobiSN}\left (z , x\right )^{2}+a \right ) y^{\prime \prime }+b y^{\prime }+\left (\alpha \operatorname {JacobiSN}\left (z , x\right )^{2}+\beta \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
❇ |
N/A |
0.102 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-3 y^{\prime \prime }-4 y^{\prime }+4 y-32 \sin \left (2 x \right )+24 \cos \left (2 x \right ) = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.201 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 a x y^{\prime \prime \prime }+6 a^{2} x^{2} y^{\prime \prime }+4 a^{3} x^{3} y^{\prime }+a^{4} x^{4} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.318 |
|
|
\[ {}4 y^{\prime \prime \prime \prime }-12 y^{\prime \prime \prime }+11 y^{\prime \prime }-3 y^{\prime }-4 \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.68 |
|
|
\[ {}x y^{\prime \prime \prime \prime }+5 y^{\prime \prime \prime }-24 = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.759 |
|
|
\[ {}x y^{\prime \prime \prime \prime }-\left (6 x^{2}+1\right ) y^{\prime \prime \prime }+12 x^{3} y^{\prime \prime }-\left (9 x^{2}-7\right ) x^{2} y^{\prime }+2 \left (x^{2}-3\right ) x^{3} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.694 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }-2 \left (\nu ^{2} x^{2}+6\right ) y^{\prime \prime }+\nu ^{2} \left (\nu ^{2} x^{2}+4\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.376 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }+2 x y^{\prime \prime \prime }+a y-b \,x^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.102 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }+4 x y^{\prime \prime \prime }+2 y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.64 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }+6 x y^{\prime \prime \prime }+6 y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.507 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }+6 x y^{\prime \prime \prime }+6 y^{\prime \prime }-\lambda ^{2} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.244 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }+8 x y^{\prime \prime \prime }+12 y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.574 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }+8 x y^{\prime \prime \prime }+12 y^{\prime \prime }-\lambda ^{2} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.26 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }+\left (2 n -2 \nu +4\right ) x y^{\prime \prime \prime }+\left (n -\nu +1\right ) \left (n -\nu +2\right ) y^{\prime \prime }-\frac {b^{4} y}{16} = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.291 |
|
|
\[ {}x^{3} y^{\prime \prime \prime \prime }+2 x^{2} y^{\prime \prime \prime }-x y^{\prime \prime }+y^{\prime }-a^{4} x^{3} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.276 |
|
|
\[ {}x^{3} y^{\prime \prime \prime \prime }+6 x^{2} y^{\prime \prime \prime }+6 x y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_missing_y |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.477 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }-2 n \left (n +1\right ) x^{2} y^{\prime \prime }+4 n \left (n +1\right ) x y^{\prime }+\left (a \,x^{4}+n \left (n +1\right ) \left (n +3\right ) \left (n -2\right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.325 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }+\left (4 n^{2}-1\right ) x y^{\prime }-4 x^{4} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.332 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }-\left (4 n^{2}-1\right ) x y^{\prime }+\left (-4 x^{4}+4 n^{2}-1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.319 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}+3\right ) x^{2} y^{\prime \prime }+\left (12 n^{2}-3\right ) x y^{\prime }-\left (4 x^{4}+12 n^{2}-3\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.326 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-\rho ^{2}-\sigma ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-\rho ^{2}-\sigma ^{2}+1\right ) x \right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.347 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-2 \mu ^{2}-2 \nu ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-2 \mu ^{2}-2 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (8 x^{2}+\left (\mu ^{2}-\nu ^{2}\right )^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.369 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+8 x^{3} y^{\prime \prime \prime }+12 x^{2} y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _missing_y]] |
✓ |
✓ |
0.447 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+8 x^{3} y^{\prime \prime \prime }+12 x^{2} y^{\prime \prime }+a y = 0 \] |
1 |
1 |
1 |
higher_order_ODE_non_constant_coefficients_of_type_Euler |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
0.291 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a \right ) x^{3} y^{\prime \prime \prime }+\left (4 b^{2} c^{2} x^{2 c}+6 \left (a -1\right )^{2}-2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )+1\right ) x^{2} y^{\prime \prime }+\left (4 \left (3 c -2 a +1\right ) b^{2} c^{2} x^{2 c}+\left (2 a -1\right ) \left (2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )-2 a \left (a -1\right )-1\right )\right ) x y^{\prime }+\left (4 \left (a -c \right ) \left (-2 c +a \right ) b^{2} c^{2} x^{2 c}+\left (c \mu +c \nu +a \right ) \left (c \mu +c \nu -a \right ) \left (c \mu -c \nu +a \right ) \left (c \mu -c \nu -a \right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.126 |
|
|
\[ {}x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a -4 c \right ) x^{3} y^{\prime \prime \prime }+\left (-2 \nu ^{2} c^{2}+2 a^{2}+4 \left (a +c -1\right )^{2}+4 \left (a -1\right ) \left (c -1\right )-1\right ) x^{2} y^{\prime \prime }+\left (2 \nu ^{2} c^{2}-2 a^{2}-\left (2 a -1\right ) \left (2 c -1\right )\right ) \left (2 a +2 c -1\right ) x y^{\prime }+\left (\left (-\nu ^{2} c^{2}+a^{2}\right ) \left (-\nu ^{2} c^{2}+a^{2}+4 a c +4 c^{2}\right )-b^{4} c^{4} x^{4 c}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.128 |
|
|
\[ {}\nu ^{4} x^{4} y^{\prime \prime \prime \prime }+\left (4 \nu -2\right ) \nu ^{3} x^{3} y^{\prime \prime \prime }+\left (\nu -1\right ) \left (2 \nu -1\right ) \nu ^{2} x^{2} y^{\prime \prime }-\frac {b^{4} x^{\frac {2}{\nu }} y}{16} = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.108 |
|
|
\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime \prime \prime }+10 x \left (x^{2}-1\right ) y^{\prime \prime \prime }+\left (24 x^{2}-8-2 \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )\right ) \left (x^{2}-1\right )\right ) y^{\prime \prime }-6 x \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )-2\right ) y^{\prime }+\left (\left (\mu \left (\mu +1\right )-\nu \left (\nu +1\right )\right )^{2}-2 \mu \left (\mu +1\right )-2 \nu \left (\nu +1\right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.6 |
|
|
\[ {}\left ({\mathrm e}^{x}+2 x \right ) y^{\prime \prime \prime \prime }+4 \left ({\mathrm e}^{x}+2\right ) y^{\prime \prime \prime }+6 \,{\mathrm e}^{x} y^{\prime \prime }+4 \,{\mathrm e}^{x} y^{\prime }+{\mathrm e}^{x} y-\frac {1}{x^{5}} = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _fully, _exact, _linear]] |
✗ |
N/A |
0.109 |
|
|
\[ {}y^{\prime \prime \prime \prime } \sin \left (x \right )^{4}+2 y^{\prime \prime \prime } \sin \left (x \right )^{3} \cos \left (x \right )+y^{\prime \prime } \sin \left (x \right )^{2} \left (\sin \left (x \right )^{2}-3\right )+y^{\prime } \sin \left (x \right ) \cos \left (x \right ) \left (2 \sin \left (x \right )^{2}+3\right )+\left (a^{4} \sin \left (x \right )^{4}-3\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.121 |
|
|
\[ {}y^{\prime \prime \prime \prime } \sin \left (x \right )^{6}+4 y^{\prime \prime \prime } \sin \left (x \right )^{5} \cos \left (x \right )-6 y^{\prime \prime } \sin \left (x \right )^{6}-4 y^{\prime } \sin \left (x \right )^{5} \cos \left (x \right )+y \sin \left (x \right )^{6}-f = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.112 |
|
|
\[ {}f \left (y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+a^{4} y\right )+2 \operatorname {df} \left (y^{\prime \prime \prime }-a^{2} y^{\prime }\right ) = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.595 |
|
|
\[ {}f y^{\prime \prime \prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _quadrature]] |
✓ |
✓ |
0.154 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+a^{4} y-\lambda \left (x a -b \right ) \left (y^{\prime \prime }-a^{2} y\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.108 |
|
|
\[ {}y^{\left (5\right )}+2 y^{\prime \prime \prime }+y^{\prime }-x a -b \sin \left (x \right )-c \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_y]] |
✓ |
✓ |
1.259 |
|
|
\[ {}y^{\left (6\right )}+y-\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
81.605 |
|
|
\[ {}y^{\left (5\right )}-y a x -b = 0 \] |
1 |
0 |
0 |
unknown |
[[_high_order, _linear, _nonhomogeneous]] |
❇ |
N/A |
0.095 |
|
|
\[ {}y^{\left (5\right )}+a \,x^{\nu } y^{\prime }+a \nu \,x^{\nu -1} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
❇ |
N/A |
0.102 |
|
|
