|
# |
ODE |
CAS classification |
Maple solved? |
Mma solved? |
|
\[
{}y^{\prime } = x^{2}-y^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y y^{\prime \prime } = 6 x^{4}
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+3 y^{\prime \prime }-54 y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}3 y^{\prime \prime \prime }-2 y^{\prime \prime }+12 y^{\prime }-8 y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}6 y^{\prime \prime \prime \prime }+5 y^{\prime \prime \prime }+25 y^{\prime \prime }+20 y^{\prime }+4 y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}9 y^{\prime \prime \prime }+11 y^{\prime \prime }+4 y^{\prime }-14 y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }-5 y^{\prime \prime }+100 y^{\prime }-500 y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime }+y = 0
\] |
[_separable] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime } = 2 y
\] |
[_separable] |
✗ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\cos \left (x \right ) y^{\prime }+y x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}3 x^{3} y^{\prime \prime }+2 x^{2} y^{\prime }+\left (-x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{3} \left (1-x \right ) y^{\prime \prime }+\left (3 x +2\right ) y^{\prime }+y x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }-y^{\prime } x +y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = x^{2}+y^{2}
\] |
[[_Riccati, _special]] |
✓ |
✓ |
|
|
\[
{}t x^{\prime \prime }+\left (t -2\right ) x^{\prime }+x = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
|
|
\[
{}t x^{\prime \prime }+\left (3 t -1\right ) x^{\prime }+3 x = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
|
|
\[
{}t x^{\prime \prime }-\left (4 t +1\right ) x^{\prime }+2 \left (2 t +1\right ) x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}t x^{\prime \prime }+2 \left (t -1\right ) x^{\prime }-2 x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}t x^{\prime \prime }-2 x^{\prime }+x t = 0
\] |
[_Lienard] |
✓ |
✓ |
|
|
\[
{}t x^{\prime \prime }+\left (4 t -2\right ) x^{\prime }+\left (13 t -4\right ) x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) t -{\mathrm e}^{t} y \left (t \right )+\cos \left (t \right ) \\ y^{\prime }\left (t \right )={\mathrm e}^{-t} x \left (t \right )+t^{2} y \left (t \right )-\sin \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) t -y \left (t \right )+{\mathrm e}^{t} z \left (t \right ) \\ y^{\prime }\left (t \right )=2 x \left (t \right )+t^{2} y \left (t \right )-z \left (t \right ) \\ z^{\prime }\left (t \right )={\mathrm e}^{-t} x \left (t \right )+3 t y \left (t \right )+t^{3} z \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}y^{\prime } = 1+x^{2}+y^{2}+x^{2} y^{4}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime }+y = 0
\] |
[_separable] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime } = 2 y
\] |
[_separable] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-{\mathrm e}^{-x}+x}{{\mathrm e}^{y}+x}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}{\mathrm e}^{x} \sin \left (y\right )+3 y-\left (3 x -{\mathrm e}^{x} \sin \left (y\right )\right ) y^{\prime } = 0
\] |
[‘x=_G(y,y’)‘] |
✗ |
✗ |
|
|
\[
{}x \ln \left (x \right )+y x +\left (y \ln \left (x \right )+y x \right ) y^{\prime } = 0
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}u^{\prime \prime }+u^{\prime }+\frac {u^{3}}{5} = \cos \left (t \right )
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}t \left (t -1\right ) y^{\prime \prime \prime \prime }+{\mathrm e}^{t} y^{\prime \prime }+4 t^{2} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}t y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }+t y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (2-t \right ) y^{\prime \prime \prime }+\left (2 t -3\right ) y^{\prime \prime }-y^{\prime } t +y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}t^{2} \left (t +3\right ) y^{\prime \prime \prime }-3 t \left (t +2\right ) y^{\prime \prime }+6 \left (1+t \right ) y^{\prime }-6 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{2}+y^{2}}{\sin \left (x \right )}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \frac {{\mathrm e}^{x}+y}{x^{2}+y^{2}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \tan \left (y x \right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{2}+y^{2}}{\ln \left (y x \right )}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \left (x^{2}+y^{2}\right ) y^{{1}/{3}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \ln \left (1+x^{2}+y^{2}\right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \sqrt {x^{2}+y^{2}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \left (x^{2}+y^{2}\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}2 x^{2}+8 y x +y^{2}+\left (2 x^{2}+\frac {x y^{3}}{3}\right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
✗ |
|
|
\[
{}y \sin \left (y x \right )+x y^{2} \cos \left (y x \right )+\left (x \sin \left (y x \right )+x y^{2} \cos \left (y x \right )\right ) y^{\prime } = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}3 x^{2} \cos \left (x \right ) y-x^{3} y \sin \left (x \right )+4 x +\left (8 y-x^{4} \sin \left (x \right ) y\right ) y^{\prime } = 0
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}3 x^{2} y^{3}-y^{2}+y+\left (-y x +2 x \right ) y^{\prime } = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\left (6 x -8\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (2 x +1\right ) y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}\sin \left (x \right ) y^{\prime \prime }+\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime }+\left (\sin \left (x \right )-\cos \left (x \right )\right ) y = {\mathrm e}^{-x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\cos \left (t^{2}\right )
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = 1+y+y^{2} \cos \left (t \right )
\] |
[_Riccati] |
✓ |
✗ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{-t^{2}}+y^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{-t^{2}}+y^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{-t^{2}}+y^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y+{\mathrm e}^{-y}+{\mathrm e}^{-t}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{3}+{\mathrm e}^{-5 t}
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \left (4 y+{\mathrm e}^{-t^{2}}\right ) {\mathrm e}^{2 y}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{-t}+\ln \left (1+y^{2}\right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}t \left (t -2\right )^{2} y^{\prime \prime }+y^{\prime } t +y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}\left (-t^{2}+1\right ) y^{\prime \prime }+\frac {y^{\prime }}{\sin \left (1+t \right )}+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}t^{3} y^{\prime \prime }-y^{\prime } t -\left (t^{2}+\frac {5}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}2 t \cos \left (y\right )+3 t^{2} y+\left (2 y+2 t^{2}\right ) y^{\prime } = 0
\] |
[‘x=_G(y,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = t^{2}+y^{2}
\] |
[[_Riccati, _special]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = 1+y+y^{2} \cos \left (t \right )
\] |
[_Riccati] |
✓ |
✗ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{-t^{2}}+y^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{-t^{2}}+y^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{-t^{2}}+y^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y+{\mathrm e}^{-y}+{\mathrm e}^{-t}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{3}+{\mathrm e}^{-5 t}
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \left (4 y+{\mathrm e}^{-t^{2}}\right ) {\mathrm e}^{2 y}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{-t}+\ln \left (1+y^{2}\right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y+{\mathrm e}^{-y}+2 t
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \frac {t^{2}+y^{2}}{1+t +y^{2}}
\] |
[_rational] |
✗ |
✗ |
|
|
\[
{}t \left (t -2\right )^{2} y^{\prime \prime }+y^{\prime } t +y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}\left (-t^{2}+1\right ) y^{\prime \prime }+\frac {y^{\prime }}{\sin \left (1+t \right )}+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}t^{2} y^{\prime \prime }+\left (-t^{2}+1\right ) y^{\prime }+4 t y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+y^{\prime \prime }+2 y^{\prime }-2 y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}x y^{2}+2 y+\left (2 y^{3}-x^{2} y+2 x \right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
✗ |
|
|
\[
{}\left (x -x \sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y-x^{2} \sqrt {x^{2}-y^{2}}-y^{\prime } x = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
✗ |
|
|
\[
{}1+x y \left (1+x y^{2}\right ) y^{\prime } = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✗ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime }+y x = x \left (-x^{2}+1\right ) \sqrt {y}
\] |
[_rational, _Bernoulli] |
✓ |
✓ |
|
|
\[
{}x^{3} \left (x^{2}+3\right ) y^{\prime \prime }+5 y^{\prime } x -\left (x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}-\frac {{y^{\prime }}^{2}}{y^{2}}+\frac {y^{\prime \prime }}{y}+\frac {2 a \coth \left (2 a x \right ) y^{\prime }}{y} = 2 a^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}2 y y^{\prime \prime \prime }+2 \left (y+3 y^{\prime }\right ) y^{\prime \prime }+2 {y^{\prime }}^{2} = \sin \left (x \right )
\] |
[[_3rd_order, _exact, _nonlinear]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\frac {y}{z^{3}} = 0
\] |
[[_Emden, _Fowler]] |
✗ |
✓ |
|
|
\[
{}y^{\prime }+p \left (x \right ) y+q \left (x \right ) y^{2} = r \left (x \right )
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y \,{\mathrm e}^{y x}+\left (2 y-x \,{\mathrm e}^{y x}\right ) y^{\prime } = 0
\] |
[‘x=_G(y,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x \left (x -3\right )}-\frac {y}{x^{3} \left (x +3\right )} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}y^{2} \left (x^{2}+1\right )+y+\left (2 y x +1\right ) y^{\prime } = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}2 y^{2}-4 x +5 = \left (4-2 y+4 y x \right ) y^{\prime }
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}x^{2}+y^{3}+y+\left (x^{3}+y^{2}-x \right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = a \,x^{n -1}+b \,x^{2 n}+c y^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = f \left (x \right )+a y+b y^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = f \left (x \right )+g \left (x \right ) y+a y^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = f \left (x \right )+g \left (x \right ) y+h \left (x \right ) y^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime }+\left (a x +y\right ) y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \left (a \,{\mathrm e}^{x}+y\right ) y^{2}
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime }+3 a \left (2 x +y\right ) y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \operatorname {f0} \left (x \right )+\operatorname {f1} \left (x \right ) y+\operatorname {f2} \left (x \right ) y^{2}+\operatorname {f3} \left (x \right ) y^{3}
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = f \left (x \right )+g \left (x \right ) y+h \left (x \right ) y^{n}
\] |
[_Chini] |
✗ |
✗ |
|
|
\[
{}y^{\prime }+\left (f \left (x \right )-y\right ) g \left (x \right ) \sqrt {\left (y-a \right ) \left (y-b \right )} = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime }+f \left (x \right )+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) \cos \left (a y\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime }+x \left (\sin \left (2 y\right )-x^{2} \cos \left (y\right )^{2}\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \tan \left (x \right ) \left (\tan \left (y\right )+\sec \left (x \right ) \sec \left (y\right )\right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \left (1+\cos \left (x \right ) \sin \left (y\right )\right ) \tan \left (y\right )
\] |
unknown |
✗ |
✓ |
|
|
\[
{}y^{\prime }+f \left (x \right )+g \left (x \right ) \tan \left (y\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = x^{m -1} y^{1-n} f \left (a \,x^{m}+b y^{n}\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}2 y^{\prime } = 2 \sin \left (y\right )^{2} \tan \left (y\right )-x \sin \left (2 y\right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
✓ |
|
|
\[
{}y^{\prime } x = y+x \sqrt {x^{2}+y^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x = y-x \left (x -y\right ) \sqrt {x^{2}+y^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x = \sin \left (x -y\right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } x +n y = f \left (x \right ) g \left (x^{n} y\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}-a y^{3}
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime }+a y^{2}+b \,x^{2} y^{3} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime } = \sec \left (y\right )+3 x \tan \left (y\right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime } = n \left (1-2 y x +y^{2}\right )
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime } = 1+y^{2}-2 x y \left (1+y^{2}\right )
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right ) = x \left (x^{2}+1\right ) \cos \left (y\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}\left (b x +a \right )^{2} y^{\prime }+c y^{2}+\left (b x +a \right ) y^{3} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime } = \cos \left (y\right ) \left (\cos \left (y\right )-2 x^{2} \sin \left (y\right )\right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}x^{7} y^{\prime }+5 x^{3} y^{2}+2 \left (x^{2}+1\right ) y^{3} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}x^{n} y^{\prime }+x^{2 n -2}+y^{2}+\left (1-n \right ) x^{n -1} = 0
\] |
[_Riccati] |
✓ |
✗ |
|
|
\[
{}x^{k} y^{\prime } = a \,x^{m}+b y^{n}
\] |
[_Chini] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+x^{3}+y = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+f \left (x \right ) = g \left (x \right ) y
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+x +f \left (x^{2}+y^{2}\right ) g \left (x \right ) = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}\left (\tan \left (x \right ) \sec \left (x \right )-2 y\right ) y^{\prime }+\sec \left (x \right ) \left (1+2 y \sin \left (x \right )\right ) = 0
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}\left (x +4 x^{3}+5 y\right ) y^{\prime }+7 x^{3}+3 x^{2} y+4 y = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}x \left (a +y\right ) y^{\prime }+b x +c y = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}\left (a +x \left (x +y\right )\right ) y^{\prime } = b \left (x +y\right ) y
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}\left (1-x^{2} y\right ) y^{\prime }-1+x y^{2} = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}x^{7} y y^{\prime } = 2 x^{2}+2+5 x^{3} y
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}\left (\operatorname {g0} \left (x \right )+y \operatorname {g1} \left (x \right )\right ) y^{\prime } = \operatorname {f0} \left (x \right )+\operatorname {f1} \left (x \right ) y+\operatorname {f2} \left (x \right ) y^{2}+\operatorname {f3} \left (x \right ) y^{3}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
✗ |
|
|
\[
{}\left (x +2 y+y^{2}\right ) y^{\prime }+y \left (1+y\right )+\left (x +y\right )^{2} y^{2} = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}\left (\cot \left (x \right )-2 y^{2}\right ) y^{\prime } = y^{3} \csc \left (x \right ) \sec \left (x \right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
✓ |
|
|
\[
{}x \left (x^{3}-3 x^{3} y+4 y^{2}\right ) y^{\prime } = 6 y^{3}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
|
|
\[
{}\left (a -3 x^{2}-y^{2}\right ) y y^{\prime }+x \left (a -x^{2}+y^{2}\right ) = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-x^{3}+3 x y^{2}+2 y^{3}\right ) y^{\prime }+2 x^{3}+3 x^{2} y+y^{2}-y^{3} = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}\left (x +2 y+2 x^{2} y^{3}+x y^{4}\right ) y^{\prime }+\left (1+y^{4}\right ) y = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}f \left (x \right ) y^{m} y^{\prime }+g \left (x \right ) y^{m +1}+h \left (x \right ) y^{n} = 0
\] |
[_Bernoulli] |
✓ |
✓ |
|
|
\[
{}x \left (1-\sqrt {x^{2}-y^{2}}\right ) y^{\prime } = y
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}a x \sqrt {1+{y^{\prime }}^{2}}+y^{\prime } x -y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✗ |
✓ |
|
|
\[
{}\frac {x^{n} y^{\prime }}{b y^{2}-c \,x^{2 a}}-\frac {a y x^{a -1}}{b y^{2}-c \,x^{2 a}}+x^{a -1} = 0
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y = y^{\prime } x +a x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✗ |
✓ |
|
|
\[
{}x +y y^{\prime } = a \sqrt {1+{y^{\prime }}^{2}}
\] |
unknown |
✓ |
✓ |
|
|
\[
{}\left (x y \sqrt {x^{2}-y^{2}}+x \right ) y^{\prime } = y-x^{2} \sqrt {x^{2}-y^{2}}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (1+y\right ) y^{\prime } = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime }+x y^{3}+a y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime }+y^{\prime } x +y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }-\left (2 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }+\frac {a y}{x^{{3}/{2}}} = 0
\] |
[[_Emden, _Fowler]] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+y = x^{{3}/{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x} = \sqrt {x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}s^{\prime } = t \ln \left (s^{2 t}\right )+8 t^{2}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✓ |
|
|
\[
{}s^{2}+s^{\prime } = \frac {s+1}{s t}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
✗ |
|
|
\[
{}x^{\prime }+x t = {\mathrm e}^{x}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}x x^{\prime }+t^{2} x = \sin \left (t \right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+3 y^{\prime }-y x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+m y = 0
\] |
[_Laguerre] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✗ |
✓ |
|
|
\[
{}x -2 \sin \left (y\right )+3+\left (2 x -4 \sin \left (y\right )-3\right ) \cos \left (y\right ) y^{\prime } = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}\left (x -x \sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}4 x^{2} y y^{\prime } = 3 x \left (3 y^{2}+2\right )+2 \left (3 y^{2}+2\right )^{3}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}\left (2 x -3\right ) y^{\prime \prime \prime }-\left (6 x -7\right ) y^{\prime \prime }+4 y^{\prime } x -4 y = 8
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (1+2 y+3 y^{2}\right ) y^{\prime \prime \prime }+6 y^{\prime } \left (y^{\prime \prime }+{y^{\prime }}^{2}+3 y y^{\prime \prime }\right ) = x
\] |
[[_3rd_order, _exact, _nonlinear]] |
✓ |
✓ |
|
|
\[
{}3 x \left (y^{2} y^{\prime \prime \prime }+6 y y^{\prime } y^{\prime \prime }+2 {y^{\prime }}^{3}\right )-3 y \left (y y^{\prime \prime }+2 {y^{\prime }}^{2}\right ) = -\frac {2}{x}
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime \prime \prime }+3 y^{\prime } y^{\prime \prime }-2 y y^{\prime \prime }-2 {y^{\prime }}^{2}+y y^{\prime } = {\mathrm e}^{2 x}
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} \left (x +1\right ) y^{\prime \prime \prime }-\left (2+4 x \right ) x^{2} y^{\prime \prime }+\left (4+10 x \right ) x y^{\prime }-\left (4+12 x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} \left (x^{2}+1\right ) y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-\left (12 x^{2}+4\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+y = \frac {1}{x^{4}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}x y^{\prime \prime }-2 y^{\prime }+y = \cos \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}y^{\prime }-\frac {y}{x} = \cos \left (x \right )
\] |
[_linear] |
✗ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+4 x^{2} y^{\prime }+3 y = 0
\] |
[[_Emden, _Fowler]] |
✗ |
✓ |
|
|
\[
{}x^{3} \left (x^{2}-25\right ) \left (x -2\right )^{2} y^{\prime \prime }+3 x \left (x -2\right ) y^{\prime }+7 \left (x +5\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime }+\lambda y = 0
\] |
[[_Emden, _Fowler]] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✗ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+\cos \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }-2 x y^{\prime \prime }+4 x^{2} y^{\prime }+8 x^{3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+{\mathrm e}^{x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y = x +\frac {1}{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}\left (\cos \left (x \right )-\sin \left (x \right )\right ) y^{\prime \prime }-2 \sin \left (x \right ) y^{\prime }+\left (\cos \left (x \right )+\sin \left (x \right )\right ) y = \left (\cos \left (x \right )-\sin \left (x \right )\right )^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}x +y y^{\prime } = a {y^{\prime }}^{2}
\] |
unknown |
✗ |
✓ |
|
|
\[
{}x^{2} y y^{\prime \prime } = x^{2} {y^{\prime }}^{2}-y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) z^{\prime \prime }+\left (1-3 x \right ) z^{\prime }+k z = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) \eta ^{\prime \prime }-\left (x +1\right ) \eta ^{\prime }+\left (k +1\right ) \eta = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x -y = x \sqrt {x^{2}-y^{2}}\, y^{\prime }
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }-y x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime } x +2 \alpha y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }-5 y^{\prime }+3 x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=3 y_{1} \left (x \right )+x y_{3} \left (x \right ) \\ y_{2}^{\prime }\left (x \right )=y_{2} \left (x \right )+x^{3} y_{3} \left (x \right ) \\ y_{3}^{\prime }\left (x \right )=2 x y_{1} \left (x \right )-y_{2} \left (x \right )+{\mathrm e}^{x} y_{3} \left (x \right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}x \ln \left (x \right ) y^{\prime }+y = 3 x^{3}
\] |
[_linear] |
✗ |
✗ |
|
|
\[
{}2 y^{2}-4 x +5 = \left (4-2 y+4 y x \right ) y^{\prime }
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}x y y^{\prime \prime } = y^{\prime }+{y^{\prime }}^{3}
\] |
[NONE] |
✗ |
✓ |
|
|
\[
{}x^{2} y^{\prime } = y
\] |
[_separable] |
✗ |
✓ |
|
|
\[
{}x^{3} \left (x -1\right ) y^{\prime \prime }-2 \left (x -1\right ) y^{\prime }+3 y x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (2-x \right ) y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✗ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime }+y \sin \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x^{2}}-\frac {y}{x^{3}} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+y x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-3 y^{\prime } x +\left (x -1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }-2 x^{2} y^{\prime \prime }+\left (x^{2}+2 x \right ) y^{\prime }-y x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+\left (2 x^{3}-x^{2}\right ) y^{\prime \prime }-y^{\prime } x +y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) y \left (t \right )+1 \\ y^{\prime }\left (t \right )=-x \left (t \right )+y \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=t y \left (t \right )+1 \\ y^{\prime }\left (t \right )=-x \left (t \right ) t +y \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y+x \,{\mathrm e}^{y}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+5 y^{\prime } x +\sqrt {x}\, y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{3} y^{\prime \prime }+4 x^{2} y^{\prime }+3 y = 0