\[ {}y^{\left (5\right )}+a y^{\prime \prime \prime \prime }-f = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.149 |
|
|
\[ {}x y^{\left (5\right )}-m n y^{\prime \prime \prime \prime }+y a x = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.365 |
|
|
\[ {}x \left (a y^{\prime }+b y^{\prime \prime }+c y^{\prime \prime \prime }+e y^{\prime \prime \prime \prime }\right ) y = 0 \] |
1 |
0 |
2 |
unknown |
[[_high_order, _missing_x]] |
✗ |
N/A |
0.0 |
|
|
\[ {}x y^{\left (5\right )}-\left (a A_{1} -A_{0} \right ) x -A_{1} -\left (\left (a A_{2} -A_{1} \right ) x +A_{2} \right ) y^{\prime } = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _missing_y]] |
❇ |
N/A |
0.168 |
|
|
\[ {}x^{2} y^{\prime \prime \prime \prime }-a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.237 |
|
|
\[ {}x^{10} y^{\left (5\right )}-a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.056 |
|
|
\[ {}x^{\frac {5}{2}} y^{\left (5\right )}-a y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.096 |
|
|
\[ {}\left (x -a \right )^{5} \left (-b +x \right )^{5} y^{\left (5\right )}-c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_high_order, _with_linear_symmetries]] |
❇ |
N/A |
55.836 |
|
|
\[ {}y^{\prime \prime }-y^{2} = 0 \] |
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]] |
✓ |
✓ |
4.3 |
|
|
\[ {}y^{\prime \prime }-6 y^{2} = 0 \] |
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]] |
✓ |
✓ |
7.651 |
|
|
\[ {}y^{\prime \prime }-6 y^{2}-x = 0 \] |
1 |
0 |
0 |
unknown |
[[_Painleve, ‘1st‘]] |
❇ |
N/A |
0.071 |
|
|
\[ {}y^{\prime \prime }-6 y^{2}+4 y = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
3.329 |
|
|
\[ {}y^{\prime \prime }+a y^{2}+b x +c = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.081 |
|
|
\[ {}y^{\prime \prime }-2 y^{3}-x y+a = 0 \] |
1 |
0 |
0 |
unknown |
[[_Painleve, ‘2nd‘]] |
❇ |
N/A |
0.077 |
|
|
\[ {}y^{\prime \prime }-a y^{3} = 0 \] |
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]] |
✓ |
✓ |
1.924 |
|
|
\[ {}y^{\prime \prime }-2 a^{2} y^{3}+2 a b x y-b = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.083 |
|
|
\[ {}y^{\prime \prime }+d +b x y+c y+a y^{3} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.078 |
|
|
\[ {}y^{\prime \prime }+d +b y^{2}+c y+a y^{3} = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
68.896 |
|
|
\[ {}y^{\prime \prime }+a \,x^{r} y^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.078 |
|
|
\[ {}y^{\prime \prime }+6 a^{10} y^{11}-y = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
39.587 |
|
|
\[ {}y^{\prime \prime }-\frac {1}{\left (a y^{2}+b x y+c \,x^{2}+\alpha y+\beta x +\gamma \right )^{\frac {3}{2}}} = 0 \] |
1 |
0 |
2 |
unknown |
[NONE] |
✗ |
N/A |
0.122 |
|
|
\[ {}y^{\prime \prime }-{\mathrm e}^{y} = 0 \] |
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]] |
✓ |
✓ |
1.605 |
|
|
\[ {}y^{\prime \prime }+a \,{\mathrm e}^{x} \sqrt {y} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.082 |
|
|
\[ {}y^{\prime \prime }+{\mathrm e}^{x} \sin \left (y\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.158 |
|
|
\[ {}y^{\prime \prime }+a \sin \left (y\right ) = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.068 |
|
|
\[ {}y^{\prime \prime }+a^{2} \sin \left (y\right )-\beta \sin \left (x \right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.322 |
|
|
\[ {}y^{\prime \prime }+a^{2} \sin \left (y\right )-\beta f \left (x \right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.169 |
|
|
\[ {}y^{\prime \prime } = \frac {f \left (\frac {y}{\sqrt {x}}\right )}{x^{\frac {3}{2}}} \] |
1 |
0 |
3 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.198 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-y^{2}-2 y = 0 \] |
1 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.406 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }-y^{\frac {3}{2}}+12 y = 0 \] |
1 |
0 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
1.02 |
|
|
\[ {}y^{\prime \prime }+5 a y^{\prime }-6 y^{2}+6 a^{2} y = 0 \] |
1 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
0.483 |
|
|
\[ {}y^{\prime \prime }+3 a y^{\prime }-2 y^{3}+2 a^{2} y = 0 \] |
1 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
0.484 |
|
|
|
|||||||||
|
\[ {}y^{\prime \prime }-\frac {\left (3 n +4\right ) y^{\prime }}{n}-\frac {2 \left (n +1\right ) \left (n +2\right ) y \left (y^{\frac {n}{n +1}}-1\right )}{n^{2}} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.122 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b y^{n}+\frac {\left (a^{2}-1\right ) y}{4} = 0 \] |
1 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.894 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b \,x^{v} y^{n} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.082 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{y}-2 a = 0 \] |
1 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.676 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+f \left (x \right ) \sin \left (y\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.118 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime }-y^{3} = 0 \] |
1 |
3 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.498 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime }-y^{3}+a y = 0 \] |
1 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
0.935 |
|
|
\[ {}y^{\prime \prime }+\left (y+3 a \right ) y^{\prime }-y^{3}+a y^{2}+2 a^{2} y = 0 \] |
1 |
0 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
1.095 |
|
|
\[ {}y^{\prime \prime }+\left (y+3 f \left (x \right )\right ) y^{\prime }-y^{3}+y^{2} f \left (x \right )+y \left (f^{\prime }\left (x \right )+2 f \left (x \right )^{2}\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime }-y^{3}-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+f \left (x \right )\right ) \left (3 y^{\prime }+y^{2}\right )+\left (a f \left (x \right )^{2}+3 f^{\prime }\left (x \right )+\frac {3 {f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}\right ) y+b f \left (x \right )^{3} = 0 \] |
1 |
0 |
1 |
unknown |
[NONE] |
❇ |
N/A |
0.113 |
|
|
\[ {}y^{\prime \prime }+\left (y-\frac {3 f^{\prime }\left (x \right )}{2 f \left (x \right )}\right ) y^{\prime }-y^{3}-\frac {f^{\prime }\left (x \right ) y^{2}}{2 f \left (x \right )}+\frac {\left (f \left (x \right )+\frac {{f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}-f^{\prime \prime }\left (x \right )\right ) y}{2 f \left (x \right )} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.105 |
|
|
\[ {}y^{\prime \prime }+2 y y^{\prime }+f \left (x \right ) y^{\prime }+f^{\prime }\left (x \right ) y = 0 \] |
1 |
1 |
1 |
second_order_integrable_as_is, exact nonlinear second order ode |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.593 |
|
|
\[ {}y^{\prime \prime }+2 y y^{\prime }+f \left (x \right ) \left (y^{\prime }+y^{2}\right )-g \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
❇ |
N/A |
0.081 |
|
|
\[ {}y^{\prime \prime }+3 y y^{\prime }+y^{3}+f \left (x \right ) y-g \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[NONE] |
❇ |
N/A |
0.078 |
|
|
\[ {}y^{\prime \prime }+\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_potential_symmetries]] |
✗ |
N/A |
0.079 |
|
|
\[ {}y^{\prime \prime }-3 y y^{\prime }-3 a y^{2}-4 a^{2} y-b = 0 \] |
1 |
0 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.779 |
|
|
\[ {}y^{\prime \prime }-\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_potential_symmetries]] |
✗ |
N/A |
0.078 |
|
|
\[ {}y^{\prime \prime }-2 a y y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
0.697 |
|
|
\[ {}y^{\prime \prime }+a y y^{\prime }+b y^{3} = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✗ |
18.211 |
|
|
\[ {}y^{\prime \prime }+f \left (x , y\right ) y^{\prime }+g \left (x , y\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.085 |
|
|
\[ {}y^{\prime \prime }+a {y^{\prime }}^{2}+b y = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.632 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime } {| y^{\prime }|}+b y^{\prime }+c y = 0 \] |
0 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
3.27 |
|
|
\[ {}y^{\prime \prime }+a {y^{\prime }}^{2}+b y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.451 |
|
|
\[ {}y^{\prime \prime }+a {y^{\prime }}^{2}+b \sin \left (y\right ) = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
8.279 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime } {| y^{\prime }|}+b \sin \left (y\right ) = 0 \] |
0 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
1.54 |
|
|
\[ {}y^{\prime \prime }+a y {y^{\prime }}^{2}+b y = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✗ |
0.865 |
|
|
\[ {}y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.237 |
|
|
\[ {}y^{\prime \prime }-\frac {D\left (f \right )\left (y\right ) {y^{\prime }}^{3}}{f \left (y\right )}+g \left (x \right ) y^{\prime }+h \left (x \right ) f \left (y\right ) = 0 \] |
0 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime }+\phi \left (y\right ) {y^{\prime }}^{2}+f \left (x \right ) y^{\prime }+g \left (x \right ) \Phi \left (y\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.119 |
|
|
\[ {}y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (y\right ) y^{\prime }+h \left (y\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.134 |
|
|
\[ {}y^{\prime \prime }+\left (1+{y^{\prime }}^{2}\right ) \left (f \left (x , y\right ) y^{\prime }+g \left (x , y\right )\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.085 |
|
|
\[ {}y^{\prime \prime }+a y \left (1+{y^{\prime }}^{2}\right )^{2} = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
5.75 |
|
|
\[ {}y^{\prime \prime }-a \left (-y+x y^{\prime }\right )^{v} = 0 \] |
0 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.083 |
|
|
\[ {}y^{\prime \prime }-k \,x^{a} y^{b} {y^{\prime }}^{r} = 0 \] |
0 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.083 |
|
|
\[ {}y^{\prime \prime }+\left (y^{\prime }-\frac {y}{x}\right )^{a} f \left (x , y\right ) = 0 \] |
0 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.122 |
|
|
\[ {}y^{\prime \prime } = a \sqrt {1+{y^{\prime }}^{2}} \] |
2 |
2 |
3 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.01 |
|
|
\[ {}y^{\prime \prime } = a \sqrt {1+{y^{\prime }}^{2}}+b \] |
2 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
194.412 |
|
|
\[ {}y^{\prime \prime } = a \sqrt {{y^{\prime }}^{2}+b y^{2}} \] |
2 |
0 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
12.945 |
|
|
\[ {}y^{\prime \prime } = a \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} \] |
2 |
2 |
3 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.207 |
|
|
\[ {}y^{\prime \prime }-2 a x \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} = 0 \] |
2 |
1 |
3 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
2.19 |
|
|
\[ {}y^{\prime \prime }-a y \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} = 0 \] |
2 |
2 |
4 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
6.375 |
|
|
\[ {}y^{\prime \prime } = 2 a \left (c +b x +y\right ) \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} \] |
2 |
0 |
4 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.119 |
|
|
\[ {}y^{\prime \prime }+y^{3} y^{\prime }-y y^{\prime } \sqrt {y^{4}+4 y^{\prime }} = 0 \] |
2 |
2 |
5 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
15.044 |
|
|
\[ {}y^{\prime \prime }-f \left (y^{\prime }, x a +b y\right ) = 0 \] |
0 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime }-y f \left (x , \frac {y^{\prime }}{y}\right ) = 0 \] |
0 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.08 |
|
|
\[ {}y^{\prime \prime }-x^{n -2} f \left (y x^{-n}, y^{\prime } x^{-n +1}\right ) = 0 \] |
0 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.102 |
|
|
\[ {}8 y^{\prime \prime }+9 {y^{\prime }}^{4} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x, second_order_ode_missing_y |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.146 |
|
|
\[ {}a y^{\prime \prime }+h \left (y^{\prime }\right )+c y = 0 \] |
0 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.671 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-x y^{n} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.08 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+a \,x^{v} y^{n} = 0 \] |
1 |
0 |
1 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.082 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x \,{\mathrm e}^{y} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.077 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b x \,{\mathrm e}^{y} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.079 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b \,x^{5-2 a} {\mathrm e}^{y} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.085 |
|
|
\[ {}x y^{\prime \prime }+\left (y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.769 |
|
|
\[ {}x y^{\prime \prime }-x^{2} {y^{\prime }}^{2}+2 y^{\prime }+y^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.081 |
|
|
\[ {}x y^{\prime \prime }+a \left (-y+x y^{\prime }\right )^{2}-b = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.083 |
|
|
\[ {}2 x y^{\prime \prime }+{y^{\prime }}^{3}+y^{\prime } = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
2.002 |
|
|
\[ {}x^{2} y^{\prime \prime } = a \left (y^{n}-y\right ) \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.083 |
|
|
\[ {}x^{2} y^{\prime \prime }+a \left ({\mathrm e}^{y}-1\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.085 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 a +b -1\right ) x y^{\prime }+\left (c^{2} b^{2} x^{2 b}+a \left (a +b \right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.458 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (1+a \right ) x y^{\prime }-x^{k} f \left (x^{k} y, x y^{\prime }+k y\right ) = 0 \] |
0 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.105 |
|
|
\[ {}x^{2} y^{\prime \prime }+a \left (-y+x y^{\prime }\right )^{2}-b \,x^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.083 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y {y^{\prime }}^{2}+b x = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.075 |
|
|
\[ {}x^{2} y^{\prime \prime }-\sqrt {a \,x^{2} {y^{\prime }}^{2}+b y^{2}} = 0 \] |
2 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.125 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.842 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-x^{4} {y^{\prime }}^{2}+4 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.079 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+a y^{3}+2 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.105 |
|
|
\[ {}x^{3} \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )+12 x y+24 = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.082 |
|
|
\[ {}x^{3} y^{\prime \prime }-a \left (-y+x y^{\prime }\right )^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.082 |
|
|
\[ {}2 x^{3} y^{\prime \prime }+x^{2} \left (9+2 x y\right ) y^{\prime }+b +x y \left (a +3 x y-2 x^{2} y^{2}\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.086 |
|
|
\[ {}2 \left (-x^{k}+4 x^{3}\right ) \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )-\left (k \,x^{k -1}-12 x^{2}\right ) \left (3 y^{\prime }+y^{2}\right )+y a x +b = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.101 |
|
|
\[ {}x^{4} y^{\prime \prime }+a^{2} y^{n} = 0 \] |
1 |
0 |
1 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.082 |
|
|
\[ {}x^{4} y^{\prime \prime }-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.08 |
|
|
\[ {}x^{4} y^{\prime \prime }-x^{2} \left (x +y^{\prime }\right ) y^{\prime }+4 y^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.081 |
|
|
\[ {}x^{4} y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{3} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.083 |
|
|
\[ {}y^{\prime \prime } \sqrt {x}-y^{\frac {3}{2}} = 0 \] |
1 |
0 |
2 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.084 |
|
|
\[ {}\left (x^{2} a +b x +c \right )^{\frac {3}{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {x^{2} a +b x +c}}\right ) = 0 \] |
1 |
0 |
3 |
unknown |
[NONE] |
✗ |
N/A |
11.394 |
|
|
\[ {}x^{\frac {n}{n +1}} y^{\prime \prime }-y^{\frac {2 n +1}{n +1}} = 0 \] |
1 |
0 |
1 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.118 |
|
|
\[ {}f \left (x \right )^{2} y^{\prime \prime }+f \left (x \right ) f^{\prime }\left (x \right ) y^{\prime }-h \left (y, f \left (x \right ) y^{\prime }\right ) = 0 \] |
0 |
0 |
1 |
unknown |
[NONE] |
❇ |
N/A |
0.093 |
|
|
\[ {}y^{\prime \prime } y-a = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.039 |
|
|
\[ {}y^{\prime \prime } y-x a = 0 \] |
1 |
0 |
1 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.073 |
|
|
\[ {}y^{\prime \prime } y-x^{2} a = 0 \] |
1 |
0 |
1 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.075 |
|
|
\[ {}y^{\prime \prime } y+{y^{\prime }}^{2}-a = 0 \] |
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]] |
✓ |
✓ |
4.072 |
|
|
\[ {}y^{\prime \prime } y+y^{2}-x a -b = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.083 |
|
|
\[ {}y^{\prime \prime } y+{y^{\prime }}^{2}-y^{\prime } = 0 \] |
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]] |
✓ |
✓ |
2.237 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+1 = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
4.772 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-1 = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
4.089 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+{\mathrm e}^{x} y \left (c y^{2}+d \right )+{\mathrm e}^{2 x} \left (b +a y^{4}\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.106 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-y^{2} \ln \left (y\right ) = 0 \] |
1 |
2 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.711 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-y^{\prime }+f \left (x \right ) y^{3}+y^{2} \left (\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {{f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.105 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+f \left (x \right ) y^{\prime }-f^{\prime }\left (x \right ) y-y^{3} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+f^{\prime }\left (x \right ) y^{\prime }-f^{\prime \prime }\left (x \right ) y+f \left (x \right ) y^{3}-y^{4} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.099 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+a y y^{\prime }+b y^{2} = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.473 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+a y y^{\prime }-2 a y^{2}+b y^{3} = 0 \] |