\] |
[[_Emden, _Fowler]] |
✗ |
✓ |
|
|
\[
{}x^{3} \left (x^{2}-25\right ) \left (x -2\right )^{2} y^{\prime \prime }+3 x \left (x -2\right ) y^{\prime }+7 \left (x +5\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✗ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✗ |
✓ |
|
|
\[
{}{y^{\prime \prime }}^{2}-2 y^{\prime \prime }+{y^{\prime }}^{2}-2 y^{\prime } x +x^{2} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+5 y^{\prime } x +3 y = 0
\] |
[[_3rd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \sqrt {1-x^{2}-y^{2}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\frac {\left (t^{2}-1\right ) y^{\prime }}{t}+\frac {t^{2} y}{\left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime \prime } = x
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{2} y^{\prime \prime } = x
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}3 y y^{\prime \prime } = \sin \left (x \right )
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-y y^{\prime } = 2 x
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-a x y^{\prime }-b x y-c x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{3} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }-x^{3} y^{\prime }-x^{3} y-x^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }-y x -x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{3}-x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }-x^{3} y-x^{4}-x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }-x^{3} y^{\prime }-y x -x^{3}-x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-x^{3} y^{\prime }-x^{3} y-x^{4}-x^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3} = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y x = x
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = x
\] |
[[_2nd_order, _missing_y]] |
✓ |
✗ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+y {y^{\prime }}^{2} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (-x^{2}+1\right ) y = x +1
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (-x^{2}+1\right ) y = x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (-x^{2}+1\right ) y = x^{2}+x +1
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (-x^{2}+1\right ) y = \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (-x^{2}+1\right ) y = 1+\sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (-x^{2}+1\right ) y = \cos \left (x \right )+\sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}2 x^{2} y^{\prime \prime }+2 y^{\prime } x -y x = 1
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}\left (y-2 y^{\prime } x \right )^{2} = {y^{\prime }}^{3}
\] |
unknown |
✓ |
✗ |
|
|
\[
{}\frac {x y^{\prime \prime }}{1-x}+y = \frac {1}{1-x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}\frac {x y^{\prime \prime }}{1-x}+y = \cos \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}\frac {x y^{\prime \prime }}{-x^{2}+1}+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime }+y = \frac {1}{x}
\] |
[[_linear, ‘class A‘]] |
✗ |
✓ |
|
|
\[
{}y^{\prime }+y = \frac {1}{x^{2}}
\] |
[[_linear, ‘class A‘]] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = \frac {1}{x}
\] |
[_quadrature] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime } = \frac {1}{x}
\] |
[[_2nd_order, _quadrature]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }+y^{\prime } = \frac {1}{x}
\] |
[[_2nd_order, _missing_y]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }+y = \frac {1}{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }+y^{\prime }+y = \frac {1}{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y x +3 x -2 y+6}{y x -3 x -2 y+6}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \cos \left (x \right )+\frac {y^{2}}{x}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}{y^{\prime \prime }}^{2}+y^{\prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y^{2} {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (\sin \left (x \right )+2 x \right ) y^{\prime }+\cos \left (y\right ) y {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (x +3\right ) y^{\prime }+\left (3+y^{2}\right ) {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}10 y^{\prime \prime }+\left ({\mathrm e}^{x}+3 x \right ) y^{\prime }+\frac {3 \,{\mathrm e}^{y} {y^{\prime }}^{2}}{\sin \left (y\right )} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }+y = 4 \cos \left (\ln \left (x +1\right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (y^{\prime } x -y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }-y x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime }+a y^{2}-b \,x^{2 \nu }-c \,x^{\nu -1} = 0
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}+\frac {g^{\prime }\left (x \right )}{f \left (x \right )} = 0
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime }+y^{3}+a x y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-y^{3}-a \,{\mathrm e}^{x} y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime }+3 a y^{3}+6 a x y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-x \left (x +2\right ) y^{3}-\left (x +3\right ) y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime }+\left (4 a^{2} x +3 a \,x^{2}+b \right ) y^{3}+3 x y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime }+2 \left (a^{2} x^{3}-b^{2} x \right ) y^{3}+3 b y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-a \left (x^{n}-x \right ) y^{3}-y^{2} = 0
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}y^{\prime }-\left (a \,x^{n}+b x \right ) y^{3}-c y^{2} = 0
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}y^{\prime }+a \phi ^{\prime }\left (x \right ) y^{3}+6 a \phi \left (x \right ) y^{2}+\frac {\left (2 a +1\right ) y \phi ^{\prime \prime }\left (x \right )}{\phi ^{\prime }\left (x \right )}+2 a +2 = 0
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}y^{\prime }-f_{3} \left (x \right ) y^{3}-f_{2} \left (x \right ) y^{2}-f_{1} \left (x \right ) y-f_{0} \left (x \right ) = 0
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0
\] |
unknown |
✓ |
✓ |
|
|
\[
{}y^{\prime }-a^{n} f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0
\] |
[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-f \left (x \right ) y^{n}-g \left (x \right ) y-h \left (x \right ) = 0
\] |
[_Chini] |
✗ |
✗ |
|
|
\[
{}y^{\prime }-f \left (x \right ) y^{a}-g \left (x \right ) y^{b} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime }-\frac {y-x^{2} \sqrt {x^{2}-y^{2}}}{x y \sqrt {x^{2}-y^{2}}+x} = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-f \left (x \right ) \left (y-g \left (x \right )\right ) \sqrt {\left (y-a \right ) \left (y-b \right )} = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime }+f \left (x \right ) \cos \left (a y\right )+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )-f^{\prime }\left (x \right )-1 = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )-1 = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-a \left (1+\tan \left (y\right )^{2}\right )+\tan \left (y\right ) \tan \left (x \right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime }-\tan \left (y x \right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-x^{a -1} y^{1-b} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime }-\frac {y a f \left (x^{c} y\right )+c \,x^{a} y^{b}}{x b f \left (x^{c} y\right )-x^{a} y^{b}} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}2 y^{\prime }-3 y^{2}-4 a y-b -c \,{\mathrm e}^{-2 a x} = 0
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x +f \left (x \right ) \left (y^{2}-x^{2}\right ) = 0
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } x +y^{3}+3 x y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x -x \sqrt {x^{2}+y^{2}}-y = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x -x \left (-x +y\right ) \sqrt {x^{2}+y^{2}}-y = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x -y \left (x \ln \left (\frac {x^{2}}{y}\right )+2\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x -\sin \left (x -y\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } x +a y-f \left (x \right ) g \left (x^{a} y\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime }+a y^{3}-a \,x^{2} y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime }+x y^{3}+a y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime }+a \,x^{2} y^{3}+b y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+\left (1+y^{2}\right ) \left (2 y x -1\right ) = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right )-x \left (x^{2}+1\right ) \cos \left (y\right )^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime }+a \left (1-2 y x +y^{2}\right ) = 0
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}x^{7} y^{\prime }+5 x^{3} y^{2}+2 \left (x^{2}+1\right ) y^{3} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}f \left (x \right ) y^{\prime }+g \left (x \right ) s \left (y\right )+h \left (x \right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+x^{3}+y = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+a y+\frac {\left (a^{2}-1\right ) x}{4}+b \,x^{n} = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+a y+b \,{\mathrm e}^{x}-2 a = 0
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+x +f \left (x^{2}+y^{2}\right ) g \left (x \right ) = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}\left (y+g \left (x \right )\right ) y^{\prime }-f_{2} \left (x \right ) y^{2}-f_{1} \left (x \right ) y-f_{0} \left (x \right ) = 0
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}x y y^{\prime }-y^{2}+y x +x^{3}-2 x^{2} = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}x \left (a +y\right ) y^{\prime }+b y+c x = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}\left (B x y+A \,x^{2}+a x +b y+c \right ) y^{\prime }-B g \left (x \right )^{2}+A x y+\alpha x +\beta y+\gamma = 0
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}\left (x^{2} y-1\right ) y^{\prime }-x y^{2}+1 = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2} y-1\right ) y^{\prime }+8 x y^{2}-8 = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}x \left (y x +x^{4}-1\right ) y^{\prime }-y \left (y x -x^{4}-1\right ) = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}\left (x^{n \left (n +1\right )} y-1\right ) y^{\prime }+2 \left (n +1\right )^{2} x^{n -1} \left (x^{n^{2}} y^{2}-1\right ) = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}\left (g_{1} \left (x \right ) y+g_{0} \left (x \right )\right ) y^{\prime }-f_{1} \left (x \right ) y-f_{2} \left (x \right ) y^{2}-f_{3} \left (x \right ) y^{3}-f_{0} \left (x \right ) = 0
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✗ |
✗ |
|
|
\[
{}\left (x +2 y+y^{2}\right ) y^{\prime }+y \left (1+y\right )+\left (x +y\right )^{2} y^{2} = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}\left (\frac {y^{2}}{b}+\frac {x^{2}}{a}\right ) \left (x +y y^{\prime }\right )+\frac {\left (a -b \right ) \left (y y^{\prime }-x \right )}{a +b} = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}\left (2 a y^{3}+3 a x y^{2}-b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 b \,x^{2} y+2 b \,x^{3} = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}\left (x^{2} y^{3}+y x \right ) y^{\prime }-1 = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
|
|
\[
{}\left (x +2 y+2 x^{2} y^{3}+x y^{4}\right ) y^{\prime }+y^{5}+y = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right )^{2}-\sin \left (y\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } \cos \left (y\right )+x \sin \left (y\right ) \cos \left (y\right )^{2}-\sin \left (y\right )^{3} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}x y^{\prime } \ln \left (x \right ) \sin \left (y\right )+\cos \left (y\right ) \left (1-x \cos \left (y\right )\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}f \left (x^{2}+a y^{2}\right ) \left (a y y^{\prime }+x \right )-y-y^{\prime } x = 0
\] |
[_exact] |
✓ |
✓ |
|
|
\[
{}f \left (x^{c} y\right ) \left (b x y^{\prime }-a \right )-x^{a} y^{b} \left (y^{\prime } x +c y\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}a {y^{\prime }}^{2}+y y^{\prime }-x = 0
\] |
unknown |
✗ |
✓ |
|
|
\[
{}a {y^{\prime }}^{2}-y y^{\prime }-x = 0
\] |
unknown |
✗ |
✓ |
|
|
\[
{}\left (\operatorname {a2} x +\operatorname {c2} \right ) {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {a0} x +\operatorname {b0} y+\operatorname {c0} = 0
\] |
[_rational, _dAlembert] |
✓ |
✗ |
|
|
\[
{}y^{\prime } = \frac {1+2 F \left (\frac {4 x^{2} y+1}{4 x^{2}}\right ) x}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {1+F \left (\frac {y a x +1}{a x}\right ) a \,x^{2}}{a \,x^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {\left (a \,x^{2}-2 F \left (y+\frac {a \,x^{4}}{8}\right )\right ) x}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {y}{\sqrt {x^{2}+1}}\right ) x}{\sqrt {x^{2}+1}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x^{{3}/{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x +F \left (-\left (x -y\right ) \left (x +y\right )\right )}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x}{-y+F \left (x^{2}+y^{2}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {6 x^{3}+5 \sqrt {x}+5 F \left (y-\frac {2 x^{3}}{5}-2 \sqrt {x}\right )}{5 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {F \left (y^{{3}/{2}}-\frac {3 \,{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{x}}{\sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {F \left (-\frac {-y^{2}+b}{x^{2}}\right ) x}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {1+x y^{2}}{x}\right )}{y x^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-x +F \left (x^{2}+y^{2}\right )}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {F \left (-\left (x -y\right ) \left (x +y\right )\right ) x}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y^{2} \left (2+F \left (\frac {x^{2}-y}{y x^{2}}\right ) x^{2}\right )}{x^{3}}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 F \left (y+\ln \left (2 x +1\right )\right ) x +F \left (y+\ln \left (2 x +1\right )\right )-2}{2 x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 y^{3}}{1+2 F \left (\frac {1+4 x y^{2}}{y^{2}}\right ) y}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\left (-{\mathrm e}^{-x^{2}}+x^{2} {\mathrm e}^{-x^{2}}-F \left (y-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}\right )\right ) x
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 y+F \left (\frac {y}{x^{2}}\right ) x^{3}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-3 x^{2} y+F \left (x^{3} y\right )}{x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-2 x -y+F \left (x \left (x +y\right )\right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x +y+F \left (-\frac {-y+x \ln \left (x \right )}{x}\right ) x^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x \left (a -1\right ) \left (a +1\right )}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y}{x \left (-1+F \left (y x \right ) y\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {\left (3+y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (1+y\right ) \left (\left (y-\ln \left (1+y\right )-\ln \left (x \right )\right ) x +1\right )}{y x}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {6 y}{8 y^{4}+9 y^{3}+12 y^{2}+6 y-F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {i x^{2} \left (i-2 \sqrt {-x^{3}+6 y}\right )}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x}{y+\sqrt {x^{2}+1}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {1+2 x^{5} \sqrt {4 x^{2} y+1}}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {{\mathrm e}^{-x^{2}} x}{y \,{\mathrm e}^{x^{2}}+1}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right ) y
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right )^{2} y
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y}{\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )+1}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{4}}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-y^{2}+4 a x \right )^{2}}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {x^{2} \left (a x -2 \sqrt {a \left (a \,x^{4}+8 y\right )}\right )}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (a y^{2}+b \,x^{2}\right )^{2} x}{a^{{5}/{2}} y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 a +x \sqrt {-y^{2}+4 a x}}{y}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 a +x^{2} \sqrt {-y^{2}+4 a x}}{y}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {\left (\sqrt {a}\, x^{4}+\sqrt {a}\, x^{3}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-2 y^{{3}/{2}}+3 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{x}}{4 \sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {i x \left (i-2 \sqrt {-x^{2}+4 \ln \left (a \right )+4 \ln \left (y\right )}\right ) y}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (1+x y^{2}\right )^{2}}{y x^{4}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{2} \left (3 x +\sqrt {-9 x^{4}+4 y^{3}}\right )}{y^{2}}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x +1+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3} \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{2} \left (x +1+2 x \sqrt {x^{3}-6 y}\right )}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+\sqrt {x^{2}+y^{2}}\, x^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y^{3} x \,{\mathrm e}^{2 x^{2}}}{y \,{\mathrm e}^{x^{2}}+1}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-x^{2}+1+4 x^{3} \sqrt {x^{2}-2 x +1+8 y}}{4 x +4}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+x^{3} \sqrt {x^{2}+y^{2}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x +1+2 \sqrt {4 x^{2} y+1}\, x^{3}}{2 x^{3} \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x \left (-2 x -2+3 x^{2} \sqrt {x^{2}+3 y}\right )}{3 x +3}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {1}{x \left (x y^{2}+1+x \right ) y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 x \,{\mathrm e}^{x}-2 x -\ln \left (x \right )-1+x^{4} \ln \left (x \right )+x^{4}-2 y x^{2} \ln \left (x \right )-2 x^{2} y+y^{2} \ln \left (x \right )+y^{2}}{{\mathrm e}^{x}-1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {\left (-\ln \left (-1+y\right )+\ln \left (1+y\right )+2 \ln \left (x \right )\right ) x \left (1+y\right )^{2}}{8}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-\ln \left (-1+y\right )+\ln \left (1+y\right )+2 \ln \left (x \right )\right )^{2} x \left (1+y\right )^{2}}{16}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-y^{2}+4 a x \right )^{3}}{\left (-y^{2}+4 a x -1\right ) y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 a x +2 a +x^{3} \sqrt {-y^{2}+4 a x}}{\left (x +1\right ) y}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-1\right ) y}{x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{2}+2 x +1+2 x^{3} \sqrt {x^{2}+2 x +1-4 y}}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-x^{2}+x +2+2 x^{3} \sqrt {x^{2}-4 x +4 y}}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {3 x^{4}+3 x^{3}+\sqrt {9 x^{4}-4 y^{3}}}{\left (x +1\right ) y^{2}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{3} \left (3 x +3+\sqrt {9 x^{4}-4 y^{3}}\right )}{\left (x +1\right ) y^{2}}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (2 x +2+y\right ) y}{\left (\ln \left (y\right )+2 x -1\right ) \left (x +1\right )}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (2 y^{{3}/{2}}-3 \,{\mathrm e}^{x}\right )^{3} {\mathrm e}^{x}}{4 \left (2 y^{{3}/{2}}-3 \,{\mathrm e}^{x}+2\right ) \sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-x^{2}-x -a x -a +2 x^{3} \sqrt {x^{2}+2 a x +a^{2}+4 y}}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 x \sin \left (x \right )-\ln \left (2 x \right )+\ln \left (2 x \right ) x^{4}-2 \ln \left (2 x \right ) x^{2} y+\ln \left (2 x \right ) y^{2}}{\sin \left (x \right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-\ln \left (y\right ) x -\ln \left (y\right )+x^{3}\right ) y}{x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x \left (-1+x -2 y x +2 x^{3}\right )}{x^{2}-y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x +y^{4}-2 x^{2} y^{2}+x^{4}}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (a y^{2}+b \,x^{2}\right )^{3} x}{a^{{5}/{2}} \left (a y^{2}+b \,x^{2}+a \right ) y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {\cos \left (y\right ) \left (x -\cos \left (y\right )+1\right )}{\left (x \sin \left (y\right )-1\right ) \left (x +1\right )}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {i \left (8 i x +16 y^{4}+8 x^{2} y^{2}+x^{4}\right )}{32 y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x}{-y+x^{4}+2 x^{2} y^{2}+y^{4}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {i \left (i x +x^{4}+2 x^{2} y^{2}+y^{4}\right )}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x -y\right )^{2} \left (x +y\right )^{2} x}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\cos \left (y\right ) \left (\cos \left (y\right ) x^{3}-x -1\right )}{\left (x \sin \left (y\right )-1\right ) \left (x +1\right )}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x +1+x^{4} \ln \left (y\right )\right ) y \ln \left (y\right )}{x \left (x +1\right )}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 x^{3} y+x^{6}+x^{2} y^{2}+y^{3}}{x^{4}}
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (2 x +2+x^{3} y\right ) y}{\left (\ln \left (y\right )+2 x -1\right ) \left (x +1\right )}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {i \left (54 i x^{2}+81 y^{4}+18 x^{4} y^{2}+x^{8}\right ) x}{243 y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (1+x y^{2}\right )^{3}}{x^{4} \left (x y^{2}+1+x \right ) y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-x \right ) y}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-\ln \left (y\right ) x -\ln \left (y\right )+x^{4}\right ) y}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {i \left (16 i x^{2}+16 y^{4}+8 x^{4} y^{2}+x^{8}\right ) x}{32 y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 y^{6}}{y^{3}+2+16 x y^{2}+32 x^{2} y^{4}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x +1+\ln \left (y\right ) x \right ) \ln \left (y\right ) y}{x \left (x +1\right )}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-3 x^{2} y+1+y^{2} x^{6}+y^{3} x^{9}}{x^{3}}
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y x +y+x \sqrt {x^{2}+y^{2}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x \left (-x -1+x^{2}-2 x^{2} y+2 x^{4}\right )}{\left (x^{2}-y\right ) \left (x +1\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 x \ln \left (\frac {1}{x -1}\right )-\coth \left (\frac {x +1}{x -1}\right )+\coth \left (\frac {x +1}{x -1}\right ) y^{2}-2 \coth \left (\frac {x +1}{x -1}\right ) x^{2} y+\coth \left (\frac {x +1}{x -1}\right ) x^{4}}{\ln \left (\frac {1}{x -1}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 x^{2} \cosh \left (\frac {1}{x -1}\right )-2 x \cosh \left (\frac {1}{x -1}\right )-1+y^{2}-2 x^{2} y+x^{4}-x +x y^{2}-2 x^{3} y+x^{5}}{\left (x -1\right ) \cosh \left (\frac {1}{x -1}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y}{x \left (-1+y+x^{2} y^{3}+y^{4} x^{3}\right )}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y^{3} x \,{\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+3 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+9 y}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x +y+1\right ) y}{\left (2 y^{3}+y+x \right ) \left (x +1\right )}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {-\frac {1}{x}-\textit {\_F1} \left (y+\frac {1}{x}\right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\textit {\_F1} \left (y^{2}-2 \ln \left (x \right )\right )}{\sqrt {y^{2}}\, x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-x \sin \left (2 y\right )-\sin \left (2 y\right )+\cos \left (2 y\right ) x^{4}+x^{4}}{2 x \left (x +1\right )}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y x +y+x^{4} \sqrt {x^{2}+y^{2}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-x \sin \left (2 y\right )-\sin \left (2 y\right )+x \cos \left (2 y\right )+x}{2 x \left (x +1\right )}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {1}{-x -\textit {\_F1} \left (y-\ln \left (x \right )\right ) y \,{\mathrm e}^{y}}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{3} {\mathrm e}^{y}+x^{4}+{\mathrm e}^{y} y-{\mathrm e}^{y} \ln \left ({\mathrm e}^{y}+x \right )+y x -\ln \left ({\mathrm e}^{y}+x \right ) x +x}{x^{2}}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{2}}{2}+\sqrt {x^{3}-6 y}+x^{2} \sqrt {x^{3}-6 y}+x^{3} \sqrt {x^{3}-6 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-\sqrt {a}\, x^{3}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (-3 x^{3} y-3+y^{2} x^{7}\right )}{x \left (x^{3} y+1\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (3+y\right )^{3} {\mathrm e}^{\frac {9 x^{2}}{2}} x \,{\mathrm e}^{\frac {3 x^{2}}{2}} {\mathrm e}^{-3 x^{2}}}{243 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+243 y}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x -y\right )^{3} \left (x +y\right )^{3} x}{\left (-y^{2}+x^{2}-1\right ) y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-2 \cos \left (y\right )+x^{3} \cos \left (2 y\right ) \ln \left (x \right )+x^{3} \ln \left (x \right )}{2 \sin \left (y\right ) \ln \left (x \right ) x}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {2 x}{3}+\sqrt {x^{2}+3 y}+x^{2} \sqrt {x^{2}+3 y}+x^{3} \sqrt {x^{2}+3 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-2 \cos \left (y\right )+x^{2} \cos \left (2 y\right ) \ln \left (x \right )+x^{2} \ln \left (x \right )}{2 \sin \left (y\right ) \ln \left (x \right ) x}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (\left (x^{2}+1\right )^{{3}/{2}} x^{2}+\left (x^{2}+1\right )^{{3}/{2}}+y^{2} \left (x^{2}+1\right )^{{3}/{2}}+x^{2} y^{3}+y^{3}\right ) x}{\left (x^{2}+1\right )^{3}}
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (3 x y^{2}+x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {-y+x^{3} \sqrt {x^{2}+y^{2}}-x^{2} \sqrt {x^{2}+y^{2}}\, y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{3}+2 x^{5} \sqrt {4 x^{2} y+1}+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 a +\sqrt {-y^{2}+4 a x}+x^{2} \sqrt {-y^{2}+4 a x}+x^{3} \sqrt {-y^{2}+4 a x}}{y}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x +y+1\right ) y}{\left (y^{4}+y^{3}+y^{2}+x \right ) \left (x +1\right )}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {-y+x^{4} \sqrt {x^{2}+y^{2}}-x^{3} \sqrt {x^{2}+y^{2}}\, y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (x^{4}+3 x y^{2}+3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {1}{-\left (y^{3}\right )^{{2}/{3}} x -\textit {\_F1} \left (y^{3}-3 \ln \left (x \right )\right ) \left (y^{3}\right )^{{1}/{3}} x}
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \frac {y \left (x -y\right ) \left (1+y\right )}{x \left (y x +x -y\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {1}{-\ln \left (x \right ) \left (y^{3}\right )^{{2}/{3}}-\textit {\_F1} \left (y^{3}+3 \,\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )\right ) \ln \left (x \right ) \left (y^{3}\right )^{{1}/{3}}}
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{{3}/{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{{3}/{2}} b \,x^{2}+a^{{5}/{2}} y^{4}}{a \,x^{2} y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (x +y\right ) \left (1+y\right )}{x \left (y x +x +y\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {3 x^{3}+\sqrt {-9 x^{4}+4 y^{3}}+x^{2} \sqrt {-9 x^{4}+4 y^{3}}+x^{3} \sqrt {-9 x^{4}+4 y^{3}}}{y^{2}}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {1}{-x +\left (\frac {1}{y}+1\right ) x +\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2}-\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2} \left (\frac {1}{y}+1\right )}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x}{2}+\frac {1}{2}+\sqrt {x^{2}+2 x +1-4 y}+x^{2} \sqrt {x^{2}+2 x +1-4 y}+x^{3} \sqrt {x^{2}+2 x +1-4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\cosh \left (x \right )}{\sinh \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sinh \left (x \right )\right )\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {x}{2}+1+\sqrt {x^{2}-4 x +4 y}+x^{2} \sqrt {x^{2}-4 x +4 y}+x^{3} \sqrt {x^{2}-4 x +4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {1}{\sin \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sin \left (x \right )\right )+\ln \left (\cos \left (x \right )+1\right )\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+x^{2} \ln \left (x \right )^{2}+2 x^{2} \ln \left (y\right ) \ln \left (x \right )+x^{2} \ln \left (y\right )^{2}\right )}{x}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (\ln \left (y\right )-1+\ln \left (x \right )+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (y\right ) \ln \left (x \right )+x^{3} \ln \left (y\right )^{2}\right )}{x}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {\left (-\frac {1}{x}-\textit {\_F1} \left (y^{2}-2 x \right )\right ) x}{\sqrt {y^{2}}}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+\sqrt {x^{2}-2 x +1+8 y}+x^{2} \sqrt {x^{2}-2 x +1+8 y}+x^{3} \sqrt {x^{2}-2 x +1+8 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {-x -\textit {\_F1} \left (y^{2}-2 x \right )}{\sqrt {y^{2}}\, x}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {\left (-\frac {y \,{\mathrm e}^{\frac {1}{x}}}{x}-\textit {\_F1} \left (y \,{\mathrm e}^{\frac {1}{x}}\right )\right ) {\mathrm e}^{-\frac {1}{x}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y+x \sqrt {x^{2}+y^{2}}+x^{3} \sqrt {x^{2}+y^{2}}+x^{4} \sqrt {x^{2}+y^{2}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \left (\frac {\ln \left (-1+y\right ) y}{\left (1-y\right ) \ln \left (x \right ) x}-\frac {\ln \left (-1+y\right )}{\left (1-y\right ) \ln \left (x \right ) x}-f \left (x \right )\right ) \left (1-y\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+\sqrt {x^{2}+2 a x +a^{2}+4 y}+x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y}+x^{3} \sqrt {x^{2}+2 a x +a^{2}+4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-x +1-2 y+3 x^{2}-2 x^{2} y+2 x^{4}+x^{3}-2 x^{3} y+2 x^{5}}{x^{2}-y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x +x^{3}+x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {-y x -y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {1+y^{4}-8 a x y^{2}+16 a^{2} x^{2}+y^{6}-12 y^{4} a x +48 y^{2} a^{2} x^{2}-64 a^{3} x^{3}}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {-y x -y+\sqrt {x^{2}+y^{2}}\, x^{2}-x \sqrt {x^{2}+y^{2}}\, y}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (a^{3}+y^{4} a^{3}+2 y^{2} a^{2} b \,x^{2}+a \,x^{4} b^{2}+y^{6} a^{3}+3 y^{4} a^{2} b \,x^{2}+3 y^{2} a \,b^{2} x^{4}+b^{3} x^{6}\right ) x}{a^{{7}/{2}} y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {\left (-1-y^{4}+2 x^{2} y^{2}-x^{4}-y^{6}+3 x^{2} y^{4}-3 x^{4} y^{2}+x^{6}\right ) x}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {i \left (32 i x +64+64 y^{4}+32 x^{2} y^{2}+4 x^{4}+64 y^{6}+48 x^{2} y^{4}+12 x^{4} y^{2}+x^{6}\right )}{128 y}
\] |
[_rational] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = -\frac {\left (-8-8 y^{3}+24 y^{{3}/{2}} {\mathrm e}^{x}-18 \,{\mathrm e}^{2 x}-8 y^{{9}/{2}}+36 y^{3} {\mathrm e}^{x}-54 y^{{3}/{2}} {\mathrm e}^{2 x}+27 \,{\mathrm e}^{3 x}\right ) {\mathrm e}^{x}}{8 \sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x}{-y+1+y^{4}+2 x^{2} y^{2}+x^{4}+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y^{2} \left (-2 y+2 x^{2}+2 x^{2} y+y x^{4}\right )}{x^{3} \left (x^{2}-y+x^{2} y\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {6 x +x^{3}+x^{3} y^{2}+4 x^{2} y+x^{3} y^{3}+6 x^{2} y^{2}+12 y x +8}{x^{3}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {i \left (i x +1+x^{4}+2 x^{2} y^{2}+y^{4}+x^{6}+3 x^{4} y^{2}+3 x^{2} y^{4}+y^{6}\right )}{y}
\] |
[_rational] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \frac {\left (-256 a \,x^{2} y-32 a^{2} x^{6}-256 a \,x^{2}+512 y^{3}+192 x^{4} a y^{2}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512 y+64 a \,x^{4}+512}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x +1+y^{4}-2 x^{2} y^{2}+x^{4}+y^{6}-3 x^{2} y^{4}+3 x^{4} y^{2}-x^{6}}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-108 x^{{3}/{2}} y+18 x^{{9}/{2}}-108 x^{{3}/{2}}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{-216 y+36 x^{3}-216}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {32 x^{5} y+8 x^{3}+32 x^{5}+64 x^{6} y^{3}+48 x^{4} y^{2}+12 x^{2} y+1}{16 x^{6} \left (4 x^{2} y+1+4 x^{2}\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-x y^{2}+x^{3}-x -y^{6}+3 x^{2} y^{4}-3 x^{4} y^{2}+x^{6}}{\left (-y^{2}+x^{2}-1\right ) y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {a^{2} x +a^{3} x^{3}+a^{3} x^{3} y^{2}+2 a^{2} x^{2} y+a x +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 y a x +1}{a^{3} x^{3}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x \left (1+x^{2}+y^{2}\right )}{-y^{3}-x^{2} y-y+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {4 x \left (a -1\right ) \left (a +1\right )}{4 y+a^{2} y^{4}-2 a^{4} y^{2} x^{2}+4 y^{2} a^{2} x^{2}+a^{6} x^{4}-3 a^{4} x^{4}+3 a^{2} x^{4}-y^{4}-2 x^{2} y^{2}-x^{4}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x^{3}+y^{4} x^{3}+2 x^{2} y^{2}+x +x^{3} y^{6}+3 x^{2} y^{4}+3 x y^{2}+1}{x^{5} y}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-2 x -y+1+x^{2} y^{2}+2 x^{3} y+x^{4}+x^{3} y^{3}+3 x^{4} y^{2}+3 x^{5} y+x^{6}}{x}
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\left (-\frac {\ln \left (y\right )}{x}+\frac {\cos \left (x \right ) \ln \left (y\right )}{\sin \left (x \right )}-\textit {\_F1} \left (x \right )\right ) y
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (\ln \left (y\right ) x +\ln \left (y\right )-x -1+x \ln \left (x \right )+\ln \left (x \right )+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x \left (x +1\right )}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (x \ln \left (x \right )+\ln \left (x \right )+\ln \left (y\right ) x +\ln \left (y\right )-x -1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}\right )}{x \left (x +1\right )}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 y^{8}}{y^{5}+2 y^{6}+2 y^{2}+16 x y^{4}+32 y^{6} x^{2}+2+24 x y^{2}+96 x^{2} y^{4}+128 x^{3} y^{6}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 y^{6} \left (1+4 x y^{2}+y^{2}\right )}{y^{3}+4 y^{5} x +y^{5}+2+24 x y^{2}+96 x^{2} y^{4}+128 x^{3} y^{6}}
\] |
[_rational] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = -\left (-\frac {\ln \left (y\right )}{x}+\frac {\ln \left (y\right )}{x \ln \left (x \right )}-\textit {\_F1} \left (x \right )\right ) y
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {\left (-\frac {\ln \left (y\right )^{2}}{2 x}-\textit {\_F1} \left (x \right )\right ) y}{\ln \left (y\right )}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x \right ) {\mathrm e}^{\frac {y}{x}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (27 y^{3}+27 \,{\mathrm e}^{3 x^{2}} y+18 \,{\mathrm e}^{3 x^{2}} y^{2}+3 y^{3} {\mathrm e}^{3 x^{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y+9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y^{2}+{\mathrm e}^{\frac {9 x^{2}}{2}} y^{3}\right ) {\mathrm e}^{3 x^{2}} x \,{\mathrm e}^{-\frac {9 x^{2}}{2}}}{243 y}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {-x^{2}-y x -x^{3}-x y^{2}+2 y x^{2} \ln \left (x \right )-x^{3} \ln \left (x \right )^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x^{2}}
\] |
[_Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-2 y-2 \ln \left (2 x +1\right )-2+2 x y^{3}+y^{3}+6 y^{2} \ln \left (2 x +1\right ) x +3 y^{2} \ln \left (2 x +1\right )+6 y \ln \left (2 x +1\right )^{2} x +3 y \ln \left (2 x +1\right )^{2}+2 \ln \left (2 x +1\right )^{3} x +\ln \left (2 x +1\right )^{3}}{\left (2 x +1\right ) \left (y+\ln \left (2 x +1\right )+1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \ln \left (x \right ) x +x^{2} \ln \left (x \right )-2 y x -x^{2}-y^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x \left (-y+x \ln \left (x \right )-x \right )}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {\left (-8 \,{\mathrm e}^{-x^{2}} y+4 x^{2} {\mathrm e}^{-2 x^{2}}-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}} y-4 x^{4} {\mathrm e}^{-2 x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{-8 y+4 x^{2} {\mathrm e}^{-x^{2}}-8}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {216 y}{-216 y^{4}-252 y^{3}-396 y^{2}-216 y+36 x^{2}-72 y x +60 y^{5}-36 x y^{3}-72 x y^{2}-24 x y^{4}+4 y^{8}+12 y^{7}+33 y^{6}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {-y+\sqrt {x^{2}+y^{2}}\, x^{2}-x \sqrt {x^{2}+y^{2}}\, y+x^{4} \sqrt {x^{2}+y^{2}}-x^{3} \sqrt {x^{2}+y^{2}}\, y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (y\right ) \ln \left (x \right )+x^{3} \ln \left (y\right )^{2}+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x}
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {-150 x^{3} y+60 x^{6}+350 x^{{7}/{2}}-150 x^{3}-125 \sqrt {x}\, y+250 x -125 \sqrt {x}-125 y^{3}+150 x^{3} y^{2}+750 y^{2} \sqrt {x}-60 y x^{6}-600 y x^{{7}/{2}}-1500 y x +8 x^{9}+120 x^{{13}/{2}}+600 x^{4}+1000 x^{{3}/{2}}}{25 \left (-5 y+2 x^{3}+10 \sqrt {x}-5\right ) x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (-1-x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2}-x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} \ln \left (x \right )+x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y+2 x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y \ln \left (x \right )+x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y \ln \left (x \right )^{2}\right )}{\left (1+\ln \left (x \right )\right ) x}
\] |
[_Bernoulli] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (-1-x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}}-x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} \ln \left (x \right )+x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y+2 x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y \ln \left (x \right )+x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y \ln \left (x \right )^{2}\right )}{\left (1+\ln \left (x \right )\right ) x}
\] |
[_Bernoulli] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {2 x +4 y \ln \left (2 x +1\right ) x +6 y^{2} \ln \left (2 x +1\right ) x +6 y \ln \left (2 x +1\right )^{2} x +2 \ln \left (2 x +1\right )^{3} x +2 x y^{3}+2 \ln \left (2 x +1\right )^{2} x +2 x y^{2}-1+3 y^{2} \ln \left (2 x +1\right )+3 y \ln \left (2 x +1\right )^{2}+y^{2}+y^{3}+2 y \ln \left (2 x +1\right )+\ln \left (2 x +1\right )^{2}+\ln \left (2 x +1\right )^{3}}{2 x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {4 x \left (a -1\right ) \left (a +1\right ) \left (-y^{2}+a^{2} x^{2}-x^{2}-2\right )}{-4 y^{3}+4 a^{2} x^{2} y-4 x^{2} y-8 y-a^{2} y^{6}+3 a^{4} y^{4} x^{2}-6 y^{4} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}-4 a^{2} x^{6}+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {8 x \left (a -1\right ) \left (a +1\right )}{8+x^{6}+2 x^{4}+2 y^{4}-8 y+4 a^{4} y^{2} x^{2}-8 y^{2} a^{2} x^{2}-a^{2} y^{6}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}+y^{6}+3 x^{2} y^{4}+3 a^{4} y^{4} x^{2}-6 y^{4} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}-2 a^{2} y^{4}-2 a^{6} x^{4}+6 a^{4} x^{4}-6 a^{2} x^{4}-4 a^{2} x^{6}+3 x^{4} y^{2}-8 a^{2}+4 x^{2} y^{2}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {1296 y}{216+216 x^{3}-1944 y^{4}-2376 y^{2}-1728 y^{3}-612 y^{5}-1296 y-432 y x +216 x^{2}-882 y^{6}+1152 x y^{4}-216 x^{2} y^{4}-570 y^{8}-324 x^{2} y^{3}+1080 x y^{3}+216 x y^{2}-648 x^{2} y+1080 y^{5} x -126 y^{10}-315 y^{9}-8 y^{12}-36 y^{11}+72 y^{8} x +216 y^{7} x +594 x y^{6}-846 y^{7}-648 x^{2} y^{2}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\frac {216 y \left (-2 y^{4}-3 y^{3}-6 y^{2}-6 y+6 x +6\right )}{216 x^{3}+2808 y^{4}-1296 y^{2}+1728 y^{3}+4428 y^{5}-1296 y-1296 y x +2484 y^{6}-432 x y^{4}-216 x^{2} y^{4}-18 y^{8}-324 x^{2} y^{3}-648 x y^{3}-1944 x y^{2}-648 x^{2} y+1080 y^{5} x -126 y^{10}-315 y^{9}-8 y^{12}-36 y^{11}+72 y^{8} x +216 y^{7} x +594 x y^{6}+594 y^{7}-648 x^{2} y^{2}}
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {x \left (-x^{2}+2 x^{2} y-2 x^{4}+1\right )}{y-x^{2}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (y^{2} x^{7}+y x^{4}+x -3\right )}{x}
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \frac {y \left (x^{2} y^{2}+y x \,{\mathrm e}^{x}+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x} \left (x -1\right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Abel] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (a^{2} x^{2 n}-1\right ) y = 0
\] |
[_Titchmarsh] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a \,x^{2 c}+b \,x^{c -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (4 a^{2} b^{2} x^{2} {\mathrm e}^{2 b \,x^{2}}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{x}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \cosh \left (x \right )^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \cos \left (2 x \right )+b \right ) y = 0
\] |
[_ellipsoidal] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \cos \left (x \right )^{2}+b \right ) y = 0
\] |
[_ellipsoidal] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (1+2 \tan \left (x \right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (\frac {m \left (m -1\right )}{\cos \left (x \right )^{2}}+\frac {n \left (n -1\right )}{\sin \left (x \right )^{2}}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (n \left (n +1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+B \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-\left (n \left (n +1\right ) k^{2} \operatorname {JacobiSN}\left (x , k\right )^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (\frac {p^{\prime \prime \prime \prime }\left (x \right )}{30}+\frac {7 p^{\prime \prime }\left (x \right )}{3}+a p \left (x \right )+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-\left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (P \left (x \right )+l \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+2 a y^{\prime }+f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x +\left (n +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x -n y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-y^{\prime } x -a y = 0
\] |
[_Hermite] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime } x +a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime } x +\left (3 x^{2}+2 n -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a x y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a \,x^{q -1} y^{\prime }+b \,x^{q -2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+2 n y^{\prime } \cot \left (x \right )+\left (-a^{2}+n^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\cot \left (x \right ) y^{\prime }+v \left (v +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime } \tan \left (x \right )+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a p^{\prime \prime }\left (x \right ) y^{\prime }+\left (a +b p \left (x \right )-4 n a p \left (x \right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\frac {\left (11 \operatorname {WeierstrassP}\left (x , a , b\right ) \operatorname {WeierstrassPPrime}\left (x , a , b\right )-6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}+\frac {a}{2}\right ) y^{\prime }}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}}+\frac {\left (\operatorname {WeierstrassPPrime}\left (x , a , b\right )^{2}-\operatorname {WeierstrassP}\left (x , a , b\right )^{2} \operatorname {WeierstrassPPrime}\left (x , a , b\right )-\operatorname {WeierstrassP}\left (x , a , b\right ) \left (6 \operatorname {WeierstrassP}\left (x , a , b\right )^{2}-\frac {a}{2}\right )\right ) y}{\operatorname {WeierstrassPPrime}\left (x , a , b\right )+\operatorname {WeierstrassP}\left (x , a , b\right )^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+f \left (x \right ) y^{\prime }+g \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right )+a \right ) y-g \left (x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a f \left (x \right )+b \right ) y^{\prime }+\left (c f \left (x \right )+d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+2 a \right ) y^{\prime }+\left (\frac {a f^{\prime }\left (x \right )}{f \left (x \right )}+a^{2}-b^{2} f \left (x \right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\frac {f \left (x \right ) f^{\prime \prime \prime }\left (x \right ) y^{\prime }}{f \left (x \right )^{2}+b^{2}}-\frac {a^{2} {f^{\prime }\left (x \right )}^{2} y}{f \left (x \right )^{2}+b^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-\left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right ) y^{\prime }+\left (\frac {f^{\prime }\left (x \right ) \left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right )}{f \left (x \right )}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {v^{2} {g^{\prime }\left (x \right )}^{2}}{g \left (x \right )^{2}}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right ) y^{\prime }+\left (\frac {h^{\prime }\left (x \right ) \left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right )}{h \left (x \right )}-\frac {h^{\prime \prime }\left (x \right )}{h \left (x \right )}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}4 y^{\prime \prime }-\left (x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}a y^{\prime \prime }-\left (a b +c +x \right ) y^{\prime }+\left (b \left (x +c \right )+d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }+x^{3} \left ({\mathrm e}^{x^{2}}-v^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (x +b \right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (x +a +b \right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }-y^{\prime } x -a y = 0
\] |
[_Laguerre] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (b -x \right ) y^{\prime }-a y = 0