1 |
0 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.585 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-\left (-1+a y\right ) y^{\prime }+2 y^{2} a^{2}-2 b^{2} y^{3}+a y = 0 \] |
1 |
0 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
1.108 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+\left (-1+a y\right ) y^{\prime }-y \left (y+1\right ) \left (y^{2} b^{2}-a^{2}\right ) = 0 \] |
1 |
0 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
2.427 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+\left (\tan \left (x \right )+\cot \left (x \right )\right ) y y^{\prime }+\left (\cos \left (x \right )^{2}-n^{2} \cot \left (x \right )^{2}\right ) y^{2} \ln \left (y\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
1.138 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-f \left (x \right ) y y^{\prime }-g \left (x \right ) y^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+\left (g \left (x \right )+y^{2} f \left (x \right )\right ) y^{\prime }-y \left (g^{\prime }\left (x \right )-f^{\prime }\left (x \right ) y^{2}\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
❇ |
N/A |
0.101 |
|
|
\[ {}y^{\prime \prime } y-3 {y^{\prime }}^{2}+3 y y^{\prime }-y^{2} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.593 |
|
|
\[ {}y^{\prime \prime } y-a {y^{\prime }}^{2} = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.01 |
|
|
\[ {}y^{\prime \prime } y+a \left (1+{y^{\prime }}^{2}\right ) = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
5.073 |
|
|
\[ {}y^{\prime \prime } y+a {y^{\prime }}^{2}+b y^{3} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.191 |
|
|
\[ {}y^{\prime \prime } y+a {y^{\prime }}^{2}+b y y^{\prime }+c y^{2}+d y^{1-a} = 0 \] |
1 |
0 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
1.409 |
|
|
\[ {}y^{\prime \prime } y+a {y^{\prime }}^{2}+f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime } y+a {y^{\prime }}^{2}+b y^{2} y^{\prime }+c y^{4} = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.517 |
|
|
\[ {}y^{\prime \prime } y-\frac {\left (a -1\right ) {y^{\prime }}^{2}}{a}-f \left (x \right ) y^{2} y^{\prime }+\frac {a f \left (x \right )^{2} y^{4}}{\left (2+a \right )^{2}}-\frac {a f^{\prime }\left (x \right ) y^{3}}{2+a} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.118 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-1-2 a y \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} = 0 \] |
2 |
2 |
4 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.839 |
|
|
\[ {}y^{\prime \prime } \left (x +y\right )+{y^{\prime }}^{2}-y^{\prime } = 0 \] |
1 |
1 |
1 |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.436 |
|
|
\[ {}y^{\prime \prime } \left (x -y\right )+2 y^{\prime } \left (y^{\prime }+1\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.082 |
|
|
\[ {}y^{\prime \prime } \left (x -y\right )-\left (y^{\prime }+1\right ) \left (1+{y^{\prime }}^{2}\right ) = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.081 |
|
|
\[ {}y^{\prime \prime } \left (x -y\right )-h \left (y^{\prime }\right ) = 0 \] |
0 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.155 |
|
|
\[ {}2 y^{\prime \prime } y+{y^{\prime }}^{2}+1 = 0 \] |
1 |
2 |
4 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
4.296 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+a = 0 \] |
1 |
2 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.714 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+y^{2} f \left (x \right )+a = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.083 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}-8 y^{3} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.088 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}-8 y^{3}-4 y^{2} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.093 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}-4 \left (2 y+x \right ) y^{2} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.085 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+\left (a y+b \right ) y^{2} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.28 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+1+2 x y^{2}+a y^{3} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.086 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+\left (b x +a y\right ) y^{2} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.086 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}-3 y^{4} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.87 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+b -4 \left (x^{2}+a \right ) y^{2}-8 x y^{3}-3 y^{4} = 0 \] |
1 |
0 |
0 |
unknown |
[[_Painleve, ‘4th‘]] |
❇ |
N/A |
0.091 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+3 f \left (x \right ) y y^{\prime }+2 \left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y^{2}-8 y^{3} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.099 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+4 y^{2} y^{\prime }+1+y^{2} f \left (x \right )+y^{4} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.09 |
|
|
\[ {}2 y^{\prime \prime } y-3 {y^{\prime }}^{2} = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.977 |
|
|
\[ {}2 y^{\prime \prime } y-3 {y^{\prime }}^{2}-4 y^{2} = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.038 |
|
|
\[ {}2 y^{\prime \prime } y-3 {y^{\prime }}^{2}+y^{2} f \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.085 |
|
|
\[ {}2 y^{\prime \prime } y-6 {y^{\prime }}^{2}+\left (1+a y^{3}\right ) y^{2} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.665 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2} \left (1+{y^{\prime }}^{2}\right ) = 0 \] |
1 |
1 |
5 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.901 |
|
|
\[ {}2 \left (y-a \right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
11.312 |
|
|
\[ {}3 y^{\prime \prime } y-2 {y^{\prime }}^{2}-x^{2} a -b x -c = 0 \] |
1 |
0 |
2 |
unknown |
[NONE] |
✗ |
N/A |
0.089 |
|
|
\[ {}3 y^{\prime \prime } y-5 {y^{\prime }}^{2} = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.968 |
|
|
\[ {}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+4 y = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.537 |
|
|
\[ {}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}-12 y^{3} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.859 |
|
|
\[ {}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+a y^{3}+b y^{2}+c y = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.375 |
|
|
\[ {}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+\left (6 y^{2}-\frac {2 f^{\prime }\left (x \right ) y}{f \left (x \right )}\right ) y^{\prime }+y^{4}-2 y^{2} y^{\prime }+g \left (x \right ) y^{2}+f \left (x \right ) y = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.11 |
|
|
\[ {}4 y^{\prime \prime } y-5 {y^{\prime }}^{2}+a y^{2} = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.512 |
|
|
\[ {}12 y^{\prime \prime } y-15 {y^{\prime }}^{2}+8 y^{3} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.827 |
|
|
\[ {}n y y^{\prime \prime }-\left (n -1\right ) {y^{\prime }}^{2} = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.852 |
|
|
\[ {}a y y^{\prime \prime }+b {y^{\prime }}^{2}+\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y+\operatorname {c0} = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.141 |
|
|
\[ {}a y y^{\prime \prime }+b {y^{\prime }}^{2}-\frac {y y^{\prime }}{\sqrt {c^{2}+x^{2}}} = 0 \] |
1 |
1 |
2 |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.536 |
|
|
\[ {}a y y^{\prime \prime }-\left (a -1\right ) {y^{\prime }}^{2}+\left (2+a \right ) f \left (x \right ) y^{2} y^{\prime }+f \left (x \right )^{2} y^{4}+a f^{\prime }\left (x \right ) y^{3} = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.106 |
|
|
\[ {}\left (a y+b \right ) y^{\prime \prime }+c {y^{\prime }}^{2} = 0 \] |
1 |
1 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.493 |
|
|
\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0 \] |
1 |
1 |
3 |
second_order_integrable_as_is, second_order_nonlinear_solved_by_mainardi_lioville_method |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.993 |
|
|
\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}+a y y^{\prime }+f \left (x \right ) = 0 \] |
1 |
1 |
2 |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.302 |
|
|
\[ {}x y y^{\prime \prime }-x {y^{\prime }}^{2}+y y^{\prime }+x \left (d +a y^{4}\right )+y \left (c +b y^{2}\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[[_Painleve, ‘3rd‘]] |
❇ |
N/A |
0.093 |
|
|
\[ {}x y y^{\prime \prime }-x {y^{\prime }}^{2}+a y y^{\prime }+b x y^{3} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.094 |
|
|
\[ {}x y y^{\prime \prime }+2 x {y^{\prime }}^{2}+a y y^{\prime } = 0 \] |
1 |
1 |
4 |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.272 |
|
|
\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime } = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.087 |
|
|
\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+a y y^{\prime } = 0 \] |
1 |
1 |
2 |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.247 |
|
|
\[ {}x y y^{\prime \prime }-4 x {y^{\prime }}^{2}+4 y y^{\prime } = 0 \] |
1 |
1 |
4 |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.246 |
|
|
\[ {}x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) {y^{\prime }}^{2}-y y^{\prime } = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.247 |
|
|
\[ {}x \left (x +y\right ) y^{\prime \prime }+x {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }-y = 0 \] |
1 |
1 |
3 |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.931 |
|
|
\[ {}2 x y y^{\prime \prime }-x {y^{\prime }}^{2}+y y^{\prime } = 0 \] |
1 |
1 |
2 |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.382 |
|
|
\[ {}x^{2} \left (y-1\right ) y^{\prime \prime }-2 x^{2} {y^{\prime }}^{2}-2 x \left (y-1\right ) y^{\prime }-2 y \left (y-1\right )^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.144 |
|
|
\[ {}x^{2} \left (x +y\right ) y^{\prime \prime }-\left (-y+x y^{\prime }\right )^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.096 |
|
|
\[ {}x^{2} \left (x -y\right ) y^{\prime \prime }+a \left (-y+x y^{\prime }\right )^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.096 |
|
|
\[ {}2 x^{2} y y^{\prime \prime }-x^{2} \left (1+{y^{\prime }}^{2}\right )+y^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.093 |
|
|
\[ {}a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.102 |
|
|
\[ {}x \left (1+x \right )^{2} y y^{\prime \prime }-x \left (1+x \right )^{2} {y^{\prime }}^{2}+2 \left (1+x \right )^{2} y y^{\prime }-a \left (2+x \right ) y^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.115 |
|
|
\[ {}8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 x^{2} y y^{\prime }+3 x y^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.131 |
|
|
\[ {}\operatorname {f0} \left (x \right ) y y^{\prime \prime }+\operatorname {f1} \left (x \right ) {y^{\prime }}^{2}+\operatorname {f2} \left (x \right ) y y^{\prime }+\operatorname {f3} \left (x \right ) y^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.115 |
|
|
\[ {}y^{2} y^{\prime \prime }-a = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.557 |
|
|
\[ {}y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}+x a = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.094 |
|
|
\[ {}y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}-x a -b = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.086 |
|
|
\[ {}\left (1+y^{2}\right ) y^{\prime \prime }+\left (1-2 y\right ) {y^{\prime }}^{2} = 0 \] |
1 |
1 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.92 |
|
|
\[ {}\left (1+y^{2}\right ) y^{\prime \prime }-3 y {y^{\prime }}^{2} = 0 \] |
1 |
1 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.019 |
|
|
\[ {}\left (x +y^{2}\right ) y^{\prime \prime }-2 \left (x -y^{2}\right ) {y^{\prime }}^{3}+y^{\prime } \left (1+4 y y^{\prime }\right ) = 0 \] |
1 |
0 |
3 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.105 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right ) \left (-y+x y^{\prime }\right ) = 0 \] |
1 |
0 |
3 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.1 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime \prime }-2 \left (1+{y^{\prime }}^{2}\right ) \left (-y+x y^{\prime }\right ) = 0 \] |
1 |
0 |
4 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.099 |
|
|
\[ {}2 y \left (1-y\right ) y^{\prime \prime }-\left (1-2 y\right ) {y^{\prime }}^{2}+y \left (1-y\right ) y^{\prime } f \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.115 |
|
|
\[ {}2 y \left (1-y\right ) y^{\prime \prime }-\left (-3 y+1\right ) {y^{\prime }}^{2}+h \left (y\right ) = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
0.141 |
|
|
\[ {}2 y \left (y-1\right ) y^{\prime \prime }-\left (3 y-1\right ) {y^{\prime }}^{2}+4 y y^{\prime } \left (f \left (x \right ) y+g \left (x \right )\right )+4 y^{2} \left (y-1\right ) \left (g \left (x \right )^{2}-f \left (x \right )^{2}-g^{\prime }\left (x \right )-f^{\prime }\left (x \right )\right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _reducible, _mu_xy]] |
❇ |
N/A |
0.133 |
|
|
\[ {}-2 y \left (1-y\right ) y^{\prime \prime }+\left (-3 y+1\right ) {y^{\prime }}^{2}-4 y y^{\prime } \left (f \left (x \right ) y+g \left (x \right )\right )+\left (1-y\right )^{3} \left (\operatorname {f0} \left (x \right )^{2} y^{2}-\operatorname {f1} \left (x \right )^{2}\right )+4 y^{2} \left (1-y\right ) \left (f \left (x \right )^{2}-g \left (x \right )^{2}-g^{\prime }\left (x \right )-f^{\prime }\left (x \right )\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.145 |
|
|
\[ {}3 y \left (1-y\right ) y^{\prime \prime }-2 \left (1-2 y\right ) {y^{\prime }}^{2}-h \left (y\right ) = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
0.151 |
|
|
\[ {}\left (1-y\right ) y^{\prime \prime }-3 \left (1-2 y\right ) {y^{\prime }}^{2}-h \left (y\right ) = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
0.128 |
|
|
\[ {}a y \left (y-1\right ) y^{\prime \prime }+\left (b y+c \right ) {y^{\prime }}^{2}+h \left (y\right ) = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
0.135 |
|
|
\[ {}a y \left (y-1\right ) y^{\prime \prime }-\left (a -1\right ) \left (2 y-1\right ) {y^{\prime }}^{2}+f y \left (y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
63.229 |
|
|
\[ {}a b y \left (y-1\right ) y^{\prime \prime }-\left (\left (2 a b -a -b \right ) y+\left (1-a \right ) b \right ) {y^{\prime }}^{2}+f y \left (y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
48.549 |
|
|
\[ {}x y^{2} y^{\prime \prime }-a = 0 \] |
1 |
0 |
4 |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.095 |
|
|
\[ {}\left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime }+\left (a^{2}-x^{2}\right ) y {y^{\prime }}^{2}-x \left (a^{2}-y^{2}\right ) y^{\prime } = 0 \] |
1 |
0 |
3 |
unknown |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.112 |
|
|
\[ {}2 x^{2} y \left (y-1\right ) y^{\prime \prime }-x^{2} \left (3 y-1\right ) {y^{\prime }}^{2}+2 x y \left (y-1\right ) y^{\prime }+\left (a y^{2}+b \right ) \left (y-1\right )^{3}+c x y^{2} \left (y-1\right )+d \,x^{2} y^{2} \left (y+1\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[[_Painleve, ‘5th‘]] |
❇ |
N/A |
0.124 |
|
|
\[ {}x^{3} y^{2} y^{\prime \prime }+\left (x +y\right ) \left (-y+x y^{\prime }\right )^{3} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.116 |
|
|
\[ {}y^{3} y^{\prime \prime }-a = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.622 |
|
|
\[ {}y \left (1+y^{2}\right ) y^{\prime \prime }+\left (1-3 y^{2}\right ) {y^{\prime }}^{2} = 0 \] |
1 |
2 |
5 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.546 |
|
|
\[ {}2 y^{3} y^{\prime \prime }+y^{4}-a^{2} x y^{2}-1 = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.112 |
|
|
\[ {}2 y^{3} y^{\prime \prime }+y^{2} {y^{\prime }}^{2}-x^{2} a -b x -c = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.108 |
|
|
\[ {}2 \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) y^{\prime \prime }-\left (\left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right )+\left (y-b \right ) \left (y-c \right )\right ) {y^{\prime }}^{2}+\left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2} \left (A_{0} +\frac {B_{0}}{\left (y-a \right )^{2}}+\frac {C_{1}}{\left (y-b \right )^{2}}+\frac {D_{0}}{\left (y-c \right )^{2}}\right ) = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✗ |
5.548 |
|
|
\[ {}\left (4 y^{3}-a y-b \right ) y^{\prime \prime }-\left (6 y^{2}-\frac {a}{2}\right ) {y^{\prime }}^{2} = 0 \] |
1 |
1 |
4 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
11.477 |
|
|
\[ {}\left (4 y^{3}-a y-b \right ) \left (y^{\prime \prime }+f y^{\prime }\right )-\left (6 y^{2}-\frac {a}{2}\right ) {y^{\prime }}^{2} = 0 \] |
1 |
1 |
4 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✗ |
1887.566 |
|
|
\[ {}-2 x y \left (1-x \right ) \left (1-y\right ) \left (x -y\right ) y^{\prime \prime }+x \left (1-x \right ) \left (x -2 x y-2 y+3 y^{2}\right ) {y^{\prime }}^{2}+2 y \left (1-y\right ) \left (x^{2}+y-2 x y\right ) y^{\prime }-y^{2} \left (1-y\right )^{2}-f \left (y \left (y-1\right ) \left (y-x \right )\right )^{\frac {3}{2}} = 0 \] |
1 |
0 |
0 |
unknown |
unknown |
❇ |
N/A |
0.214 |
|
|
\[ {}2 x^{2} y \left (1-x \right )^{2} \left (1-y\right ) \left (x -y\right ) y^{\prime \prime }-x^{2} \left (1-x \right )^{2} \left (x -2 x y-2 y+3 y^{2}\right ) {y^{\prime }}^{2}-2 x y \left (1-x \right ) \left (1-y\right ) \left (x^{2}+y-2 x y\right ) y^{\prime }+b x \left (1-y\right )^{2} \left (x -y\right )^{2}-c \left (1-x \right ) y^{2} \left (x -y\right )^{2}-d x y^{2} \left (1-x \right ) \left (1-y\right )^{2}+a y^{2} \left (x -y\right )^{2} \left (1-y\right )^{2} = 0 \] |
1 |
0 |
0 |
unknown |
[[_Painleve, ‘6th‘]] |
❇ |
N/A |
0.208 |
|
|
\[ {}\left (y^{2}-1\right ) \left (y^{2} a^{2}-1\right ) y^{\prime \prime }+b \sqrt {\left (1-y^{2}\right ) \left (1-y^{2} a^{2}\right )}\, {y^{\prime }}^{2}+\left (1+a^{2}-2 y^{2} a^{2}\right ) y {y^{\prime }}^{2} = 0 \] |
1 |
1 |
0 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
14.163 |
|
|
\[ {}\left (c +2 b x +x^{2} a +y^{2}\right )^{2} y^{\prime \prime }+d y = 0 \] |
1 |
0 |
2 |
unknown |
[NONE] |
✗ |
N/A |
0.107 |
|
|
\[ {}\sqrt {y}\, y^{\prime \prime }-a = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.237 |