\] |
[_Laguerre] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }-2 \left (x -1\right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }-\left (3 x -2\right ) y^{\prime }-\left (2 x -3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (a x +b +n \right ) y^{\prime }+n a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }-\left (a +b \right ) \left (x +1\right ) y^{\prime }+a b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (\left (a +b \right ) x +m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }-2 \left (x^{2}-a \right ) y^{\prime }+2 n x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 x y^{\prime \prime }-\left (x -1\right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+a y = 0
\] |
[_Laguerre] |
✓ |
✓ |
|
|
\[
{}4 x y^{\prime \prime }-\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}4 x y^{\prime \prime }+4 y-\left (x +2\right ) y+l y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}4 x y^{\prime \prime }+4 m y^{\prime }-\left (x -2 m -4 n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}16 x y^{\prime \prime }+8 y^{\prime }-\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}5 \left (a x +b \right ) y^{\prime \prime }+8 a y^{\prime }+c \left (a x +b \right )^{{1}/{5}} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 a x y^{\prime \prime }+\left (b x +a \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (\operatorname {a2} x +\operatorname {b2} \right ) y^{\prime \prime }+\left (\operatorname {a1} x +\operatorname {b1} \right ) y^{\prime }+\left (\operatorname {a0} x +\operatorname {b0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}-x \,{\mathrm e}^{x} \left (2+x \ln \left (x \right )\right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+a y^{\prime }-y x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+2 y^{\prime } x +\left (l \,x^{2}+a x -n \left (n +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+2 \left (x +a \right ) y^{\prime }-b \left (b -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x +3\right ) x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (x^{2}-2 x \right ) y^{\prime }-\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-v \left (v -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (2 a x +b \right ) x y^{\prime }+\left (a b x +c \,x^{2}+d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) x y^{\prime }+f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{3}+1\right ) x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (-x^{4}+\left (2 n +2 a +1\right ) x^{2}+\left (-1\right )^{n} a -a^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2 n}+\operatorname {b1} \,x^{n}+\operatorname {c1} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{\operatorname {a1}}+b \right ) x y^{\prime }+\left (A \,x^{2 \operatorname {a1}}+B \,x^{\operatorname {a1}}+C \,x^{\operatorname {b1}}+\operatorname {DD} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (2 x^{2} \tan \left (x \right )-x \right ) y^{\prime }-\left (x \tan \left (x \right )+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (2 x^{2} \cot \left (x \right )+x \right ) y^{\prime }+\left (x \cot \left (x \right )+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right ) x +f \left (x \right )^{2}-f \left (x \right )+a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x^{2} f \left (x \right ) y^{\prime }+\left (x^{2} \left (f^{\prime }\left (x \right )+f \left (x \right )^{2}+a \right )-v \left (v -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x -2 x^{2} f \left (x \right )\right ) y^{\prime }+\left (x^{2} \left (1+f \left (x \right )^{2}-f^{\prime }\left (x \right )\right )-f \left (x \right ) x -v^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x -v \left (v -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-v \left (v +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-n \left (n +1\right ) y+\frac {d}{d x}\operatorname {LegendreP}\left (n , x\right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime } x +f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 y^{\prime } x -l y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 y^{\prime } x -v \left (v +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-2 y^{\prime } x -\left (v +2\right ) \left (v -1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (v +n +1\right ) \left (v -n \right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (v -n +1\right ) \left (v +n \right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x \left (x +1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }-v \left (v +1\right ) y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\[
{}x \left (x -1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\[
{}x \left (x -1\right ) y^{\prime \prime }+\left (\left (a +1\right ) x +b \right ) y^{\prime }-l y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\[
{}x \left (x +2\right ) y^{\prime \prime }+2 \left (n +1+\left (n +1-2 l \right ) x -l \,x^{2}\right ) y^{\prime }+\left (2 l \left (p -n -1\right ) x +2 p l +m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x -1\right ) \left (x -2\right ) y^{\prime \prime }-\left (2 x -3\right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\[
{}2 x \left (x -1\right ) y^{\prime \prime }+\left (\left (2 v +5\right ) x -2 v -3\right ) y^{\prime }+\left (v +1\right ) y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\[
{}4 x^{2} y^{\prime \prime }-\left (-4 k x +4 m^{2}+x^{2}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 y^{\prime } x +\left (-x^{2}+2 \left (1-m +2 l \right ) x -m^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 y^{\prime } x +f \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x \left (4 x -1\right ) y^{\prime \prime }+\left (\left (4 a +2\right ) x -a \right ) y^{\prime }+a \left (a -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}48 x \left (x -1\right ) y^{\prime \prime }+\left (152 x -40\right ) y^{\prime }+53 y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\[
{}144 x \left (x -1\right ) y^{\prime \prime }+\left (120 x -48\right ) y^{\prime }+y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\[
{}144 x \left (x -1\right ) y^{\prime \prime }+\left (168 x -96\right ) y^{\prime }+y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\[
{}a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\operatorname {a2} \,x^{2} y^{\prime \prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x \right ) y^{\prime }+\left (\operatorname {a0} \,x^{2}+\operatorname {b0} x +\operatorname {c0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\operatorname {A2} \left (a x +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (a x +b \right ) y^{\prime }+\operatorname {A0} \left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +f \right ) y^{\prime }+g y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+2 y^{\prime } x -y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+x^{2} y^{\prime }+\left (a \,x^{2}+b x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+y x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 x^{2}+1\right ) y^{\prime }-v \left (v +1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 \left (n +1\right ) x^{2}+2 n +1\right ) y^{\prime }-\left (v -n \right ) \left (v +n +1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }-\left (2 \left (n -1\right ) x^{2}+2 n -1\right ) y^{\prime }+\left (v +n \right ) \left (-v +n -1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-y x = 0
\] |
[[_elliptic, _class_II]] |
✓ |
✓ |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (3 x^{2}-1\right ) y^{\prime }+y x = 0
\] |
[[_elliptic, _class_I]] |
✓ |
✓ |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (a +b +1\right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (x -1\right )}-\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (x -1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = \frac {2 y^{\prime }}{x \left (x -2\right )}-\frac {y}{x^{2} \left (x -2\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (x -1\right ) \left (x -a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{x \left (x -1\right ) \left (x -a \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}-\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {v \left (v +1\right ) y}{4 x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (a +1\right ) x -1\right ) y^{\prime }}{x \left (x -1\right )}-\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (x -1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (a x +b \right ) y}{4 x \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \left (b +2\right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (a x +1\right ) x^{2}}-\frac {\left (a b x -c d \right ) y}{\left (a x +1\right ) x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}+A x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {v \left (v +1\right ) y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 a \,x^{2}+n \left (n +1\right ) \left (x^{2}-1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {b y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (a -1\right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (a -1\right )-v \left (v +1\right )\right )-a \left (a +1\right )\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (n +1\right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {a x y^{\prime }}{x^{2}+1}-\frac {b y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (\left (x^{2}-1\right ) \left (a \,x^{2}+b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = \frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}-\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (v \left (v +1\right ) \left (x -1\right )-a^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (-v \left (v +1\right ) \left (x -1\right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (b \,x^{2}+c x +d \right ) y}{a \,x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (a +1\right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime } = -\left (\frac {1-\operatorname {a1} -\operatorname {b1}}{x -\operatorname {c1}}+\frac {1-\operatorname {a2} -\operatorname {b2}}{x -\operatorname {c2}}+\frac {1-\operatorname {a3} -\operatorname {b3}}{x -\operatorname {c3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {a1} \operatorname {b1} \left (\operatorname {c1} -\operatorname {c3} \right ) \left (\operatorname {c1} -\operatorname {c2} \right )}{x -\operatorname {c1}}+\frac {\operatorname {a2} \operatorname {b2} \left (\operatorname {c2} -\operatorname {c1} \right ) \left (\operatorname {c2} -\operatorname {c3} \right )}{x -\operatorname {c2}}+\frac {\operatorname {a3} \operatorname {b3} \left (\operatorname {c3} -\operatorname {c2} \right ) \left (\operatorname {c3} -\operatorname {c1} \right )}{x -\operatorname {c3}}\right ) y}{\left (x -\operatorname {c1} \right ) \left (x -\operatorname {c2} \right ) \left (x -\operatorname {c3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2} \left (\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right )+\left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )+\left (x^{2}-\operatorname {a3} \right ) \left (x^{2}-\operatorname {a1} \right )\right )-\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}-\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}-\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = \frac {y}{{\mathrm e}^{x}+1}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}-\left (-a^{2}+n^{2}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 n +1\right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\left (v +n +1\right ) \left (v -n \right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = \frac {2 y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\sin \left (x \right )^{2} y^{\prime \prime }-\left (a \sin \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\sin \left (x \right )^{2} y^{\prime \prime }-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-\left (a^{2} b^{2}-\left (a +1\right )^{2}\right ) \sin \left (x \right )^{2}-a \left (a +1\right ) b \sin \left (2 x \right )-a \left (a -1\right )\right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (v \left (v +1\right ) \sin \left (x \right )^{2}-n^{2}\right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = \frac {\cos \left (2 x \right ) y^{\prime }}{\sin \left (2 x \right )}-2 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\sin \left (x \right ) y^{\prime }}{\cos \left (x \right )}-\frac {\left (2 x^{2}+x^{2} \sin \left (x \right )^{2}-24 \cos \left (x \right )^{2}\right ) y}{4 x^{2} \cos \left (x \right )^{2}}+\sqrt {\cos \left (x \right )}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}-\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime } = -\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-a \cos \left (x \right )^{2} \sin \left (x \right )^{2}-m \left (m -1\right ) \sin \left (x \right )^{2}-n \left (n -1\right ) \cos \left (x \right )^{2}\right ) y}{\cos \left (x \right )^{2} \sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } = \frac {\phi ^{\prime }\left (x \right ) y^{\prime }}{\phi \left (x \right )-\phi \left (a \right )}-\frac {\left (-n \left (n +1\right ) \left (\phi \left (x \right )-\phi \left (a \right )\right )^{2}+D^{\left (2\right )}\left (\phi \right )\left (a \right )\right ) y}{\phi \left (x \right )-\phi \left (a \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\phi \left (x^{3}\right )-\phi \left (x \right ) \phi ^{\prime }\left (x \right )-\phi ^{\prime \prime }\left (x \right )\right ) y^{\prime }}{\phi ^{\prime }\left (x \right )+\phi \left (x \right )^{2}}-\frac {\left ({\phi ^{\prime }\left (x \right )}^{2}-\phi \left (x \right )^{2} \phi ^{\prime }\left (x \right )-\phi \left (x \right ) \phi ^{\prime \prime }\left (x \right )\right ) y}{\phi ^{\prime }\left (x \right )+\phi \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime } = \frac {2 \,\operatorname {JacobiSN}\left (x , k\right ) \operatorname {JacobiCN}\left (x , k\right ) \operatorname {JacobiDN}\left (x , k\right ) y^{\prime }-2 \left (1-2 \left (k^{2}+1\right ) \operatorname {JacobiSN}\left (a , k\right )^{2}+3 k^{2} \operatorname {JacobiSN}\left (a , k\right )^{4}\right ) y}{\operatorname {JacobiSN}\left (x , k\right )^{2}-\operatorname {JacobiSN}\left (a , k\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime } = -\frac {f^{\prime }\left (x \right ) y^{\prime }}{2 f \left (x \right )}-\frac {g \left (x \right ) y}{f \left (x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 f \left (x \right ) {g^{\prime }\left (x \right )}^{2} g \left (x \right )-\left (g \left (x \right )^{2}-1\right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )\right ) y^{\prime }}{f \left (x \right ) g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}-\frac {\left (\left (g \left (x \right )^{2}-1\right ) \left (f^{\prime }\left (x \right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )-f \left (x \right ) f^{\prime \prime }\left (x \right ) g^{\prime }\left (x \right )\right )-\left (2 f^{\prime }\left (x \right ) g \left (x \right )+v \left (v +1\right ) f \left (x \right ) g^{\prime }\left (x \right )\right ) f \left (x \right ) {g^{\prime }\left (x \right )}^{2}\right ) y}{f \left (x \right )^{2} g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }+y a \,x^{3}-b x = 0
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }-a \,x^{b} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }+2 a x y^{\prime }+a y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+\left (a +b -1\right ) x y^{\prime }-b y a = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }+x^{2 c -2} y^{\prime }+\left (c -1\right ) x^{2 c -3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }-3 \left (2 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }+b y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+\left (-n^{2}+1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }+\frac {\left (\left (-n^{2}+1\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right )-a \right ) y}{2} = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }-\left (4 n \left (n +1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }-2 n \left (n +1\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+\left (A \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }+B \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }-\left (3 k^{2} \operatorname {JacobiSN}\left (z , x\right )^{2}+a \right ) y^{\prime }+\left (b +c \operatorname {JacobiSN}\left (z , x\right )^{2}-3 k^{2} \operatorname {JacobiSN}\left (z , x\right ) \operatorname {JacobiCN}\left (z , x\right ) \operatorname {JacobiDN}\left (z , x\right )\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }-\left (6 k^{2} \sin \left (x \right )^{2}+a \right ) y^{\prime }+b y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+2 f \left (x \right ) y^{\prime }+f^{\prime }\left (x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }-6 x y^{\prime \prime }+2 \left (4 x^{2}+2 a -1\right ) y^{\prime }-8 y a x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }+3 a x y^{\prime \prime }+3 a^{2} x^{2} y^{\prime }+a^{3} x^{3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }-y^{\prime \prime } \sin \left (x \right )-2 y^{\prime } \cos \left (x \right )+y \sin \left (x \right )-\ln \left (x \right ) = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime }+y^{\prime }+f \left (x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }+f \left (x \right ) \left (x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y\right ) = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime }+g \left (x \right ) y^{\prime }+\left (f \left (x \right ) g \left (x \right )+g^{\prime }\left (x \right )\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+3 f \left (x \right ) y^{\prime \prime }+\left (f^{\prime }\left (x \right )+2 f \left (x \right )^{2}+4 g \left (x \right )\right ) y^{\prime }+\left (4 f \left (x \right ) g \left (x \right )+2 g^{\prime }\left (x \right )\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}27 y^{\prime \prime \prime }-36 n^{2} \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }-2 n \left (n +3\right ) \left (4 n -3\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime \prime }+3 y^{\prime \prime }+y x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime \prime }+3 y^{\prime \prime }-a \,x^{2} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime \prime }+\left (a +b \right ) y^{\prime \prime }-y^{\prime } x -a y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime \prime }-\left (x +2 v \right ) y^{\prime \prime }-\left (x -2 v -1\right ) y^{\prime }+\left (x -1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime \prime }+\left (x^{2}-3\right ) y^{\prime \prime }+4 y^{\prime } x +2 y-f \left (x \right ) = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
✓ |
|
|
\[
{}2 x y^{\prime \prime \prime }+3 y^{\prime \prime }+y a x -b = 0
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}2 x y^{\prime \prime \prime }-4 \left (x +\nu -1\right ) y^{\prime \prime }+\left (2 x +6 \nu -5\right ) y^{\prime }+\left (1-2 \nu \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 x y^{\prime \prime \prime }+3 \left (2 a x +k \right ) y^{\prime \prime }+6 \left (a k +b x \right ) y^{\prime }+\left (3 b k +2 c x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (x -2\right ) x y^{\prime \prime \prime }-\left (x -2\right ) x y^{\prime \prime }-2 y^{\prime }+2 y = 0
\] |
[[_3rd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
|
|
\[
{}\left (2 x -1\right ) y^{\prime \prime \prime }-8 y^{\prime } x +8 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-6 y^{\prime }+a \,x^{2} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+\left (x +1\right ) y^{\prime \prime }-y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+3 x y^{\prime \prime }+\left (4 a^{2} x^{2 a}+1-4 \nu ^{2} a^{2}\right ) y^{\prime } = 4 a^{3} x^{2 a -1} y
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-3 \left (x -m \right ) x y^{\prime \prime }+\left (2 x^{2}+4 \left (n -m \right ) x +m \left (2 m -1\right )\right ) y^{\prime }-2 n \left (2 x -2 m +1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+4 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }+3 y x -f \left (x \right ) = 0
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+6 x y^{\prime \prime }+6 y^{\prime }+a \,x^{2} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-3 \left (p +q \right ) x y^{\prime \prime }+3 p \left (3 q +1\right ) y^{\prime }-x^{2} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-2 \left (n +1\right ) x y^{\prime \prime }+\left (a \,x^{2}+6 n \right ) y^{\prime }-2 y a x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-\left (x^{2}-2 x \right ) y^{\prime \prime }-\left (x^{2}+\nu ^{2}-\frac {1}{4}\right ) y^{\prime }+\left (x^{2}-2 x +\nu ^{2}-\frac {1}{4}\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-\left (x +\nu \right ) x y^{\prime \prime }+\nu \left (2 x +1\right ) y^{\prime }-\nu \left (x +1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-2 \left (x^{2}-x \right ) y^{\prime \prime }+\left (x^{2}-2 x +\frac {1}{4}-\nu ^{2}\right ) y^{\prime }+\left (\nu ^{2}-\frac {1}{4}\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-\left (x^{4}-6 x \right ) y^{\prime \prime }-\left (2 x^{3}-6\right ) y^{\prime }+2 x^{2} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+2\right ) y^{\prime \prime \prime }-2 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }-2 y x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 x \left (x -1\right ) y^{\prime \prime \prime }+3 \left (2 x -1\right ) y^{\prime \prime }+\left (2 a x +b \right ) y^{\prime }+a y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+\left (-\nu ^{2}+1\right ) x y^{\prime }+\left (a \,x^{3}+\nu ^{2}-1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+\left (4 x^{3}+\left (-4 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (4 \nu ^{2}-1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+\left (a \,x^{2 \nu }+1-\nu ^{2}\right ) x y^{\prime }+\left (b \,x^{3 \nu }+a \left (\nu -1\right ) x^{2 \nu }+\nu ^{2}-1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }-4 x^{2} y^{\prime \prime }+\left (x^{2}+8\right ) x y^{\prime }-2 \left (x^{2}+4\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+6 x^{2} y^{\prime \prime }+\left (a \,x^{3}-12\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+3 \left (1-a \right ) x^{2} y^{\prime \prime }+\left (4 b^{2} c^{2} x^{2 c +1}+1-4 \nu ^{2} c^{2}+3 a \left (a -1\right ) x \right ) y^{\prime }+\left (4 b^{2} c^{2} \left (c -a \right ) x^{2 c}+a \left (4 \nu ^{2} c^{2}-a^{2}\right )\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+\left (x +3\right ) x^{2} y^{\prime \prime }+5 \left (x -6\right ) x y^{\prime }+\left (4 x +30\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}\left (x^{2}+1\right ) x y^{\prime \prime \prime }+3 \left (2 x^{2}+1\right ) y^{\prime \prime }-12 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x +3\right ) x^{2} y^{\prime \prime \prime }-3 x \left (x +2\right ) y^{\prime \prime }+6 \left (x +1\right ) y^{\prime }-6 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 \left (x -\operatorname {a1} \right ) \left (x -\operatorname {a2} \right ) \left (x -\operatorname {a3} \right ) y^{\prime \prime \prime }+\left (9 x^{2}-6 \left (\operatorname {a1} +\operatorname {a2} +\operatorname {a3} \right ) x +3 \operatorname {a1} \operatorname {a2} +3 \operatorname {a1} \operatorname {a3} +3 \operatorname {a2} \operatorname {a3} \right ) y^{\prime \prime }-2 \left (\left (n^{2}+n -3\right ) x +b \right ) y^{\prime }-n \left (n +1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x +1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x +2\right ) x^{2} y^{\prime \prime }+\left (10 x +4\right ) x y^{\prime }-4 \left (3 x +1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-4 \left (3 x^{2}+1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{6} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{6} y^{\prime \prime \prime }+6 x^{5} y^{\prime \prime }+a y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (x^{4}+2 x^{2}+2 x +1\right ) y^{\prime \prime \prime }-\left (2 x^{6}+3 x^{4}-6 x^{2}-6 x -1\right ) y^{\prime \prime }+\left (x^{6}-6 x^{3}-15 x^{2}-12 x -2\right ) y^{\prime }+\left (x^{4}+4 x^{3}+8 x^{2}+6 x +1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x -a \right )^{3} \left (x -b \right )^{3} y^{\prime \prime \prime }-c y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (\sin \left (x \right )+x \right ) y^{\prime \prime \prime }+3 \left (\cos \left (x \right )+1\right ) y^{\prime \prime }-3 y^{\prime } \sin \left (x \right )-y \cos \left (x \right )+\sin \left (x \right ) = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime } \sin \left (x \right )^{2}+3 y^{\prime \prime } \sin \left (x \right ) \cos \left (x \right )+\left (\cos \left (2 x \right )+4 \nu \left (\nu +1\right ) \sin \left (x \right )^{2}\right ) y^{\prime }+2 \nu \left (\nu +1\right ) y \sin \left (2 x \right ) = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}f^{\prime }\left (x \right ) y^{\prime \prime }+f \left (x \right ) y^{\prime \prime \prime }+g^{\prime }\left (x \right ) y^{\prime }+g \left (x \right ) y^{\prime \prime }+h^{\prime }\left (x \right ) y+h \left (x \right ) y^{\prime }+A \left (x \right ) \left (f \left (x \right ) y^{\prime \prime }+g \left (x \right ) y^{\prime }+h \left (x \right ) y\right ) = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+y^{\prime } x +n y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }-y^{\prime } x -n y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime \prime }+a \left (b x -1\right ) y^{\prime \prime }+a b y^{\prime }+\lambda y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime \prime }+\left (a \,x^{2}+b \lambda +c \right ) y^{\prime \prime }+\left (a \,x^{2}+\beta \lambda +\gamma \right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime \prime }+a \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime \prime }+b \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }+\left (c \left (6 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )^{2}-\frac {\operatorname {g2}}{2}\right )+d \right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime \prime }-\left (12 k^{2} \operatorname {JacobiSN}\left (z , x\right )^{2}+a \right ) y^{\prime \prime }+b y^{\prime }+\left (\alpha \operatorname {JacobiSN}\left (z , x\right )^{2}+\beta \right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime \prime }+4 a x y^{\prime \prime \prime }+6 a^{2} x^{2} y^{\prime \prime }+4 a^{3} x^{3} y^{\prime }+a^{4} x^{4} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime \prime \prime }-\left (6 x^{2}+1\right ) y^{\prime \prime \prime }+12 x^{3} y^{\prime \prime }-\left (9 x^{2}-7\right ) x^{2} y^{\prime }+2 \left (x^{2}-3\right ) x^{3} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime }-2 \left (\nu ^{2} x^{2}+6\right ) y^{\prime \prime }+\nu ^{2} \left (\nu ^{2} x^{2}+4\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime }+2 x y^{\prime \prime \prime }+a y-b \,x^{2} = 0
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime }+6 x y^{\prime \prime \prime }+6 y^{\prime \prime }-\lambda ^{2} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime }+8 x y^{\prime \prime \prime }+12 y^{\prime \prime }-\lambda ^{2} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime }+\left (2 n -2 \nu +4\right ) x y^{\prime \prime \prime }+\left (n -\nu +1\right ) \left (n -\nu +2\right ) y^{\prime \prime }-\frac {b^{4} y}{16} = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime \prime \prime }+2 x^{2} y^{\prime \prime \prime }-x y^{\prime \prime }+y^{\prime }-a^{4} x^{3} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }-2 n \left (n +1\right ) x^{2} y^{\prime \prime }+4 n \left (n +1\right ) x y^{\prime }+\left (a \,x^{4}+n \left (n +1\right ) \left (n +3\right ) \left (n -2\right )\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }+\left (4 n^{2}-1\right ) x y^{\prime }-4 y x^{4} = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }-\left (4 n^{2}-1\right ) x y^{\prime }+\left (-4 x^{4}+4 n^{2}-1\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}+3\right ) x^{2} y^{\prime \prime }+\left (12 n^{2}-3\right ) x y^{\prime }-\left (4 x^{4}+12 n^{2}-3\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-\rho ^{2}-\sigma ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-\rho ^{2}-\sigma ^{2}+1\right ) x \right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-2 \mu ^{2}-2 \nu ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-2 \mu ^{2}-2 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (8 x^{2}+\left (\mu ^{2}-\nu ^{2}\right )^{2}\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a \right ) x^{3} y^{\prime \prime \prime }+\left (4 b^{2} c^{2} x^{2 c}+6 \left (a -1\right )^{2}-2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )+1\right ) x^{2} y^{\prime \prime }+\left (4 \left (3 c -2 a +1\right ) b^{2} c^{2} x^{2 c}+\left (2 a -1\right ) \left (2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )-2 a \left (a -1\right )-1\right )\right ) x y^{\prime }+\left (4 \left (a -c \right ) \left (a -2 c \right ) b^{2} c^{2} x^{2 c}+\left (c \mu +c \nu +a \right ) \left (c \mu +c \nu -a \right ) \left (c \mu -c \nu +a \right ) \left (c \mu -c \nu -a \right )\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a -4 c \right ) x^{3} y^{\prime \prime \prime }+\left (-2 \nu ^{2} c^{2}+2 a^{2}+4 \left (a +c -1\right )^{2}+4 \left (a -1\right ) \left (c -1\right )-1\right ) x^{2} y^{\prime \prime }+\left (2 \nu ^{2} c^{2}-2 a^{2}-\left (2 a -1\right ) \left (2 c -1\right )\right ) \left (2 a +2 c -1\right ) x y^{\prime }+\left (\left (-\nu ^{2} c^{2}+a^{2}\right ) \left (-\nu ^{2} c^{2}+a^{2}+4 a c +4 c^{2}\right )-b^{4} c^{4} x^{4 c}\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\nu ^{4} x^{4} y^{\prime \prime \prime \prime }+\left (4 \nu -2\right ) \nu ^{3} x^{3} y^{\prime \prime \prime }+\left (\nu -1\right ) \left (2 \nu -1\right ) \nu ^{2} x^{2} y^{\prime \prime }-\frac {b^{4} x^{\frac {2}{\nu }} y}{16} = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime \prime \prime }+10 x \left (x^{2}-1\right ) y^{\prime \prime \prime }+\left (24 x^{2}-8-2 \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )\right ) \left (x^{2}-1\right )\right ) y^{\prime \prime }-6 x \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )-2\right ) y^{\prime }+\left (\left (\mu \left (\mu +1\right )-\nu \left (\nu +1\right )\right )^{2}-2 \mu \left (\mu +1\right )-2 \nu \left (\nu +1\right )\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}\left ({\mathrm e}^{x}+2 x \right ) y^{\prime \prime \prime \prime }+4 \left ({\mathrm e}^{x}+2\right ) y^{\prime \prime \prime }+6 \,{\mathrm e}^{x} y^{\prime \prime }+4 \,{\mathrm e}^{x} y^{\prime }+{\mathrm e}^{x} y-\frac {1}{x^{5}} = 0
\] |
[[_high_order, _fully, _exact, _linear]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime \prime } \sin \left (x \right )^{4}+2 y^{\prime \prime \prime } \sin \left (x \right )^{3} \cos \left (x \right )+y^{\prime \prime } \sin \left (x \right )^{2} \left (\sin \left (x \right )^{2}-3\right )+y^{\prime } \sin \left (x \right ) \cos \left (x \right ) \left (2 \sin \left (x \right )^{2}+3\right )+\left (a^{4} \sin \left (x \right )^{4}-3\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime \prime } \sin \left (x \right )^{6}+4 y^{\prime \prime \prime } \sin \left (x \right )^{5} \cos \left (x \right )-6 y^{\prime \prime } \sin \left (x \right )^{6}-4 y^{\prime } \sin \left (x \right )^{5} \cos \left (x \right )+y \sin \left (x \right )^{6}-f = 0
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+a^{4} y-\lambda \left (a x -b \right ) \left (y^{\prime \prime }-a^{2} y\right ) = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\left (5\right )}-y a x -b = 0
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
|
|
\[
{}y^{\left (5\right )}+a \,x^{\nu } y^{\prime }+a \nu \,x^{\nu -1} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x y^{\left (5\right )}-m n y^{\prime \prime \prime \prime }+y a x = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\left (5\right )}-\left (a A_{1} -A_{0} \right ) x -A_{1} -\left (\left (a A_{2} -A_{1} \right ) x +A_{2} \right ) y^{\prime } = 0
\] |
[[_high_order, _missing_y]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime }-a y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{10} y^{\left (5\right )}-a y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{{5}/{2}} y^{\left (5\right )}-a y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x -a \right )^{5} \left (x -b \right )^{5} y^{\left (5\right )}-c y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-6 y^{2}-x = 0
\] |
[[_Painleve, ‘1st‘]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a y^{2}+b x +c = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-2 y^{3}-y x +a = 0
\] |
[[_Painleve, ‘2nd‘]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-2 a^{2} y^{3}+2 a b x y-b = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+d +b x y+c y+a y^{3} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a \,x^{r} y^{2} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-\frac {1}{\left (a y^{2}+b x y+c \,x^{2}+\alpha y+\beta x +\gamma \right )^{{3}/{2}}} = 0
\] |
[NONE] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+a \,{\mathrm e}^{x} \sqrt {y} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+{\mathrm e}^{x} \sin \left (y\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a^{2} \sin \left (y\right )-\beta \sin \left (x \right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a^{2} \sin \left (y\right )-\beta f \left (x \right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime } = \frac {f \left (\frac {y}{\sqrt {x}}\right )}{x^{{3}/{2}}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-3 y^{\prime }-y^{2}-2 y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-7 y^{\prime }-y^{{3}/{2}}+12 y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+5 a y^{\prime }-6 y^{2}+6 a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+3 a y^{\prime }-2 y^{3}+2 a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\frac {\left (3 n +4\right ) y^{\prime }}{n}-\frac {2 \left (n +1\right ) \left (n +2\right ) y \left (y^{\frac {n}{n +1}}-1\right )}{n^{2}} = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b y^{n}+\frac {\left (a^{2}-1\right ) y}{4} = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b \,x^{v} y^{n} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{y}-2 a = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+f \left (x \right ) \sin \left (y\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+y y^{\prime }-y^{3}+a y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (y+3 a \right ) y^{\prime }-y^{3}+a y^{2}+2 a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (y+3 f \left (x \right )\right ) y^{\prime }-y^{3}+y^{2} f \left (x \right )+y \left (f^{\prime }\left (x \right )+2 f \left (x \right )^{2}\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+y y^{\prime }-y^{3}-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+f \left (x \right )\right ) \left (3 y^{\prime }+y^{2}\right )+\left (a f \left (x \right )^{2}+3 f^{\prime }\left (x \right )+\frac {3 {f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}\right ) y+b f \left (x \right )^{3} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (y-\frac {3 f^{\prime }\left (x \right )}{2 f \left (x \right )}\right ) y^{\prime }-y^{3}-\frac {f^{\prime }\left (x \right ) y^{2}}{2 f \left (x \right )}+\frac {\left (f \left (x \right )+\frac {{f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}-f^{\prime \prime }\left (x \right )\right ) y}{2 f \left (x \right )} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+2 y y^{\prime }+f \left (x \right ) y^{\prime }+f^{\prime }\left (x \right ) y = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+2 y y^{\prime }+f \left (x \right ) \left (y^{\prime }+y^{2}\right )-g \left (x \right ) = 0
\] |
[[_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+3 y y^{\prime }+y^{3}+f \left (x \right ) y-g \left (x \right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0
\] |
[[_2nd_order, _with_potential_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-3 y y^{\prime }-3 a y^{2}-4 a^{2} y-b = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0
\] |
[[_2nd_order, _with_potential_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+f \left (x , y\right ) y^{\prime }+g \left (x , y\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime } {| y^{\prime }|}+b y^{\prime }+c y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a {y^{\prime }}^{2}+b y^{\prime }+c y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime } {| y^{\prime }|}+b \sin \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-\frac {D\left (f \right )\left (y\right ) {y^{\prime }}^{3}}{f \left (y\right )}+g \left (x \right ) y^{\prime }+h \left (x \right ) f \left (y\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\phi \left (y\right ) {y^{\prime }}^{2}+f \left (x \right ) y^{\prime }+g \left (x \right ) \Phi \left (y\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (y\right ) y^{\prime }+h \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (1+{y^{\prime }}^{2}\right ) \left (f \left (x , y\right ) y^{\prime }+g \left (x , y\right )\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-a \left (y^{\prime } x -y\right )^{v} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-k \,x^{a} y^{b} {y^{\prime }}^{r} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (y^{\prime }-\frac {y}{x}\right )^{a} f \left (x , y\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime } = 2 a \left (c +b x +y\right ) \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-f \left (y^{\prime }, a x +b y\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-y f \left (x , \frac {y^{\prime }}{y}\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-x^{n -2} f \left (y x^{-n}, y^{\prime } x^{-n +1}\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}a y^{\prime \prime }+h \left (y^{\prime }\right )+c y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }-x y^{n} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+a \,x^{v} y^{n} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+x \,{\mathrm e}^{y} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+a y^{\prime }+b x \,{\mathrm e}^{y} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+a y^{\prime }+b \,x^{5-2 a} {\mathrm e}^{y} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }-x^{2} {y^{\prime }}^{2}+2 y^{\prime }+y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+a \left (y^{\prime } x -y\right )^{2}-b = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime } = a \left (y^{n}-y\right )
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+a \left ({\mathrm e}^{y}-1\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a +1\right ) x y^{\prime }-x^{k} f \left (x^{k} y, y^{\prime } x +k y\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+a \left (y^{\prime } x -y\right )^{2}-b \,x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+a y {y^{\prime }}^{2}+b x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }-\sqrt {a \,x^{2} {y^{\prime }}^{2}+b y^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}4 x^{2} y^{\prime \prime }-x^{4} {y^{\prime }}^{2}+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}9 x^{2} y^{\prime \prime }+a y^{3}+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )+12 y x +24 = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }-a \left (y^{\prime } x -y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}2 x^{3} y^{\prime \prime }+x^{2} \left (9+2 y x \right ) y^{\prime }+b +x y \left (a +3 y x -2 x^{2} y^{2}\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}2 \left (-x^{k}+4 x^{3}\right ) \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )-\left (k \,x^{k -1}-12 x^{2}\right ) \left (3 y^{\prime }+y^{2}\right )+y a x +b = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}x^{4} y^{\prime \prime }+a^{2} y^{n} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{4} y^{\prime \prime }-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime }-x^{2} \left (x +y^{\prime }\right ) y^{\prime }+4 y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime }+\left (y^{\prime } x -y\right )^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } \sqrt {x}-y^{{3}/{2}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (a \,x^{2}+b x +c \right )^{{3}/{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {a \,x^{2}+b x +c}}\right ) = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}x^{\frac {n}{n +1}} y^{\prime \prime }-y^{\frac {2 n +1}{n +1}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}f \left (x \right )^{2} y^{\prime \prime }+f \left (x \right ) f^{\prime }\left (x \right ) y^{\prime }-h \left (y, f \left (x \right ) y^{\prime }\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y y^{\prime \prime }-a x = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime \prime }-a \,x^{2} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime \prime }+y^{2}-a x -b = 0
\] |
[NONE] |
✗ |
✓ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}+{\mathrm e}^{x} y \left (c y^{2}+d \right )+{\mathrm e}^{2 x} \left (b +a y^{4}\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}-y^{\prime }+f \left (x \right ) y^{3}+y^{2} \left (\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {{f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}+f \left (x \right ) y^{\prime }-f^{\prime }\left (x \right ) y-y^{3} = 0
\] |
[NONE] |
✗ |
✓ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}+f^{\prime }\left (x \right ) y^{\prime }-f^{\prime \prime }\left (x \right ) y+f \left (x \right ) y^{3}-y^{4} = 0
\] |
[NONE] |
✗ |
✓ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}+a y y^{\prime }-2 a y^{2}+b y^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}-\left (a y-1\right ) y^{\prime }+2 a^{2} y^{2}-2 b^{2} y^{3}+a y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✓ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}+\left (a y-1\right ) y^{\prime }-y \left (1+y\right ) \left (b^{2} y^{2}-a^{2}\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}+\left (\tan \left (x \right )+\cot \left (x \right )\right ) y y^{\prime }+\left (\cos \left (x \right )^{2}-n^{2} \cot \left (x \right )^{2}\right ) y^{2} \ln \left (y\right ) = 0
\] |
[[_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}-f \left (x \right ) y y^{\prime }-g \left (x \right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}+\left (g \left (x \right )+y^{2} f \left (x \right )\right ) y^{\prime }-y \left (g^{\prime }\left (x \right )-f^{\prime }\left (x \right ) y^{2}\right ) = 0
\] |
[[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime \prime }+a {y^{\prime }}^{2}+b y y^{\prime }+c y^{2}+d y^{1-a} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime \prime }+a {y^{\prime }}^{2}+f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime \prime }-\frac {\left (a -1\right ) {y^{\prime }}^{2}}{a}-f \left (x \right ) y^{2} y^{\prime }+\frac {a f \left (x \right )^{2} y^{4}}{\left (a +2\right )^{2}}-\frac {a f^{\prime }\left (x \right ) y^{3}}{a +2} = 0
\] |
[NONE] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime } \left (x -y\right )+2 y^{\prime } \left (y^{\prime }+1\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } \left (x -y\right )-\left (y^{\prime }+1\right ) \left (1+{y^{\prime }}^{2}\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime } \left (x -y\right )-h \left (y^{\prime }\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\[
{}2 y y^{\prime \prime }-{y^{\prime }}^{2}+y^{2} f \left (x \right )+a = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}2 y y^{\prime \prime }-{y^{\prime }}^{2}-4 \left (x +2 y\right ) y^{2} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}2 y y^{\prime \prime }-{y^{\prime }}^{2}+1+2 x y^{2}+a y^{3} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}2 y y^{\prime \prime }-{y^{\prime }}^{2}+\left (a y+b x \right ) y^{2} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}2 y y^{\prime \prime }-{y^{\prime }}^{2}+b -4 \left (x^{2}+a \right ) y^{2}-8 x y^{3}-3 y^{4} = 0
\] |
[[_Painleve, ‘4th‘]] |
✗ |
✗ |
|
|
\[
{}2 y y^{\prime \prime }-{y^{\prime }}^{2}+3 f \left (x \right ) y y^{\prime }+2 \left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y^{2}-8 y^{3} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}2 y y^{\prime \prime }-{y^{\prime }}^{2}+4 y^{2} y^{\prime }+1+y^{2} f \left (x \right )+y^{4} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}2 y y^{\prime \prime }-3 {y^{\prime }}^{2}+y^{2} f \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}3 y y^{\prime \prime }-2 {y^{\prime }}^{2}-a \,x^{2}-b x -c = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}4 y y^{\prime \prime }-3 {y^{\prime }}^{2}+\left (6 y^{2}-\frac {2 f^{\prime }\left (x \right ) y}{f \left (x \right )}\right ) y^{\prime }+y^{4}-2 y^{2} y^{\prime }+g \left (x \right ) y^{2}+f \left (x \right ) y = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}a y y^{\prime \prime }+b {y^{\prime }}^{2}+\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y+\operatorname {c0} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}a y y^{\prime \prime }-\left (a -1\right ) {y^{\prime }}^{2}+\left (a +2\right ) f \left (x \right ) y^{2} y^{\prime }+f \left (x \right )^{2} y^{4}+a f^{\prime }\left (x \right ) y^{3} = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2}+a y y^{\prime }+f \left (x \right ) = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}x y y^{\prime \prime }-x {y^{\prime }}^{2}+y y^{\prime }+x \left (d +a y^{4}\right )+y \left (c +b y^{2}\right ) = 0
\] |
[[_Painleve, ‘3rd‘]] |
✗ |
✗ |
|
|
\[