|
|
\[ {}\sqrt {x^{2}+y^{2}}\, y^{\prime \prime }-a \left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}} = 0 \] |
2 |
0 |
3 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.17 |
|
|
\[ {}y \left (1-\ln \left (y\right )\right ) y^{\prime \prime }+\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2} = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.244 |
|
|
\[ {}\left (b +a \sin \left (y\right )^{2}\right ) y^{\prime \prime }+a {y^{\prime }}^{2} \cos \left (y\right ) \sin \left (y\right )+A y \left (c +a \sin \left (y\right )^{2}\right ) = 0 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
48.585 |
|
|
\[ {}h \left (y\right ) y^{\prime \prime }+a D\left (h \right )\left (y\right ) {y^{\prime }}^{2}+j \left (y\right ) = 0 \] |
0 |
0 |
2 |
unknown |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
0.0 |
|
|
|
|||||||||
|
\[ {}h \left (y\right ) y^{\prime \prime }-D\left (h \right )\left (y\right ) {y^{\prime }}^{2}-h \left (y\right )^{2} j \left (x , \frac {y^{\prime }}{h \left (y\right )}\right ) = 0 \] |
0 |
0 |
1 |
unknown |
[NONE] |
❇ |
N/A |
0.0 |
|
|
\[ {}y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2} = 0 \] |
1 |
0 |
2 |
second_order_integrable_as_is, exact nonlinear second order ode |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
❇ |
N/A |
2.382 |
|
|
\[ {}\left (-y+x y^{\prime }\right ) y^{\prime \prime }+4 {y^{\prime }}^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.101 |
|
|
\[ {}\left (-y+x y^{\prime }\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right )^{2} = 0 \] |
1 |
0 |
3 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.096 |
|
|
\[ {}a \,x^{3} y^{\prime } y^{\prime \prime }+b y^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.101 |
|
|
\[ {}\left (\operatorname {f1} y^{\prime }+\operatorname {f2} y\right ) y^{\prime \prime }+\operatorname {f3} {y^{\prime }}^{2}+\operatorname {f4} \left (x \right ) y y^{\prime }+\operatorname {f5} \left (x \right ) y^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.111 |
|
|
\[ {}\left (2 y^{2} y^{\prime }+x^{2}\right ) y^{\prime \prime }+2 y {y^{\prime }}^{3}+3 x y^{\prime }+y = 0 \] |
1 |
0 |
1 |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]] |
❇ |
N/A |
3.013 |
|
|
\[ {}\left ({y^{\prime }}^{2}+y^{2}\right ) y^{\prime \prime }+y^{3} = 0 \] |
1 |
1 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.789 |
|
|
\[ {}\left ({y^{\prime }}^{2}+a \left (-y+x y^{\prime }\right )\right ) y^{\prime \prime }-b = 0 \] |
1 |
0 |
4 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.1 |
|
|
\[ {}\left (a \sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }\right ) y^{\prime \prime }-{y^{\prime }}^{2}-1 = 0 \] |
2 |
2 |
4 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
5.132 |
|
|
\[ {}h \left (y^{\prime }\right ) y^{\prime \prime }+j \left (y\right ) y^{\prime }+f = 0 \] |
0 |
0 |
1 |
unknown |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1]] |
❇ |
N/A |
0.227 |
|
|
\[ {}{y^{\prime \prime }}^{2}-a y-b = 0 \] |
2 |
4 |
5 |
second_order_ode_high_degree |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
3.767 |
|
|
\[ {}a^{2} {y^{\prime \prime }}^{2}-2 a x y^{\prime \prime }+y^{\prime } = 0 \] |
2 |
3 |
2 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.154 |
|
|
\[ {}2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2}-x y^{\prime \prime } \left (x +4 y^{\prime }\right )+2 \left (x +y^{\prime }\right ) y^{\prime }-2 y = 0 \] |
2 |
0 |
2 |
unknown |
[NONE] |
✗ |
N/A |
0.197 |
|
|
\[ {}3 x^{2} {y^{\prime \prime }}^{2}-2 \left (3 x y^{\prime }+y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2} = 0 \] |
2 |
0 |
3 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.187 |
|
|
\[ {}x^{2} \left (2-9 x \right ) {y^{\prime \prime }}^{2}-6 x \left (1-6 x \right ) y^{\prime } y^{\prime \prime }+6 y^{\prime \prime } y-36 x {y^{\prime }}^{2} = 0 \] |
2 |
0 |
4 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.224 |
|
|
\[ {}F_{1,1}\left (x \right ) {y^{\prime }}^{2}+\left (\left (F_{2,1}\left (x \right )+F_{1,2}\left (x \right )\right ) y^{\prime \prime }+y \left (F_{1,0}\left (x \right )+F_{0,1}\left (x \right )\right )\right ) y^{\prime }+F_{2,2}\left (x \right ) {y^{\prime \prime }}^{2}+y \left (F_{2,0}\left (x \right )+F_{0,2}\left (x \right )\right ) y^{\prime \prime }+F_{0,0}\left (x \right ) y^{2} = 0 \] |
2 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.437 |
|
|
\[ {}y {y^{\prime \prime }}^{2}-a \,{\mathrm e}^{2 x} = 0 \] |
2 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.235 |
|
|
\[ {}\left (y^{2} a^{2}-b^{2}\right ) {y^{\prime \prime }}^{2}-2 a^{2} y {y^{\prime }}^{2} y^{\prime \prime }+\left (a^{2} {y^{\prime }}^{2}-1\right ) {y^{\prime }}^{2} = 0 \] |
2 |
6 |
6 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.497 |
|
|
\[ {}\left (y^{2}-x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime }\right )^{2}-4 x y \left (-y+x y^{\prime }\right )^{3} = 0 \] |
2 |
0 |
3 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.202 |
|
|
\[ {}\left (2 y^{\prime \prime } y-{y^{\prime }}^{2}\right )^{3}+32 y^{\prime \prime } \left (x y^{\prime \prime }-y^{\prime }\right )^{3} = 0 \] |
4 |
0 |
0 |
unknown |
unknown |
❇ |
N/A |
0.447 |
|
|
\[ {}\sqrt {a {y^{\prime \prime }}^{2}+b {y^{\prime }}^{2}}+c y y^{\prime \prime }+d {y^{\prime }}^{2} = 0 \] |
2 |
0 |
4 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
20.473 |
|
|
\[ {}y^{\prime \prime \prime }-a^{2} \left ({y^{\prime }}^{5}+2 {y^{\prime }}^{3}+y^{\prime }\right ) = 0 \] |
1 |
0 |
2 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime } y-{y^{\prime }}^{2}+1 = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime \prime }-y^{\prime \prime } y+{y^{\prime }}^{2} = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime \prime }+a y y^{\prime \prime } = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.0 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }+x y^{\prime \prime }+\left (2 x y-1\right ) y^{\prime }+y^{2}-f \left (x \right ) = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _exact, _nonlinear]] |
❇ |
N/A |
0.0 |
|
|
\[ {}x^{2} y^{\prime \prime \prime }+x \left (y-1\right ) y^{\prime \prime }+x {y^{\prime }}^{2}+\left (1-y\right ) y^{\prime } = 0 \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y y^{\prime \prime \prime }-y^{\prime } y^{\prime \prime }+y^{3} y^{\prime } = 0 \] |
1 |
0 |
3 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}4 y^{2} y^{\prime \prime \prime }-18 y y^{\prime } y^{\prime \prime }+15 {y^{\prime }}^{3} = 0 \] |
1 |
0 |
3 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}9 y^{2} y^{\prime \prime \prime }-45 y y^{\prime } y^{\prime \prime }+40 {y^{\prime }}^{3} = 0 \] |
1 |
0 |
3 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}2 y^{\prime } y^{\prime \prime \prime }-3 {y^{\prime }}^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_3rd_order, _missing_x]] |
✗ |
N/A |
0.0 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) y^{\prime \prime \prime }-3 y^{\prime } {y^{\prime \prime }}^{2} = 0 \] |
1 |
0 |
4 |
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 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) y^{\prime \prime \prime }-\left (3 y^{\prime }+a \right ) {y^{\prime \prime }}^{2} = 0 \] |
1 |
0 |
4 |
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 |
|
|
\[ {}y^{\prime \prime } y^{\prime \prime \prime }-a \sqrt {b^{2} {y^{\prime \prime }}^{2}+1} = 0 \] |
1 |
0 |
4 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime } y^{\prime \prime \prime \prime }-y^{\prime \prime } y^{\prime \prime \prime }+{y^{\prime }}^{3} y^{\prime \prime \prime } = 0 \] |
1 |
0 |
2 |
unknown |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] |
❇ |
N/A |
0.0 |
|
|
\[ {}y^{\prime } \left (f^{\prime \prime \prime }\left (x \right ) y^{\prime }+3 f^{\prime \prime }\left (x \right ) y^{\prime \prime }+3 f^{\prime }\left (x \right ) y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime \prime \prime }\right )-y^{\prime \prime } f y^{\prime \prime \prime }+{y^{\prime }}^{3} \left (f^{\prime }\left (x \right ) y^{\prime }+f \left (x \right ) y^{\prime \prime }\right )+2 q \left (x \right ) {y^{\prime }}^{2} \sin \left (y\right )+\left (q \left (x \right ) y^{\prime \prime }-q^{\prime }\left (x \right ) y^{\prime }\right ) \cos \left (y\right ) = 0 \] |
1 |
0 |
0 |
unknown |
[NONE] |
❇ |
N/A |
0.0 |
|
|
\[ {}3 y^{\prime \prime } y^{\prime \prime \prime \prime }-5 {y^{\prime \prime \prime }}^{2} = 0 \] |
1 |
0 |
2 |
unknown |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries], [_high_order, _reducible, _mu_poly_yn]] |
✗ |
N/A |
0.0 |
|
|
\[ {}9 {y^{\prime \prime }}^{2} y^{\left (5\right )}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+40 y^{\prime \prime \prime } = 0 \] |
1 |
0 |
2 |
unknown |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] |
✗ |
N/A |
0.0 |
|
|
\[ {}y^{\prime \prime }-f \left (y\right ) = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✗ |
N/A |
0.19 |
|
|
\[ {}y^{\prime \prime \prime } = f \left (y\right ) \] |
1 |
0 |
1 |
unknown |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.0 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=a x \\ y^{\prime }=b \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.379 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=a y \\ y^{\prime }=-a x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.274 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=a y \\ y^{\prime }=b x \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.333 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=a x-y \\ y^{\prime }=x+a y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.277 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=a x+b y \\ y^{\prime }=c x+b y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.655 |
|
|
\[ {}\left [\begin {array}{c} a x^{\prime }+b y^{\prime }=\alpha x+\beta y \\ b x^{\prime }-a y^{\prime }=\beta x-\alpha y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.619 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-y \\ y^{\prime }=2 x+2 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.486 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+3 x+4 y=0 \\ y^{\prime }+2 x+5 y=0 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.329 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-5 x-2 y \\ y^{\prime }=x-7 y \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.46 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=a_{1} x+b_{1} y+c_{1} \\ y^{\prime }=a_{2} x+b_{2} y+c_{2} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.484 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+2 y=3 t \\ y^{\prime }-2 x=4 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.073 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+y-t^{2}+6 t +1=0 \\ y^{\prime }-x=-3 t^{2}+3 t +1 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.967 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+3 x-y={\mathrm e}^{2 t} \\ y^{\prime }+x+5 y={\mathrm e}^{t} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.792 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+2 x+y^{\prime }+y={\mathrm e}^{2 t}+t \\ x^{\prime }-x+y^{\prime }+3 y={\mathrm e}^{t}-1 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.428 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+y^{\prime }-y={\mathrm e}^{t} \\ 2 x^{\prime }+y^{\prime }+2 y=\cos \left (t \right ) \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.898 |
|
|
\[ {}\left [\begin {array}{c} 4 x^{\prime }+9 y^{\prime }+2 x+31 y={\mathrm e}^{t} \\ 3 x^{\prime }+7 y^{\prime }+x+24 y=3 \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.455 |
|
|
\[ {}\left [\begin {array}{c} 4 x^{\prime }+9 y^{\prime }+11 x+31 y={\mathrm e}^{t} \\ 3 x^{\prime }+7 y^{\prime }+8 x+24 y={\mathrm e}^{2 t} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.793 |
|
|
\[ {}\left [\begin {array}{c} 4 x^{\prime }+9 y^{\prime }+44 x+49 y=t \\ 3 x^{\prime }+7 y^{\prime }+34 x+38 y={\mathrm e}^{t} \end {array}\right ] \] |
1 |
1 |
2 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.813 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x f \left (t \right )+y g \left (t \right ) \\ y^{\prime }=-x g \left (t \right )+y f \left (t \right ) \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.024 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+\left (a x+b y\right ) f \left (t \right )=g \left (t \right ) \\ y^{\prime }+\left (c x+d y\right ) f \left (t \right )=h \left (t \right ) \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.026 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x \cos \left (t \right ) \\ y^{\prime }=x \,{\mathrm e}^{-\sin \left (t \right )} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.023 |
|
|
\[ {}\left [\begin {array}{c} t x^{\prime }+y=0 \\ t y^{\prime }+x=0 \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.02 |
|
|
\[ {}\left [\begin {array}{c} t x^{\prime }+2 x=t \\ t y^{\prime }-\left (2+t \right ) x-t y=-t \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.023 |
|
|
\[ {}\left [\begin {array}{c} t x^{\prime }+2 x-2 y=t \\ t y^{\prime }+x+5 y=t^{2} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.026 |
|
|
\[ {}\left [\begin {array}{c} t^{2} \left (1-\sin \left (t \right )\right ) x^{\prime }=t \left (1-2 \sin \left (t \right )\right ) x+t^{2} y \\ t^{2} \left (1-\sin \left (t \right )\right ) y^{\prime }=\left (t \cos \left (t \right )-\sin \left (t \right )\right ) x+t \left (1-t \cos \left (t \right )\right ) y \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.043 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+y^{\prime }+y=f \left (t \right ) \\ x^{\prime \prime }+y^{\prime \prime }+y^{\prime }+x+y=g \left (t \right ) \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.94 |
|
|
\[ {}\left [\begin {array}{c} 2 x^{\prime }+y^{\prime }-3 x=0 \\ x^{\prime \prime }+y^{\prime }-2 y={\mathrm e}^{2 t} \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.934 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-y^{\prime }+x=2 t \\ x^{\prime \prime }+y^{\prime }-9 x+3 y=\sin \left (2 t \right ) \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.918 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-x+2 y=0 \\ x^{\prime \prime }-2 y^{\prime }=2 t -\cos \left (2 t \right ) \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.787 |
|
|
\[ {}\left [\begin {array}{c} t x^{\prime }-t y^{\prime }-2 y=0 \\ t x^{\prime \prime }+2 x^{\prime }+x t =0 \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.918 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }+a y=0 \\ y^{\prime \prime }-a^{2} y=0 \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.907 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }=a x+b y \\ y^{\prime \prime }=c x+d y \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.929 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }=a_{1} x+b_{1} y+c_{1} \\ y^{\prime \prime }=a_{2} x+b_{2} y+c_{2} \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.937 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }+x+y=-5 \\ y^{\prime \prime }-4 x-3 y=-3 \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.918 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }=\left (3 \cos \left (a t +b \right )^{2}-1\right ) c^{2} x+\frac {3 c^{2} y \sin \left (2 a t b \right )}{2} \\ y^{\prime \prime }=\left (3 \sin \left (a t +b \right )^{2}-1\right ) c^{2} y+\frac {3 c^{2} x \sin \left (2 a t b \right )}{2} \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
❇ |
N/A |
1.841 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }+6 x+7 y=0 \\ y^{\prime \prime }+3 x+2 y=2 t \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.931 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }-a y^{\prime }+b x=0 \\ y^{\prime \prime }+a x^{\prime }+b y=0 \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.925 |
|
|
\[ {}\left [\begin {array}{c} a_{1} x^{\prime \prime }+b_{1} x^{\prime }+c_{1} x-A y^{\prime }=B \,{\mathrm e}^{i \omega t} \\ a_{2} y^{\prime \prime }+b_{2} y^{\prime }+c_{2} y+A x^{\prime }=0 \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.934 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }+a \left (x^{\prime }-y^{\prime }\right )+b_{1} x=c_{1} {\mathrm e}^{i \omega t} \\ y^{\prime \prime }+a \left (y^{\prime }-x^{\prime }\right )+b_{2} y=c_{2} {\mathrm e}^{i \omega t} \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.958 |
|
|
\[ {}\left [\begin {array}{c} \operatorname {a11} x^{\prime \prime }+\operatorname {b11} x^{\prime }+\operatorname {c11} x+\operatorname {a12} y^{\prime \prime }+\operatorname {b12} y^{\prime }+\operatorname {c12} y=0 \\ \operatorname {a21} x^{\prime \prime }+\operatorname {b21} x^{\prime }+\operatorname {c21} x+\operatorname {a22} y^{\prime \prime }+\operatorname {b22} y^{\prime }+\operatorname {c22} y=0 \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.945 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }-2 x^{\prime }-y^{\prime }+y=0 \\ y^{\prime \prime \prime }-y^{\prime \prime }+2 x^{\prime }-x=t \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.928 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }+y^{\prime \prime }+y^{\prime }=\sinh \left (2 t \right ) \\ 2 x^{\prime \prime }+y^{\prime \prime }=2 t \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.92 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }-x^{\prime }+y^{\prime }=0 \\ x^{\prime \prime }+y^{\prime \prime }-x=0 \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.917 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=2 x \\ y^{\prime }=3 x-2 y \\ z^{\prime }=2 y+3 z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.443 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=4 x \\ y^{\prime }=x-2 y \\ z^{\prime }=x-4 y+z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.453 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y-z \\ y^{\prime }=x+y \\ z^{\prime }=z+x \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.419 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }-y+z=0 \\ y^{\prime }-x-y=t \\ z^{\prime }-x-z=t \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.564 |
|
|