{}x y y^{\prime \prime }-x {y^{\prime }}^{2}+a y y^{\prime }+b x y^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (1+y\right ) y^{\prime } = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) {y^{\prime }}^{2}-y y^{\prime } = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (-1+y\right ) y^{\prime \prime }-2 x^{2} {y^{\prime }}^{2}-2 x \left (-1+y\right ) y^{\prime }-2 y \left (-1+y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (x +y\right ) y^{\prime \prime }-\left (y^{\prime } x -y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (x -y\right ) y^{\prime \prime }+a \left (y^{\prime } x -y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}2 x^{2} y y^{\prime \prime }-x^{2} \left (1+{y^{\prime }}^{2}\right )+y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}x \left (x +1\right )^{2} y y^{\prime \prime }-x \left (x +1\right )^{2} {y^{\prime }}^{2}+2 \left (x +1\right )^{2} y y^{\prime }-a \left (x +2\right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 x^{2} y y^{\prime }+3 x y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✗ |
|
|
\[
{}\operatorname {f0} \left (x \right ) y y^{\prime \prime }+\operatorname {f1} \left (x \right ) {y^{\prime }}^{2}+\operatorname {f2} \left (x \right ) y y^{\prime }+\operatorname {f3} \left (x \right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}+a x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}-a x -b = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}\left (x +y^{2}\right ) y^{\prime \prime }-2 \left (x -y^{2}\right ) {y^{\prime }}^{3}+y^{\prime } \left (1+4 y y^{\prime }\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+y^{2}\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right ) \left (y^{\prime } x -y\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+y^{2}\right ) y^{\prime \prime }-2 \left (1+{y^{\prime }}^{2}\right ) \left (y^{\prime } x -y\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\[
{}2 y \left (1-y\right ) y^{\prime \prime }-\left (1-2 y\right ) {y^{\prime }}^{2}+y \left (1-y\right ) y^{\prime } f \left (x \right ) = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}2 y \left (1-y\right ) y^{\prime \prime }-\left (1-3 y\right ) {y^{\prime }}^{2}+h \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}2 y \left (-1+y\right ) y^{\prime \prime }-\left (3 y-1\right ) {y^{\prime }}^{2}+4 y y^{\prime } \left (f \left (x \right ) y+g \left (x \right )\right )+4 y^{2} \left (-1+y\right ) \left (g \left (x \right )^{2}-f \left (x \right )^{2}-g^{\prime }\left (x \right )-f^{\prime }\left (x \right )\right ) = 0
\] |
[[_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
|
|
\[
{}-2 y \left (1-y\right ) y^{\prime \prime }+\left (1-3 y\right ) {y^{\prime }}^{2}-4 y y^{\prime } \left (f \left (x \right ) y+g \left (x \right )\right )+\left (1-y\right )^{3} \left (\operatorname {f0} \left (x \right )^{2} y^{2}-\operatorname {f1} \left (x \right )^{2}\right )+4 y^{2} \left (1-y\right ) \left (f \left (x \right )^{2}-g \left (x \right )^{2}-g^{\prime }\left (x \right )-f^{\prime }\left (x \right )\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}3 y \left (1-y\right ) y^{\prime \prime }-2 \left (1-2 y\right ) {y^{\prime }}^{2}-h \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}\left (1-y\right ) y^{\prime \prime }-3 \left (1-2 y\right ) {y^{\prime }}^{2}-h \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}a y \left (-1+y\right ) y^{\prime \prime }+\left (b y+c \right ) {y^{\prime }}^{2}+h \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}x y^{2} y^{\prime \prime }-a = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime }+\left (a^{2}-x^{2}\right ) y {y^{\prime }}^{2}-x \left (a^{2}-y^{2}\right ) y^{\prime } = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}2 x^{2} y \left (-1+y\right ) y^{\prime \prime }-x^{2} \left (3 y-1\right ) {y^{\prime }}^{2}+2 x y \left (-1+y\right ) y^{\prime }+\left (a y^{2}+b \right ) \left (-1+y\right )^{3}+c x y^{2} \left (-1+y\right )+d \,x^{2} y^{2} \left (1+y\right ) = 0
\] |
[[_Painleve, ‘5th‘]] |
✗ |
✗ |
|
|
\[
{}x^{3} y^{2} y^{\prime \prime }+\left (x +y\right ) \left (y^{\prime } x -y\right )^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 y^{3} y^{\prime \prime }+y^{4}-a^{2} x y^{2}-1 = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}2 y^{3} y^{\prime \prime }+y^{2} {y^{\prime }}^{2}-a \,x^{2}-b x -c = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}-2 x y \left (1-x \right ) \left (1-y\right ) \left (x -y\right ) y^{\prime \prime }+x \left (1-x \right ) \left (x -2 y x -2 y+3 y^{2}\right ) {y^{\prime }}^{2}+2 y \left (1-y\right ) \left (x^{2}+y-2 y x \right ) y^{\prime }-y^{2} \left (1-y\right )^{2}-f \left (y \left (-1+y\right ) \left (-x +y\right )\right )^{{3}/{2}} = 0
\] |
unknown |
✗ |
✗ |
|
|
\[
{}2 x^{2} y \left (1-x \right )^{2} \left (1-y\right ) \left (x -y\right ) y^{\prime \prime }-x^{2} \left (1-x \right )^{2} \left (x -2 y x -2 y+3 y^{2}\right ) {y^{\prime }}^{2}-2 x y \left (1-x \right ) \left (1-y\right ) \left (x^{2}+y-2 y x \right ) y^{\prime }+b x \left (1-y\right )^{2} \left (x -y\right )^{2}-c \left (1-x \right ) y^{2} \left (x -y\right )^{2}-d x y^{2} \left (1-x \right ) \left (1-y\right )^{2}+a y^{2} \left (x -y\right )^{2} \left (1-y\right )^{2} = 0
\] |
[[_Painleve, ‘6th‘]] |
✗ |
✗ |
|
|
\[
{}\left (c +2 b x +a \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }+d y = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}\sqrt {x^{2}+y^{2}}\, y^{\prime \prime }-a \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}h \left (y\right ) y^{\prime \prime }+a h \left (y\right ) {y^{\prime }}^{2}+j \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\[
{}h \left (y\right ) y^{\prime \prime }-D\left (h \right )\left (y\right ) {y^{\prime }}^{2}-h \left (y\right )^{2} j \left (x , \frac {y^{\prime }}{h \left (y\right )}\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2} = 0
\] |
[[_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]] |
✗ |
✗ |
|
|
\[
{}\left (y^{\prime } x -y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (y^{\prime } x -y\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}a \,x^{3} y^{\prime } y^{\prime \prime }+b y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}\left (\operatorname {f1} y^{\prime }+\operatorname {f2} y\right ) y^{\prime \prime }+\operatorname {f3} {y^{\prime }}^{2}+\operatorname {f4} \left (x \right ) y y^{\prime }+\operatorname {f5} \left (x \right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (2 y^{2} y^{\prime }+x^{2}\right ) y^{\prime \prime }+2 y {y^{\prime }}^{3}+3 y^{\prime } x +y = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
✗ |
|
|
\[
{}\left ({y^{\prime }}^{2}+a \left (y^{\prime } x -y\right )\right ) y^{\prime \prime }-b = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \sqrt {1+{y^{\prime }}^{2}}-y^{\prime } x \right ) y^{\prime \prime }-{y^{\prime }}^{2}-1 = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
|
|
\[
{}h \left (y^{\prime }\right ) y^{\prime \prime }+j \left (y\right ) y^{\prime }+f = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1]] |
✗ |
✗ |
|
|
\[
{}{y^{\prime \prime }}^{2}-a y-b = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
|
|
\[
{}a^{2} {y^{\prime \prime }}^{2}-2 a x y^{\prime \prime }+y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
✗ |
|
|
\[
{}2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2}-x y^{\prime \prime } \left (x +4 y^{\prime }\right )+2 \left (x +y^{\prime }\right ) y^{\prime }-2 y = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\[
{}3 x^{2} {y^{\prime \prime }}^{2}-2 \left (3 y^{\prime } x +y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (2-9 x \right ) {y^{\prime \prime }}^{2}-6 x \left (1-6 x \right ) y^{\prime } y^{\prime \prime }+6 y y^{\prime \prime }-36 x {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}F_{1,1}\left (x \right ) {y^{\prime }}^{2}+\left (\left (F_{2,1}\left (x \right )+F_{1,2}\left (x \right )\right ) y^{\prime \prime }+y \left (F_{1,0}\left (x \right )+F_{0,1}\left (x \right )\right )\right ) y^{\prime }+F_{2,2}\left (x \right ) {y^{\prime \prime }}^{2}+y \left (F_{2,0}\left (x \right )+F_{0,2}\left (x \right )\right ) y^{\prime \prime }+F_{0,0}\left (x \right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y {y^{\prime \prime }}^{2}-a \,{\mathrm e}^{2 x} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (y^{2}-x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime }\right )^{2}-4 x y \left (y^{\prime } x -y\right )^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}\left (2 y y^{\prime \prime }-{y^{\prime }}^{2}\right )^{3}+32 y^{\prime \prime } \left (x y^{\prime \prime }-y^{\prime }\right )^{3} = 0
\] |
unknown |
✗ |
✓ |
|
|
\[
{}\sqrt {a {y^{\prime \prime }}^{2}+b {y^{\prime }}^{2}}+c y y^{\prime \prime }+d {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+y y^{\prime \prime }-{y^{\prime }}^{2}+1 = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }-y y^{\prime \prime }+{y^{\prime }}^{2} = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+a y y^{\prime \prime } = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+x y^{\prime \prime }+\left (2 y x -1\right ) y^{\prime }+y^{2}-f \left (x \right ) = 0
\] |
[[_3rd_order, _exact, _nonlinear]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+x \left (-1+y\right ) y^{\prime \prime }+x {y^{\prime }}^{2}+\left (1-y\right ) y^{\prime } = 0
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime \prime \prime }-y^{\prime } y^{\prime \prime }+y^{3} y^{\prime } = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}4 y^{2} y^{\prime \prime \prime }-18 y y^{\prime } y^{\prime \prime }+15 {y^{\prime }}^{3} = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}9 y^{2} y^{\prime \prime \prime }-45 y y^{\prime } y^{\prime \prime }+40 {y^{\prime }}^{3} = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } y^{\prime \prime \prime \prime }-y^{\prime \prime } y^{\prime \prime \prime }+{y^{\prime }}^{3} y^{\prime \prime \prime } = 0
\] |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime } \left (f^{\prime \prime \prime }\left (x \right ) y^{\prime }+3 f^{\prime \prime }\left (x \right ) y^{\prime \prime }+3 f^{\prime }\left (x \right ) y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime \prime \prime }\right )-y^{\prime \prime } f y^{\prime \prime \prime }+{y^{\prime }}^{3} \left (f^{\prime }\left (x \right ) y^{\prime }+f \left (x \right ) y^{\prime \prime }\right )+2 q \left (x \right ) {y^{\prime }}^{2} \sin \left (y\right )+\left (q \left (x \right ) y^{\prime \prime }-q^{\prime }\left (x \right ) y^{\prime }\right ) \cos \left (y\right ) = 0
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}9 {y^{\prime \prime }}^{2} y^{\left (5\right )}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+40 y^{\prime \prime \prime } = 0
\] |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-f \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime } = f \left (y\right )
\] |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) f \left (t \right )+y \left (t \right ) g \left (t \right ) \\ y^{\prime }\left (t \right )=-x \left (t \right ) g \left (t \right )+y \left (t \right ) f \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )+\left (a x \left (t \right )+b y \left (t \right )\right ) f \left (t \right )=g \left (t \right ) \\ y^{\prime }\left (t \right )+\left (c x \left (t \right )+d y \left (t \right )\right ) f \left (t \right )=h \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) \cos \left (t \right ) \\ y^{\prime }\left (t \right )=x \left (t \right ) {\mathrm e}^{-\sin \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} t x^{\prime }\left (t \right )+y \left (t \right )=0 \\ t y^{\prime }\left (t \right )+x \left (t \right )=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} t x^{\prime }\left (t \right )+2 x \left (t \right )=t \\ t y^{\prime }\left (t \right )-\left (t +2\right ) x \left (t \right )-t y \left (t \right )=-t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} t x^{\prime }\left (t \right )+2 x \left (t \right )-2 y \left (t \right )=t \\ t y^{\prime }\left (t \right )+x \left (t \right )+5 y \left (t \right )=t^{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} t^{2} \left (1-\sin \left (t \right )\right ) x^{\prime }\left (t \right )=t \left (1-2 \sin \left (t \right )\right ) x \left (t \right )+t^{2} y \left (t \right ) \\ t^{2} \left (1-\sin \left (t \right )\right ) y^{\prime }\left (t \right )=\left (t \cos \left (t \right )-\sin \left (t \right )\right ) x \left (t \right )+t \left (1-t \cos \left (t \right )\right ) y \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )+y \left (t \right )=f \left (t \right ) \\ x^{\prime \prime }\left (t \right )+y^{\prime \prime }\left (t \right )+y^{\prime }\left (t \right )+x \left (t \right )+y \left (t \right )=g \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} 2 x^{\prime }\left (t \right )+y^{\prime }\left (t \right )-3 x \left (t \right )=0 \\ x^{\prime \prime }\left (t \right )+y^{\prime }\left (t \right )-2 y \left (t \right )={\mathrm e}^{2 t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )-y^{\prime }\left (t \right )+x \left (t \right )=2 t \\ x^{\prime \prime }\left (t \right )+y^{\prime }\left (t \right )-9 x \left (t \right )+3 y \left (t \right )=\sin \left (2 t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )-x \left (t \right )+2 y \left (t \right )=0 \\ x^{\prime \prime }\left (t \right )-2 y^{\prime }\left (t \right )=2 t -\cos \left (2 t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} t x^{\prime }\left (t \right )-t y^{\prime }\left (t \right )-2 y \left (t \right )=0 \\ t x^{\prime \prime }\left (t \right )+2 x^{\prime }\left (t \right )+x \left (t \right ) t =0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )+a y \left (t \right )=0 \\ y^{\prime \prime }\left (t \right )-a^{2} y \left (t \right )=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=a x \left (t \right )+b y \left (t \right ) \\ y^{\prime \prime }\left (t \right )=c x \left (t \right )+d y \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=a_{1} x \left (t \right )+b_{1} y \left (t \right )+c_{1} \\ y^{\prime \prime }\left (t \right )=a_{2} x \left (t \right )+b_{2} y \left (t \right )+c_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )+x \left (t \right )+y \left (t \right )=-5 \\ y^{\prime \prime }\left (t \right )-4 x \left (t \right )-3 y \left (t \right )=-3 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=\left (3 \cos \left (a t +b \right )^{2}-1\right ) c^{2} x \left (t \right )+\frac {3 c^{2} y \left (t \right ) \sin \left (2 a t b \right )}{2} \\ y^{\prime \prime }\left (t \right )=\left (3 \sin \left (a t +b \right )^{2}-1\right ) c^{2} y \left (t \right )+\frac {3 c^{2} x \left (t \right ) \sin \left (2 a t b \right )}{2} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )+6 x \left (t \right )+7 y \left (t \right )=0 \\ y^{\prime \prime }\left (t \right )+3 x \left (t \right )+2 y \left (t \right )=2 t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )-a y^{\prime }\left (t \right )+b x \left (t \right )=0 \\ y^{\prime \prime }\left (t \right )+a x^{\prime }\left (t \right )+b y \left (t \right )=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} a_{1} x^{\prime \prime }\left (t \right )+b_{1} x^{\prime }\left (t \right )+c_{1} x \left (t \right )-A y^{\prime }\left (t \right )=B \,{\mathrm e}^{i \omega t} \\ a_{2} y^{\prime \prime }\left (t \right )+b_{2} y^{\prime }\left (t \right )+c_{2} y \left (t \right )+A x^{\prime }\left (t \right )=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )+a \left (x^{\prime }\left (t \right )-y^{\prime }\left (t \right )\right )+b_{1} x \left (t \right )=c_{1} {\mathrm e}^{i \omega t} \\ y^{\prime \prime }\left (t \right )+a \left (y^{\prime }\left (t \right )-x^{\prime }\left (t \right )\right )+b_{2} y \left (t \right )=c_{2} {\mathrm e}^{i \omega t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} \operatorname {a11} x^{\prime \prime }\left (t \right )+\operatorname {b11} x^{\prime }\left (t \right )+\operatorname {c11} x \left (t \right )+\operatorname {a12} y^{\prime \prime }\left (t \right )+\operatorname {b12} y^{\prime }\left (t \right )+\operatorname {c12} y \left (t \right )=0 \\ \operatorname {a21} x^{\prime \prime }\left (t \right )+\operatorname {b21} x^{\prime }\left (t \right )+\operatorname {c21} x \left (t \right )+\operatorname {a22} y^{\prime \prime }\left (t \right )+\operatorname {b22} y^{\prime }\left (t \right )+\operatorname {c22} y \left (t \right )=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )-2 x^{\prime }\left (t \right )-y^{\prime }\left (t \right )+y \left (t \right )=0 \\ y^{\prime \prime \prime }\left (t \right )-y^{\prime \prime }\left (t \right )+2 x^{\prime }\left (t \right )-x \left (t \right )=t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )+y^{\prime \prime }\left (t \right )+y^{\prime }\left (t \right )=\sinh \left (2 t \right ) \\ 2 x^{\prime \prime }\left (t \right )+y^{\prime \prime }\left (t \right )=2 t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )-x^{\prime }\left (t \right )+y^{\prime }\left (t \right )=0 \\ x^{\prime \prime }\left (t \right )+y^{\prime \prime }\left (t \right )-x \left (t \right )=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=h \left (t \right ) y \left (t \right )-g \left (t \right ) z \left (t \right ) \\ y^{\prime }\left (t \right )=f \left (t \right ) z \left (t \right )-h \left (t \right ) x \left (t \right ) \\ z^{\prime }\left (t \right )=x \left (t \right ) g \left (t \right )-y \left (t \right ) f \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} t x^{\prime }\left (t \right )=2 x \left (t \right )-t \\ t^{3} y^{\prime }\left (t \right )=-x \left (t \right )+t^{2} y \left (t \right )+t \\ t^{4} z^{\prime }\left (t \right )=-x \left (t \right )-t^{2} y \left (t \right )+t^{3} z \left (t \right )+t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} a t x^{\prime }\left (t \right )=b c \left (y \left (t \right )-z \left (t \right )\right ) \\ b t y^{\prime }\left (t \right )=c a \left (z \left (t \right )-x \left (t \right )\right ) \\ c t z^{\prime }\left (t \right )=a b \left (x \left (t \right )-y \left (t \right )\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }\left (t \right )=a x_{2} \left (t \right )+b x_{3} \left (t \right ) \cos \left (c t \right )+b x_{4} \left (t \right ) \sin \left (c t \right ) \\ x_{2}^{\prime }\left (t \right )=-a x_{1} \left (t \right )+b x_{3} \left (t \right ) \sin \left (c t \right )-b x_{4} \left (t \right ) \cos \left (c t \right ) \\ x_{3}^{\prime }\left (t \right )=-b x_{1} \left (t \right ) \cos \left (c t \right )-b x_{2} \left (t \right ) \sin \left (c t \right )+a x_{4} \left (t \right ) \\ x_{4}^{\prime }\left (t \right )=-b x_{1} \left (t \right ) \sin \left (c t \right )+b x_{2} \left (t \right ) \cos \left (c t \right )-a x_{3} \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=-x \left (t \right ) \left (x \left (t \right )+y \left (t \right )\right ) \\ y^{\prime }\left (t \right )=y \left (t \right ) \left (x \left (t \right )+y \left (t \right )\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=\left (a y \left (t \right )+b \right ) x \left (t \right ) \\ y^{\prime }\left (t \right )=\left (c x \left (t \right )+d \right ) y \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) \left (a \left (p x \left (t \right )+q y \left (t \right )\right )+\alpha \right ) \\ y^{\prime }\left (t \right )=y \left (t \right ) \left (\beta +b \left (p x \left (t \right )+q y \left (t \right )\right )\right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=h \left (a -x \left (t \right )\right ) \left (c -x \left (t \right )-y \left (t \right )\right ) \\ y^{\prime }\left (t \right )=k \left (b -y \left (t \right )\right ) \left (c -x \left (t \right )-y \left (t \right )\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=y \left (t \right )^{2}-\cos \left (x \left (t \right )\right ) \\ y^{\prime }\left (t \right )=-y \left (t \right ) \sin \left (x \left (t \right )\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=-x \left (t \right ) y \left (t \right )^{2}+x \left (t \right )+y \left (t \right ) \\ y^{\prime }\left (t \right )=x \left (t \right )^{2} y \left (t \right )-x \left (t \right )-y \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right )+y \left (t \right )-x \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) \\ y^{\prime }\left (t \right )=-x \left (t \right )+y \left (t \right )-y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=-y \left (t \right )+x \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}-1\right ) \\ y^{\prime }\left (t \right )=x \left (t \right )+y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}-1\right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=-y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) \\ y^{\prime }\left (t \right )=\left \{\begin {array}{cc} x \left (t \right )^{2}+y \left (t \right )^{2} & 2 x \left (t \right )\le x \left (t \right )^{2}+y \left (t \right )^{2} \\ \left (\frac {x \left (t \right )}{2}-\frac {y \left (t \right )^{2}}{2 x \left (t \right )}\right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) & \operatorname {otherwise} \end {array}\right . \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=-y \left (t \right )+\left (\left \{\begin {array}{cc} x \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}-1\right ) \sin \left (\frac {1}{x \left (t \right )^{2}+y \left (t \right )^{2}}\right ) & x \left (t \right )^{2}+y \left (t \right )^{2}\neq 1 \\ 0 & \operatorname {otherwise} \end {array}\right .\right ) \\ y^{\prime }\left (t \right )=x \left (t \right )+\left (\left \{\begin {array}{cc} y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}-1\right ) \sin \left (\frac {1}{x \left (t \right )^{2}+y \left (t \right )^{2}}\right ) & x \left (t \right )^{2}+y \left (t \right )^{2}\neq 1 \\ 0 & \operatorname {otherwise} \end {array}\right .