\[ {}\left [\begin {array}{c} a x^{\prime }=b c \left (y-z\right ) \\ b y^{\prime }=c a \left (-x+z\right ) \\ c z^{\prime }=a b \left (x-y\right ) \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.363 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=c y-b z \\ y^{\prime }=a z-c x \\ z^{\prime }=b x-a y \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
1.073 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=h \left (t \right ) y-g \left (t \right ) z \\ y^{\prime }=f \left (t \right ) z-h \left (t \right ) x \\ z^{\prime }=x g \left (t \right )-y f \left (t \right ) \end {array}\right ] \] |
1 |
0 |
3 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.032 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+y-z \\ y^{\prime }=y+z-x \\ z^{\prime }=x-y+z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.855 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-3 x+48 y-28 z \\ y^{\prime }=-4 x+40 y-22 z \\ z^{\prime }=-6 x+57 y-31 z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
0.529 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=6 x-72 y+44 z \\ y^{\prime }=4 x-4 y+26 z \\ z^{\prime }=6 x-63 y+38 z \end {array}\right ] \] |
1 |
1 |
3 |
system of linear ODEs |
system of linear ODEs |
✓ |
✓ |
23.843 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=a x+g y+\beta z \\ y^{\prime }=g x+b y+\alpha z \\ z^{\prime }=\beta x+\alpha y+c z \end {array}\right ] \] |
1 |
0 |
3 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
127.673 |
|
|
\[ {}\left [\begin {array}{c} t x^{\prime }=2 x-t \\ t^{3} y^{\prime }=-x+t^{2} y+t \\ t^{4} z^{\prime }=-x-t^{2} y+t^{3} z+t \end {array}\right ] \] |
1 |
0 |
3 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.032 |
|
|
\[ {}\left [\begin {array}{c} a t x^{\prime }=b c \left (y-z\right ) \\ b t y^{\prime }=c a \left (-x+z\right ) \\ c t z^{\prime }=a b \left (x-y\right ) \end {array}\right ] \] |
1 |
0 |
3 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.033 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime }=a x_{2}+b x_{3} \cos \left (c t \right )+b x_{4} \sin \left (c t \right ) \\ x_{2}^{\prime }=-a x_{1}+b x_{3} \sin \left (c t \right )-b x_{4} \cos \left (c t \right ) \\ x_{3}^{\prime }=-b x_{1} \cos \left (c t \right )-b x_{2} \sin \left (c t \right )+a x_{4} \\ x_{4}^{\prime }=-b x_{1} \sin \left (c t \right )+b x_{2} \cos \left (c t \right )-a x_{3} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.05 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-x \left (x+y\right ) \\ y^{\prime }=y \left (x+y\right ) \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.436 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\left (a y+b \right ) x \\ y^{\prime }=\left (c x+d \right ) y \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.308 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x \left (a \left (p x+q y\right )+\alpha \right ) \\ y^{\prime }=y \left (\beta +b \left (p x+q y\right )\right ) \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.303 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=h \left (a -x\right ) \left (c -x-y\right ) \\ y^{\prime }=k \left (b -y\right ) \left (c -x-y\right ) \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.315 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y^{2}-\cos \left (x\right ) \\ y^{\prime }=-y \sin \left (x\right ) \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.326 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-x \,y^{2}+x+y \\ y^{\prime }=x^{2} y-x-y \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.315 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x+y-x \left (x^{2}+y^{2}\right ) \\ y^{\prime }=-x+y-y \left (x^{2}+y^{2}\right ) \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.318 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-y+x \left (x^{2}+y^{2}-1\right ) \\ y^{\prime }=x+y \left (x^{2}+y^{2}-1\right ) \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.316 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-y \left (x^{2}+y^{2}\right ) \\ y^{\prime }=\left \{\begin {array}{cc} x^{2}+y^{2} & 2 x\le x^{2}+y^{2} \\ \left (\frac {x}{2}-\frac {y^{2}}{2 x}\right ) \left (x^{2}+y^{2}\right ) & \operatorname {otherwise} \end {array}\right . \end {array}\right ] \] |
1 |
0 |
0 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.34 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-y+\left \{\begin {array}{cc} x \left (x^{2}+y^{2}-1\right ) \sin \left (\frac {1}{x^{2}+y^{2}}\right ) & x^{2}+y^{2}\neq 1 \\ 0 & \operatorname {otherwise} \end {array}\right . \\ y^{\prime }=x+\left \{\begin {array}{cc} y \left (x^{2}+y^{2}-1\right ) \sin \left (\frac {1}{x^{2}+y^{2}}\right ) & x^{2}+y^{2}\neq 1 \\ 0 & \operatorname {otherwise} \end {array}\right . \end {array}\right ] \] |
1 |
0 |
0 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.341 |
|
|
\[ {}\left [\begin {array}{c} \left (t^{2}+1\right ) x^{\prime }=-x t +y \\ \left (t^{2}+1\right ) y^{\prime }=-x-t y \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.026 |
|
|
\[ {}\left [\begin {array}{c} \left (x^{2}+y^{2}-t^{2}\right ) x^{\prime }=-2 x t \\ \left (x^{2}+y^{2}-t^{2}\right ) y^{\prime }=-2 t y \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.311 |
|
|
\[ {}\left [\begin {array}{c} {x^{\prime }}^{2}+t x^{\prime }+a y^{\prime }-x=0 \\ x^{\prime } y^{\prime }+t y^{\prime }-y=0 \end {array}\right ] \] |
1 |
0 |
3 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.364 |
|
|
\[ {}\left [\begin {array}{c} x=t x^{\prime }+f \left (x^{\prime }, y^{\prime }\right ) \\ y=t y^{\prime }+g \left (x^{\prime }, y^{\prime }\right ) \end {array}\right ] \] |
1 |
0 |
2 |
unknown |
system of linear ODEs |
✗ |
N/A |
0.899 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }=a \,{\mathrm e}^{2 x}-{\mathrm e}^{-x}+{\mathrm e}^{-2 x} \cos \left (y\right )^{2} \\ y^{\prime \prime }={\mathrm e}^{-2 x} \sin \left (y\right ) \cos \left (y\right )-\frac {\sin \left (y\right )}{\cos \left (y\right )^{3}} \end {array}\right ] \] |
1 |
0 |
0 |
unknown |
system of linear ODEs |
❇ |
N/A |
1.807 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime \prime }=\frac {k x}{\left (x^{2}+y^{2}\right )^{\frac {3}{2}}} \\ y^{\prime \prime }=\frac {k y}{\left (x^{2}+y^{2}\right )^{\frac {3}{2}}} \end {array}\right ] \] |
1 |
0 |
0 |
unknown |
system of linear ODEs |
❇ |
N/A |
1.81 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=y-z \\ y^{\prime }=x^{2}+y \\ z^{\prime }=x^{2}+z \end {array}\right ] \] |
1 |
0 |
3 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.481 |
|
|
\[ {}\left [\begin {array}{c} a x^{\prime }=\left (b -c \right ) y z \\ b y^{\prime }=\left (c -a \right ) z x \\ c z^{\prime }=\left (-b +a \right ) x y \end {array}\right ] \] |
1 |
0 |
4 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.716 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x \left (y-z\right ) \\ y^{\prime }=y \left (-x+z\right ) \\ z^{\prime }=z \left (x-y\right ) \end {array}\right ] \] |
1 |
0 |
5 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.71 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }+y^{\prime }=x y \\ y^{\prime }+z^{\prime }=y z \\ x^{\prime }+z^{\prime }=x z \end {array}\right ] \] |
1 |
0 |
3 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.678 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=\frac {x^{2}}{2}-\frac {y}{24} \\ y^{\prime }=2 x y-3 z \\ z^{\prime }=3 x z-\frac {y^{2}}{6} \end {array}\right ] \] |
1 |
0 |
3 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.702 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x \left (y^{2}-z^{2}\right ) \\ y^{\prime }=y \left (z^{2}-x^{2}\right ) \\ z^{\prime }=z \left (x^{2}-y^{2}\right ) \end {array}\right ] \] |
1 |
0 |
8 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.713 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=x \left (y^{2}-z^{2}\right ) \\ y^{\prime }=-y \left (z^{2}+x^{2}\right ) \\ z^{\prime }=z \left (x^{2}+y^{2}\right ) \end {array}\right ] \] |
1 |
0 |
5 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.704 |
|
|
\[ {}\left [\begin {array}{c} x^{\prime }=-x \,y^{2}+x+y \\ y^{\prime }=x^{2} y-x-y \\ z^{\prime }=y^{2}-x^{2} \end {array}\right ] \] |
1 |
0 |
2 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
0.71 |
|
|
\[ {}\left [\begin {array}{c} \left (x-y\right ) \left (x-z\right ) x^{\prime }=f \left (t \right ) \\ \left (-x+y\right ) \left (y-z\right ) y^{\prime }=f \left (t \right ) \\ \left (-x+z\right ) \left (-y+z\right ) z^{\prime }=f \left (t \right ) \end {array}\right ] \] |
1 |
0 |
3 |
system of linear ODEs |
system of linear ODEs |
✗ |
N/A |
0.708 |
|
|
\[ {}\left [\begin {array}{c} x_{1}^{\prime } \sin \left (x_{2}\right )=x_{4} \sin \left (x_{3}\right )+x_{5} \cos \left (x_{3}\right ) \\ x_{2}^{\prime }=x_{4} \cos \left (x_{3}\right )-x_{5} \sin \left (x_{3}\right ) \\ x_{3}^{\prime }+x_{1}^{\prime } \cos \left (x_{2}\right )=a \\ x_{4}^{\prime }-\left (1-\lambda \right ) a x_{5}=-m \sin \left (x_{2}\right ) \cos \left (x_{3}\right ) \\ x_{5}^{\prime }+\left (1-\lambda \right ) a x_{4}=m \sin \left (x_{2}\right ) \sin \left (x_{3}\right ) \end {array}\right ] \] |
1 |
0 |
0 |
system of linear ODEs |
system of linear ODEs |
❇ |
N/A |
1.958 |
|
|
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