\right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} \left (t^{2}+1\right ) x^{\prime }\left (t \right )=-x \left (t \right ) t +y \left (t \right ) \\ \left (t^{2}+1\right ) y^{\prime }\left (t \right )=-x \left (t \right )-t y \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} \left (x \left (t \right )^{2}+y \left (t \right )^{2}-t^{2}\right ) x^{\prime }\left (t \right )=-2 x \left (t \right ) t \\ \left (x \left (t \right )^{2}+y \left (t \right )^{2}-t^{2}\right ) y^{\prime }\left (t \right )=-2 t y \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} {x^{\prime }\left (t \right )}^{2}+t x^{\prime }\left (t \right )+a y^{\prime }\left (t \right )-x \left (t \right )=0 \\ x^{\prime }\left (t \right ) y^{\prime }\left (t \right )+t y^{\prime }\left (t \right )-y \left (t \right )=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x \left (t \right )=t x^{\prime }\left (t \right )+f \left (x^{\prime }\left (t \right ), y^{\prime }\left (t \right )\right ) \\ y \left (t \right )=t y^{\prime }\left (t \right )+g \left (x^{\prime }\left (t \right ), y^{\prime }\left (t \right )\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=a \,{\mathrm e}^{2 x \left (t \right )}-{\mathrm e}^{-x \left (t \right )}+{\mathrm e}^{-2 x \left (t \right )} \cos \left (y \left (t \right )\right )^{2} \\ y^{\prime \prime }\left (t \right )={\mathrm e}^{-2 x \left (t \right )} \sin \left (y \left (t \right )\right ) \cos \left (y \left (t \right )\right )-\frac {\sin \left (y \left (t \right )\right )}{\cos \left (y \left (t \right )\right )^{3}} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=\frac {k x \left (t \right )}{\left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )^{{3}/{2}}} \\ y^{\prime \prime }\left (t \right )=\frac {k y \left (t \right )}{\left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )^{{3}/{2}}} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=y \left (t \right )-z \left (t \right ) \\ y^{\prime }\left (t \right )=x \left (t \right )^{2}+y \left (t \right ) \\ z^{\prime }\left (t \right )=x \left (t \right )^{2}+z \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} a x^{\prime }\left (t \right )=\left (b -c \right ) y \left (t \right ) z \left (t \right ) \\ b y^{\prime }\left (t \right )=\left (c -a \right ) z \left (t \right ) x \left (t \right ) \\ c z^{\prime }\left (t \right )=\left (a -b \right ) x \left (t \right ) y \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) \left (y \left (t \right )-z \left (t \right )\right ) \\ y^{\prime }\left (t \right )=y \left (t \right ) \left (z \left (t \right )-x \left (t \right )\right ) \\ z^{\prime }\left (t \right )=z \left (t \right ) \left (x \left (t \right )-y \left (t \right )\right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )=x \left (t \right ) y \left (t \right ) \\ y^{\prime }\left (t \right )+z^{\prime }\left (t \right )=y \left (t \right ) z \left (t \right ) \\ x^{\prime }\left (t \right )+z^{\prime }\left (t \right )=x \left (t \right ) z \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=\frac {x \left (t \right )^{2}}{2}-\frac {y \left (t \right )}{24} \\ y^{\prime }\left (t \right )=2 x \left (t \right ) y \left (t \right )-3 z \left (t \right ) \\ z^{\prime }\left (t \right )=3 x \left (t \right ) z \left (t \right )-\frac {y \left (t \right )^{2}}{6} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) \left (y \left (t \right )^{2}-z \left (t \right )^{2}\right ) \\ y^{\prime }\left (t \right )=y \left (t \right ) \left (z \left (t \right )^{2}-x \left (t \right )^{2}\right ) \\ z^{\prime }\left (t \right )=z \left (t \right ) \left (x \left (t \right )^{2}-y \left (t \right )^{2}\right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) \left (y \left (t \right )^{2}-z \left (t \right )^{2}\right ) \\ y^{\prime }\left (t \right )=-y \left (t \right ) \left (z \left (t \right )^{2}+x \left (t \right )^{2}\right ) \\ z^{\prime }\left (t \right )=z \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=-x \left (t \right ) y \left (t \right )^{2}+x \left (t \right )+y \left (t \right ) \\ y^{\prime }\left (t \right )=x \left (t \right )^{2} y \left (t \right )-x \left (t \right )-y \left (t \right ) \\ z^{\prime }\left (t \right )=y \left (t \right )^{2}-x \left (t \right )^{2} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} \left (x \left (t \right )-y \left (t \right )\right ) \left (x \left (t \right )-z \left (t \right )\right ) x^{\prime }\left (t \right )=f \left (t \right ) \\ \left (-x \left (t \right )+y \left (t \right )\right ) \left (y \left (t \right )-z \left (t \right )\right ) y^{\prime }\left (t \right )=f \left (t \right ) \\ \left (z \left (t \right )-x \left (t \right )\right ) \left (z \left (t \right )-y \left (t \right )\right ) z^{\prime }\left (t \right )=f \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }\left (t \right ) \sin \left (x_{2} \left (t \right )\right )=x_{4} \left (t \right ) \sin \left (x_{3} \left (t \right )\right )+x_{5} \left (t \right ) \cos \left (x_{3} \left (t \right )\right ) \\ x_{2}^{\prime }\left (t \right )=x_{4} \left (t \right ) \cos \left (x_{3} \left (t \right )\right )-x_{5} \left (t \right ) \sin \left (x_{3} \left (t \right )\right ) \\ x_{3}^{\prime }\left (t \right )+x_{1}^{\prime }\left (t \right ) \cos \left (x_{2} \left (t \right )\right )=a \\ x_{4}^{\prime }\left (t \right )-\left (1-\lambda \right ) a x_{5} \left (t \right )=-m \sin \left (x_{2} \left (t \right )\right ) \cos \left (x_{3} \left (t \right )\right ) \\ x_{5}^{\prime }\left (t \right )+\left (1-\lambda \right ) a x_{4} \left (t \right )=m \sin \left (x_{2} \left (t \right )\right ) \sin \left (x_{3} \left (t \right )\right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+a n \,x^{n -1}-a^{2} x^{2 n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = a y^{2}+b \,x^{2 n}+c \,x^{n -1}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+k \left (a x +b \right )^{n} \left (c x +d \right )^{-n -4}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+a \,x^{2 m} \left (b \,x^{m}+c \right )^{n}-\frac {n^{2}}{4}+\frac {1}{4}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+a \,x^{n} y+a \,x^{n -1}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \,x^{n} y-a b \,x^{n}-b^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+a \,x^{n +m +1}-a \,x^{m}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+b c \,x^{m}-a \,c^{2} x^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+c k \,x^{k -1}-b c \,x^{m +k}-a \,c^{2} x^{n +2 k}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime } = c \,x^{2} y^{2}+\left (a \,x^{2}+b x \right ) y+\alpha \,x^{2}+\beta x +\gamma
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime } = c \,x^{2} y^{2}+\left (a \,x^{n}+b \right ) x y+\alpha \,x^{2 n}+\beta \,x^{n}+\gamma
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime } = \left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2}
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime }+\lambda \left (1-2 y x +y^{2}\right ) = 0
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha } = 0
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma = 0
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (a x +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +\lambda c
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime } = a \,x^{3} y^{2}+x \left (b x +c \right ) y+\alpha x +\beta
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}a \left (x^{2}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+b x \left (x^{2}-1\right ) y+c \,x^{2}+d x +s = 0
\] |
[_rational, _Riccati] |
✗ |
✗ |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = c y^{2}-b \,x^{m -1} y+a \,x^{n -2}
\] |
[_rational, _Riccati] |
✓ |
✗ |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = a \,x^{n -2} y^{2}+b \,x^{m -1} y+c
\] |
[_rational, _Riccati] |
✗ |
✗ |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = \alpha \,x^{k} y^{2}+\beta \,x^{s} y-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \sigma y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{\lambda x} y-a b \,{\mathrm e}^{\lambda x}-b^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4}
\] |
[_Riccati] |
✓ |
✗ |
|
|
\[
{}y^{\prime } = b \,{\mathrm e}^{\mu x} y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x}
\] |
[_Riccati] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\mu x} y^{2}+a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x} y-b \lambda \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{k n x}+d \,{\mathrm e}^{k \left (2 n +1\right ) x}
\] |
[_Riccati] |
✗ |
✓ |
|
|
\[
{}\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime } = y^{2}+k \,{\mathrm e}^{\nu x} y-m^{2}+k m \,{\mathrm e}^{\nu x}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b n \,x^{n -1}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b \lambda \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) y+c \,x^{n}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } x = a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y+c \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}x^{4} \left (y^{\prime }-y^{2}\right ) = a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \sinh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sinh \left (\mu x \right ) y-d^{2}+c d \sinh \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-\lambda ^{2}+a \cosh \left (\lambda x \right )^{n} \sinh \left (\lambda x \right )^{-n -4}
\] |
[_Riccati] |
✓ |
✗ |
|
|
\[
{}\left (a \cosh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cosh \left (\mu x \right ) y-d^{2}+c d \cosh \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \tanh \left (\mu x \right ) y-d^{2}+c d \tanh \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-\lambda ^{2}+3 a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \coth \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } x = x y^{2}-a^{2} x \ln \left (\beta x \right )^{2}+a
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } x = x y^{2}-a^{2} x \ln \left (\beta x \right )^{2 k}+a k \ln \left (\beta x \right )^{k -1}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } x = a \,x^{n} y^{2}+b -a \,b^{2} x^{n} \ln \left (x \right )^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+a \left (b \ln \left (x \right )+c \right )^{n}+\frac {1}{4}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y+b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \ln \left (x \right )+b \right ) y^{\prime } = y^{2}+c \ln \left (x \right )^{n} y-\lambda ^{2}+\lambda c \ln \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \ln \left (x \right )+b \right ) y^{\prime } = \ln \left (x \right )^{n} y^{2}+c y-\lambda ^{2} \ln \left (x \right )^{n}+\lambda c
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+c \sin \left (\lambda x +a \right )^{n} \sin \left (\lambda x +b \right )^{-n -4}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+a \sin \left (b x \right )^{m} y+a \sin \left (b x \right )^{m}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \sin \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sin \left (\mu x \right ) y-d^{2}+c d \sin \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+c \cos \left (\lambda x +a \right )^{n} \cos \left (\lambda x +b \right )^{-n -4}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+a \cos \left (b x \right )^{m} y+a \cos \left (b x \right )^{m}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}\left (a \cos \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \tan \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+k \tan \left (\mu x \right ) y-d^{2}+k d \tan \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-2 a b \cot \left (a x \right ) y+b^{2}-a^{2}
\] |
[_Riccati] |
✓ |
✗ |
|
|
\[
{}y^{\prime } = a \cot \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\[
{}\left (a \cot \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+c \sin \left (\lambda x \right )^{n} \cos \left (\lambda x \right )^{-n -4}
\] |
[_Riccati] |
✓ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \cos \left (\lambda x \right )^{n} y-a \cos \left (\lambda x \right )^{n -1}
\] |
[_Riccati] |
✗ |
✓ |
|
|
\[
{}\sin \left (2 x \right )^{n +1} y^{\prime } = a y^{2} \sin \left (x \right )^{2 n}+b \cos \left (x \right )^{2 n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2}-2 \lambda ^{2} \tan \left (x \right )^{2}-2 \lambda ^{2} \cot \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arcsin \left (x \right )^{n} y+b m \,x^{m -1}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \arcsin \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arcsin \left (x \right )^{m}-n y
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda \arccos \left (x \right )^{n} y-a^{2}+a \lambda \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arccos \left (x \right )^{n} y+b m \,x^{m -1}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arccos \left (x \right )^{m}-n y
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arctan \left (x \right )^{n} y+b m \,x^{m -1}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \arctan \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arctan \left (x \right )^{m}-n y
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}-b \lambda \,x^{m} \operatorname {arccot}\left (x \right )^{n} y+b m \,x^{m -1}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } x = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \operatorname {arccot}\left (x \right )^{m}-n y
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )+a n \,x^{n -1}-a^{2} x^{2 n} f \left (x \right )
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } x = x^{2 n} f \left (x \right ) y^{2}+\left (a \,x^{n} f \left (x \right )-n \right ) y+b f \left (x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )+g \left (x \right ) y+a n \,x^{n -1}-a \,x^{n} g \left (x \right )-a^{2} x^{2 n} f \left (x \right )
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a \,x^{n} g \left (x \right ) y+a n \,x^{n -1}+a^{2} x^{2 n} \left (g \left (x \right )-f \left (x \right )\right )
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right )
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )+g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{\lambda x} g \left (x \right )-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right )
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a \,{\mathrm e}^{\lambda x} g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}+a^{2} {\mathrm e}^{2 \lambda x} \left (g \left (x \right )-f \left (x \right )\right )
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} f \left (x \right ) {\mathrm e}^{2 \lambda \,x^{2}}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )+\lambda x y+a f \left (x \right ) {\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a \tanh \left (\lambda x \right )^{2} \left (a f \left (x \right )+\lambda \right )+a \lambda
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a \coth \left (\lambda x \right )^{2} \left (a f \left (x \right )+\lambda \right )+a \lambda
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a^{2} f \left (x \right )+a \lambda \sinh \left (\lambda x \right )-a^{2} f \left (x \right ) \sinh \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } x = y^{2} f \left (x \right )+a -a^{2} f \left (x \right ) \ln \left (x \right )^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } x = f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}-a
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a x \ln \left (x \right ) f \left (x \right ) y+a \ln \left (x \right )+a
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a^{2} f \left (x \right )+a \lambda \sin \left (\lambda x \right )+a^{2} f \left (x \right ) \sin \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a^{2} f \left (x \right )+a \lambda \cos \left (\lambda x \right )+a^{2} f \left (x \right ) \cos \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a \tan \left (\lambda x \right )^{2} \left (a f \left (x \right )-\lambda \right )+a \lambda
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a \cot \left (\lambda x \right )^{2} \left (a f \left (x \right )-\lambda \right )+a \lambda
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \frac {f^{\prime }\left (x \right ) y^{2}}{g \left (x \right )}-\frac {g^{\prime }\left (x \right )}{f \left (x \right )}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\[
{}f \left (x \right )^{2} y^{\prime }-f^{\prime }\left (x \right ) y^{2}+g \left (x \right ) \left (y-f \left (x \right )\right ) = 0
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+a^{2} f \left (a x +b \right )
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+\frac {f \left (\frac {1}{x}\right )}{x^{4}}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+\frac {f \left (\frac {a x +b}{c x +d}\right )}{\left (c x +d \right )^{4}}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime } = x^{4} f \left (x \right ) y^{2}+1
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime } = x^{4} y^{2}+x^{2 n} f \left (a \,x^{n}+b \right )-\frac {n^{2}}{4}+\frac {1}{4}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )+g \left (x \right ) y+h \left (x \right )
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+{\mathrm e}^{2 \lambda x} f \left ({\mathrm e}^{\lambda x}\right )-\frac {\lambda ^{2}}{4}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}-\frac {\lambda ^{2}}{4}+\frac {{\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )}{\left (c \,{\mathrm e}^{\lambda x}+d \right )^{4}}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}-\lambda ^{2}+\frac {f \left (\coth \left (\lambda x \right )\right )}{\sinh \left (\lambda x \right )^{4}}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}-\lambda ^{2}+\frac {f \left (\tanh \left (\lambda x \right )\right )}{\cosh \left (\lambda x \right )^{4}}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+f \left (a \ln \left (x \right )+b \right )+\frac {1}{4}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\cot \left (\lambda x \right )\right )}{\sin \left (\lambda x \right )^{4}}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\tan \left (\lambda x \right )\right )}{\cos \left (\lambda x \right )^{4}}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\frac {\sin \left (\lambda x +a \right )}{\sin \left (\lambda x +b \right )}\right )}{\sin \left (\lambda x +b \right )^{4}}
\] |
[_Riccati] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {2 x}{9}+A +\frac {B}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = 2 A \left (\sqrt {x}+4 A +\frac {3 A^{2}}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = A x +\frac {B}{x}-\frac {B^{2}}{x^{3}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = A \,x^{k -1}-k B \,x^{k}+k \,B^{2} x^{2 k -1}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {A}{x}-\frac {A^{2}}{x^{3}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = A +B \,{\mathrm e}^{-\frac {2 x}{A}}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {2 \left (m +1\right )}{\left (m +3\right )^{2}}+A \,x^{m}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = \frac {2 m -2}{\left (m -3\right )^{2}}+\frac {2 A \left (m \left (m +3\right ) \sqrt {x}+\left (4 m^{2}+3 m +9\right ) A +\frac {3 m \left (m +3\right ) A^{2}}{\sqrt {x}}\right )}{\left (m -3\right )^{2}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {\left (2 m +1\right ) x}{4 m^{2}}+\frac {A}{x}-\frac {A^{2}}{x^{3}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = -\frac {3 x}{16}+\frac {5 A}{x^{{1}/{3}}}-\frac {12 A^{2}}{x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {A}{x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = -\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = 2 x +\frac {A}{x^{2}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = -\frac {6 X}{25}+\frac {2 A \left (2 \sqrt {x}+19 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{25}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {3 x}{8}+\frac {3 \sqrt {a^{2}+x^{2}}}{8}-\frac {a^{2}}{16 \sqrt {a^{2}+x^{2}}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {4 x}{25}+\frac {A}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {9 x}{100}+\frac {A}{x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (5 \sqrt {x}+34 A +\frac {15 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (25 \sqrt {x}+41 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{98}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {2 x}{9}+\frac {A}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = -\frac {5 x}{36}+\frac {A}{x^{{7}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {6 A \left (-3 \sqrt {x}+23 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {30 x}{121}+\frac {3 A \left (21 \sqrt {x}+35 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{242}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {3 x}{16}+\frac {A}{x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {4 A \left (-10 \sqrt {x}+27 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {A}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = \frac {A}{x^{2}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = A \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (n +1\right ) \left (n +3\right ) A^{2}}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = A \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (2 n +3\right ) A^{2}}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = A \sqrt {x}+2 A^{2}+\frac {B}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = 2 A^{2}-A \sqrt {x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {x}{4}+\frac {6 A \left (\sqrt {x}+8 A +\frac {5 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {6 x}{25}+\frac {6 A \left (2 \sqrt {x}+7 A +\frac {4 A^{2}}{\sqrt {x}}\right )}{25}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {3 x}{16}+\frac {3 A}{x^{{1}/{3}}}-\frac {12 A^{2}}{x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {3 x}{8}+\frac {3 \sqrt {b^{2}+x^{2}}}{8}+\frac {3 b^{2}}{16 \sqrt {b^{2}+x^{2}}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {9 x}{32}+\frac {15 \sqrt {b^{2}+x^{2}}}{32}+\frac {3 b^{2}}{64 \sqrt {b^{2}+x^{2}}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {3 x}{32}-\frac {3 \sqrt {a^{2}+x^{2}}}{32}+\frac {15 a^{2}}{64 \sqrt {a^{2}+x^{2}}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = A \,x^{2}-\frac {9}{625 A}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = -\frac {6}{25} x -A \,x^{2}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = \frac {6}{25} x -A \,x^{2}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = 12 x +\frac {A}{x^{{5}/{2}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {63 x}{4}+\frac {A}{x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = 2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = 2 x +2 A \left (-10 \sqrt {x}+19 A +\frac {30 A^{2}}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {28 x}{121}+\frac {2 A \left (5 \sqrt {x}+106 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{121}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (5 \sqrt {x}+262 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+A \sqrt {x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = 6 x +\frac {A}{x^{4}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-y = 20 x +\frac {A}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {15 x}{4}+\frac {A}{x^{7}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {10 x}{49}+\frac {2 A \left (4 \sqrt {x}+61 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (\sqrt {x}+166 A +\frac {55 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {4 x}{25}+\frac {A \left (7 \sqrt {x}+49 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{50}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {15 x}{4}+\frac {6 A}{x^{{1}/{3}}}-\frac {3 A^{2}}{x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {3 x}{16}+\frac {A}{x^{{1}/{3}}}+\frac {B}{x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {5 x}{36}+\frac {A}{x^{{3}/{5}}}-\frac {B}{x^{{7}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {k}{\sqrt {A \,x^{2}+B x +c}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+3 A \left (\frac {1}{49}+B \right ) \sqrt {x}+3 A^{2} \left (\frac {4}{49}-\frac {5 B}{2}\right )+\frac {15 A^{3} \left (\frac {1}{49}-\frac {5 B}{4}\right )}{4 \sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {6 x}{25}+\frac {4 B^{2} \left (\left (2-A \right ) x^{{1}/{3}}-\frac {3 B \left (2 A +1\right )}{2}+\frac {B^{2} \left (1-3 A \right )}{x^{{1}/{3}}}-\frac {A \,B^{3}}{x^{{2}/{3}}}\right )}{75}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-y = \frac {3 x}{4}-\frac {3 A \,x^{{1}/{3}}}{2}+\frac {3 A^{2}}{4 x^{{1}/{3}}}-\frac {27 A^{4}}{625 x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {6 x}{25}+\frac {7 A \,x^{{1}/{3}}}{5}+\frac {31 A^{2}}{3 x^{{1}/{3}}}-\frac {100 A^{4}}{3 x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {10 x}{49}+\frac {13 A^{2}}{5 x^{{1}/{5}}}-\frac {7 A^{3}}{20 x^{{4}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {33 x}{169}+\frac {286 A^{2}}{3 x^{{5}/{11}}}-\frac {770 A^{3}}{9 x^{{13}/{11}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {21 x}{100}+\frac {7 A^{2} \left (\frac {123}{x^{{1}/{7}}}+\frac {280 A}{x^{{5}/{7}}}-\frac {400 A^{2}}{x^{{9}/{7}}}\right )}{9}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = a x +b \,x^{m}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = -\frac {\left (m +1\right ) x}{\left (m +2\right )^{2}}+A \,x^{2 m +1}+B \,x^{3 m +1}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = a^{2} \lambda \,{\mathrm e}^{2 \lambda x}-a \left (b \lambda +1\right ) {\mathrm e}^{\lambda x}+b
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = a^{2} \lambda \,{\mathrm e}^{2 \lambda x}+a \lambda x \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\lambda x}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = 2 a^{2} \lambda \sin \left (2 \lambda x \right )+2 a \sin \left (\lambda x \right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-y = a^{2} f^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )-\frac {\left (f \left (x \right )+b \right )^{2} f^{\prime \prime }\left (x \right )}{{f^{\prime }\left (x \right )}^{3}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = \left (a x +b \right ) y+1
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime } = \frac {y}{\left (a x +b \right )^{2}}+1
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime } = \left (a -\frac {1}{a x}\right ) y+1
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime } = \frac {3 y}{\sqrt {a \,x^{{3}/{2}}+8 x}}+1
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime } = \left (\frac {a}{x^{{2}/{3}}}-\frac {2}{3 a \,x^{{1}/{3}}}\right ) y+1
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime } = a \,{\mathrm e}^{\lambda x} y+1
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime } = \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{-\lambda x}\right ) y+1
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = a y \cosh \left (x \right )+1
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = a y \sinh \left (x \right )+1
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = a \cos \left (\lambda x \right ) y+1
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = a \sin \left (\lambda x \right ) y+1
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = \left (a x +3 b \right ) y+c \,x^{3}-a b \,x^{2}-2 b^{2} x
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime } = \left (3 a x +b \right ) y-a^{2} x^{3}-a b \,x^{2}+c x
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}2 y y^{\prime } = \left (7 a x +5 b \right ) y-3 a^{2} x^{3}-2 c \,x^{2}-3 b^{2} x
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime } = \left (\left (3-m \right ) x -1\right ) y-\left (m -1\right ) a x
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+x \left (a \,x^{2}+b \right ) y+x = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }+a \left (1-\frac {1}{x}\right ) y = a^{2}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-a \left (1-\frac {b}{x}\right ) y = a^{2} b
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime } = x^{n -1} \left (\left (2 n +1\right ) x +a n \right ) y-n \,x^{2 n} \left (x +a \right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime } = a \left (-n b +x \right ) x^{n -1} y+c \left (x^{2}-\left (2 n +1\right ) b x +n \left (n +1\right ) b^{2}\right ) x^{2 n -1}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime } = \left (a \left (2 n +k \right ) x^{k}+b \right ) x^{n -1} y+\left (-a^{2} n \,x^{2 k}-a b \,x^{k}+c \right ) x^{2 n -1}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = \left (a \left (2 n +k \right ) x^{2 k}+b \left (2 m -k \right )\right ) x^{m -k -1} y-\frac {a^{2} m \,x^{4 k}+c \,x^{2 k}+b^{2} m}{x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = \frac {\left (\left (m +2 L -3\right ) x +n -2 L +3\right ) y}{x}+\left (\left (m -L -1\right ) x^{2}+\left (n -m -2 L +3\right ) x -n +L -2\right ) x^{1-2 L}
\] |
unknown |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = \left (a \left (2 n +1\right ) x^{2}+c x +b \left (2 n -1\right )\right ) x^{n -2} y-\left (n \,a^{2} x^{4}+a c \,x^{3}+n \,b^{2}+b c x +d \,x^{2}\right ) x^{2 n -3}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = \left (a \left (n -1\right ) x +b \left (2 \lambda +n \right )\right ) x^{\lambda -1} \left (a x +b \right )^{-\lambda -2} y-\left (a n x +b \left (\lambda +n \right )\right ) x^{2 \lambda -1} \left (a x +b \right )^{-2 \lambda -3}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (\left (m -1\right ) x +1\right ) y}{x} = \frac {a^{2} \left (m x +1\right ) \left (x -1\right )}{x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-a \left (1-\frac {b}{\sqrt {x}}\right ) y = \frac {a^{2} b}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime } = \frac {3 y}{\left (a x +b \right )^{{1}/{3}} x^{{5}/{3}}}+\frac {3}{\left (a x +b \right )^{{2}/{3}} x^{{7}/{3}}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\[
{}3 y y^{\prime } = \frac {\left (-7 \lambda s \left (3 s +4 \lambda \right ) x +6 s -2 \lambda \right ) y}{x^{{1}/{3}}}+\frac {6 \lambda s x -6}{x^{{2}/{3}}}+2 \left (\lambda s \left (3 s +4 \lambda \right ) x +5 \lambda \right ) \left (-\lambda s \left (3 s +4 \lambda \right ) x +3 s +4 \lambda \right ) x^{{1}/{3}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (6 x -1\right ) y}{2 x} = -\frac {a^{2} \left (x -1\right ) \left (4 x -1\right )}{2 x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (1+\frac {2 b}{x^{2}}\right ) y}{2} = \frac {a^{2} \left (3 x +\frac {4 b}{x}\right )}{16}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (13 x -20\right ) y}{14 x^{{9}/{7}}} = -\frac {3 a^{2} \left (x -1\right ) \left (x -8\right )}{14 x^{{11}/{17}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {5 a \left (23 x -16\right ) y}{56 x^{{9}/{7}}} = -\frac {3 a^{2} \left (x -1\right ) \left (25 x -32\right )}{56 x^{{11}/{17}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (19 x +85\right ) y}{26 x^{{18}/{13}}} = -\frac {3 a^{2} \left (x -1\right ) \left (x +25\right )}{26 x^{{23}/{13}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (13 x -18\right ) y}{15 x^{{7}/{5}}} = -\frac {4 a^{2} \left (x -1\right ) \left (x -6\right )}{15 x^{{9}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (5 x +1\right ) y}{2 \sqrt {x}} = a^{2} \left (-x^{2}+1\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {3 a \left (19 x -14\right ) x^{{7}/{5}} y}{35} = -\frac {4 a^{2} \left (x -1\right ) \left (9 x -14\right ) x^{{9}/{5}}}{35}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {3 a \left (3 x +7\right ) y}{10 x^{{13}/{10}}} = -\frac {a^{2} \left (x -1\right ) \left (x +9\right )}{5 x^{{8}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (7 x -12\right ) y}{10 x^{{7}/{5}}} = -\frac {a^{2} \left (x -1\right ) \left (x -16\right )}{10 x^{{9}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {3 a \left (13 x -8\right ) y}{20 x^{{7}/{5}}} = -\frac {a^{2} \left (x -1\right ) \left (27 x -32\right )}{20 x^{{9}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {3 a \left (3 x +11\right ) y}{14 x^{{10}/{7}}} = -\frac {a^{2} \left (x -1\right ) \left (x -27\right )}{14 x^{{13}/{7}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (x +1\right ) y}{2 x^{{7}/{4}}} = \frac {a^{2} \left (x -1\right ) \left (3 x +5\right )}{4 x^{{5}/{2}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (x +1\right ) y}{2 x^{{7}/{4}}} = \frac {a^{2} \left (x -1\right ) \left (x +5\right )}{4 x^{{5}/{2}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (4 x +3\right ) y}{14 x^{{8}/{7}}} = -\frac {a^{2} \left (x -1\right ) \left (16 x +5\right )}{14 x^{{9}/{7}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (13 x -3\right ) y}{6 x^{{2}/{3}}} = -\frac {a^{2} \left (x -1\right ) \left (5 x -1\right )}{6 x^{{1}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (8 x -1\right ) y}{28 x^{{8}/{7}}} = \frac {a^{2} \left (x -1\right ) \left (32 x +3\right )}{28 x^{{9}/{7}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (5 x -4\right ) y}{x^{4}} = \frac {a^{2} \left (x -1\right ) \left (3 x -1\right )}{x^{7}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {2 a \left (3 x -10\right ) y}{5 x^{4}} = \frac {a^{2} \left (x -1\right ) \left (8 x -5\right )}{5 x^{7}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (39 x -4\right ) y}{42 x^{{9}/{7}}} = -\frac {a^{2} \left (x -1\right ) \left (9 x -1\right )}{42 x^{{11}/{7}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (x -2\right ) y}{x} = \frac {2 a^{2} \left (x -1\right )}{x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (3 x -2\right ) y}{x} = -\frac {2 a^{2} \left (x -1\right )^{2}}{x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (1-\frac {b}{x^{2}}\right ) y}{x} = \frac {a^{2} b}{x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (3 x -4\right ) y}{4 x^{{5}/{2}}} = \frac {a^{2} \left (x -1\right ) \left (x +2\right )}{4 x^{4}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (33 x +2\right ) y}{30 x^{{6}/{5}}} = -\frac {a^{2} \left (x -1\right ) \left (9 x -4\right )}{30 x^{{7}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (x -8\right ) y}{8 x^{{5}/{2}}} = -\frac {a^{2} \left (x -1\right ) \left (3 x -4\right )}{8 x^{4}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (17 x +18\right ) y}{30 x^{{22}/{15}}} = -\frac {a^{2} \left (x -1\right ) \left (x +4\right )}{30 x^{{29}/{15}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (6 x -13\right ) y}{13 x^{{5}/{2}}} = -\frac {a^{2} \left (x -1\right ) \left (x -13\right )}{26 x^{4}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (24 x +11\right ) x^{{27}/{20}} y}{30} = -\frac {a^{2} \left (x -1\right ) \left (9 x +1\right )}{60 x^{{17}/{10}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {2 a \left (3 x +2\right ) y}{5 x^{{8}/{5}}} = \frac {a^{2} \left (x -1\right ) \left (8 x +1\right )}{5 x^{{11}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {6 a \left (4 x +1\right ) y}{5 x^{{7}/{5}}} = \frac {a^{2} \left (x -1\right ) \left (27 x +8\right )}{5 x^{{9}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (x +4\right ) y}{5 x^{{8}/{5}}} = \frac {a^{2} \left (x -1\right ) \left (3 x +7\right )}{5 x^{{3}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (x +4\right ) y}{5 x^{{8}/{5}}} = \frac {a^{2} \left (x -1\right ) \left (3 x +7\right )}{5 x^{{11}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (2 x -1\right ) y}{x^{{5}/{2}}} = \frac {a^{2} \left (x -1\right ) \left (3 x +1\right )}{2 x^{4}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (x -6\right ) y}{5 x^{{7}/{5}}} = \frac {2 a^{2} \left (x -1\right ) \left (x +4\right )}{5 x^{{9}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (21 x +19\right ) y}{5 x^{{7}/{5}}} = -\frac {2 a^{2} \left (x -1\right ) \left (9 x -4\right )}{5 x^{{9}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {3 a y}{x^{{7}/{4}}} = \frac {a^{2} \left (x -1\right ) \left (x -9\right )}{4 x^{{5}/{2}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (\left (k +1\right ) x -1\right ) y}{x^{2}} = \frac {a^{2} \left (k +1\right ) \left (x -1\right )}{x^{2}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-a \left (\left (k -2\right ) x +2 k -3\right ) x^{-k} y = a^{2} \left (k -2\right ) \left (x -1\right )^{2} x^{1-2 k}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (\left (4 k -7\right ) x -4 k +5\right ) x^{-k} y}{2} = \frac {a^{2} \left (2 k -3\right ) \left (x -1\right )^{2} x^{1-2 k}}{2}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\left (\left (2 n -1\right ) x -a n \right ) x^{-n -1} y = n \left (x -a \right ) x^{-2 n}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }-\left (\left (n +1\right ) x -a n \right ) x^{n -1} \left (x -a \right )^{-n -2} y = n \,x^{2 n} \left (x -a \right )^{-2 n -3}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-a \left (\left (2 k -3\right ) x +1\right ) x^{-k} y = a^{2} \left (k -2\right ) \left (\left (k -1\right ) x +1\right ) x^{2-2 k}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-a \left (\left (n +2 k -3\right ) x +3-2 k \right ) x^{-k} y = a^{2} \left (\left (n +k -1\right ) x^{2}-\left (n +2 k -3\right ) x +k -2\right ) x^{1-2 k}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (\left (n +2\right ) x -2\right ) x^{-\frac {2 n +1}{n}} y}{n} = \frac {a^{2} \left (\left (n +1\right ) x^{2}-2 x -n +1\right ) x^{-\frac {3 n +2}{n}}}{n}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-\frac {a \left (\frac {\left (n +4\right ) x}{n +2}-2\right ) x^{-\frac {2 n +1}{n}} y}{n} = \frac {a^{2} \left (2 x^{2}+\left (n^{2}+n -4\right ) x -\left (n -1\right ) \left (n +2\right )\right ) x^{-\frac {3 n +2}{n}}}{n \left (n +2\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+\frac {a \left (\frac {\left (3 n +5\right ) x}{2}+\frac {n -1}{n +1}\right ) x^{-\frac {n +4}{n +3}} y}{n +3} = -\frac {a^{2} \left (\left (n +1\right ) x^{2}-\frac {\left (n^{2}+2 n +5\right ) x}{n +1}+\frac {4}{n +1}\right ) x^{-\frac {n +5}{n +3}}}{2 n +6}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }-a \left (\frac {n +2}{n}+b \,x^{n}\right ) y = -\frac {a^{2} x \left (\frac {n +1}{n}+b \,x^{n}\right )}{n}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime } = \left (a \,{\mathrm e}^{x}+b \right ) y+c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime } = \left (a \left (2 \mu +\lambda \right ) {\mathrm e}^{\lambda x}+b \right ) {\mathrm e}^{\mu x} y+\left (-a^{2} \mu \,{\mathrm e}^{2 \lambda x}-a b \,{\mathrm e}^{\lambda x}+c \right ) {\mathrm e}^{2 \mu x}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = \left (a \,{\mathrm e}^{\lambda x}+b \right ) y+c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime } = {\mathrm e}^{\lambda x} \left (2 a \lambda x +a +b \right ) y-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime } = {\mathrm e}^{a x} \left (2 a \,x^{2}+b +2 x \right ) y+{\mathrm e}^{2 a x} \left (-a \,x^{4}-b \,x^{2}+c \right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+a \left (2 b x +1\right ) {\mathrm e}^{b x} y = -a^{2} b \,x^{2} {\mathrm e}^{2 b x}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y = -a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime }+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y = -a^{2} b \,x^{{3}/{2}} {\mathrm e}^{4 b \sqrt {x}}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime } = \left (a \cosh \left (x \right )+b \right ) y-a b \sinh \left (x \right )+c
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = \left (a \sinh \left (x \right )+b \right ) y-a b \cosh \left (x \right )+c
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = \left (2 \ln \left (x \right )+a +1\right ) y+x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\[
{}y y^{\prime } = \left (2 \ln \left (x \right )^{2}+2 \ln \left (x \right )+a \right ) y+x \left (-\ln \left (x \right )^{4}-a \ln \left (x \right )^{2}+b \right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = a x \cos \left (\lambda \,x^{2}\right ) y+x
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y y^{\prime } = a x \sin \left (\lambda \,x^{2}\right ) y+x
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}\left (y+a k \,x^{2}+b x +c \right ) y^{\prime } = -a y^{2}+2 a k x y+m y+k \left (k +b -m \right ) x +s
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}\left (y+a \,x^{n +1}+b \,x^{n}\right ) y^{\prime } = \left (a n \,x^{n}+c \,x^{n -1}\right ) y
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}x y y^{\prime } = a y^{2}+b y+c \,x^{n}+s
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
✗ |
|
|
\[
{}x y y^{\prime } = -n y^{2}+a \left (2 n +1\right ) x y+b y-a^{2} n \,x^{2}-a b x +c
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }-\left (a \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (a \,x^{2}+b c x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-a \,x^{n -2} \left (a \,x^{n}+n +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }-\left (b \,x^{2}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}+a \,x^{n}+n \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}-a \,x^{n}+n \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x +\left (n -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime } x +2 n y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a x y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a x y^{\prime }+b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a x y^{\prime }+\left (b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+2 a x y^{\prime }+\left (b \,x^{4}+a^{2} x^{2}+c x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (-c \,x^{2 n}+a \,x^{n +1}+b \,x^{n}+n \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+a \,x^{n} y^{\prime }+b \,x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a \,x^{n} y^{\prime }-b \left (a \,x^{n +m}+b \,x^{2 m}+m \,x^{m -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (a^{2} x^{2 n}+b \,x^{2 m}+a n \,x^{n -1}+c \,x^{m -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}+b -c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (a b \,x^{n}-a \,x^{n -1}+b^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a b \,x^{n}+b \,x^{n -1}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{n}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a b \,x^{n}+2 b \,x^{n -1}-a^{2} x \right ) y^{\prime }+a \left (a b \,x^{n}+b \,x^{n -1}-a^{2} x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+x^{n} \left (a \,x^{2}+\left (a c +b \right ) x +b c \right ) y^{\prime }-x^{n} \left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+c \left (a \,x^{n}+b \,x^{m}-c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (a b \,x^{n +m}+b \left (m +1\right ) x^{m -1}-a \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (a b \,x^{n +m}+b c \,x^{m}+a n \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+a y^{\prime }+\left (b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+a y^{\prime }+b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x y^{\prime \prime }+\left (b -x \right ) y^{\prime }-a y = 0
\] |
[_Laguerre] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (\left (a +b \right ) x +n +m \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+b \,x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x y^{\prime \prime }+\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (b -2\right ) x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{2}+B x +\operatorname {C0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b x +2\right ) y^{\prime }+\left (c \,x^{2}+d x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n}-a \,x^{n -1}-b^{2} x +2 b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}-c x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+\left (a b \,x^{n}+b -3 n +1\right ) y^{\prime }+a^{2} n \left (b -n \right ) x^{2 n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{2 n -1}+d \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \,x^{n -1}+2\right ) y^{\prime }+b \,x^{n -2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b x \right ) y^{\prime }+\left (a b \,x^{n}+a n \,x^{n -1}-b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+\left (a b \,x^{n}+b \,x^{n -1}+a x -1\right ) y^{\prime }+a^{2} b \,x^{n} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+\left (a b \,x^{n +m}+a n \,x^{n}+b \,x^{m}+1-2 n \right ) y^{\prime }+a^{2} b n \,x^{2 n +m -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a x +b \right ) y^{\prime \prime }+s \left (c x +d \right ) y^{\prime }-s^{2} \left (\left (a +c \right ) x +b +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a_{2} x +b_{2} \right ) y^{\prime \prime }+\left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x +b_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+2 a b x +b^{2}-b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (2 b -1\right ) x^{2}+b \left (b +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{2 n}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2 n} \left (b \,x^{n}+c \right )^{m}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (b -n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a b \,x^{n}+a c \,x^{n -1}+b^{2} x^{2}+2 b c x +c^{2}-c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n +2 m}-b^{2} x^{4 m +2}+a m \,x^{n -1}-m^{2}-m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (b +n -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{n +2}+b \,x^{2}+c \right ) y^{\prime }+\left (a n \,x^{n +1}+a c \,x^{n}+b c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+n \left (n -1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +\nu \left (\nu +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (\nu +n +1\right ) \left (\nu -n \right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (\nu -n +1\right ) \left (\nu +n \right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+\left (2 a +1\right ) y^{\prime }-b \left (2 a +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (2 a -3\right ) x y^{\prime }+\left (n +1\right ) \left (n +2 a -1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\beta -\alpha -\left (\alpha +\beta +2\right ) x \right ) y^{\prime }+n \left (n +\alpha +\beta +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (2 n +1\right ) a x y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\left (2 a \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x \left (x -1\right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\[
{}x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (x +k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }-\left (a \,x^{n}-a b \,x^{n -1}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) y^{\prime }+s x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (c -2 a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (k -1\right ) \left (\left (-a k +n \right ) x +m -b k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\left (m -a \right ) x^{2}+\left (2 c m -1\right ) x -c \right ) y^{\prime }+\left (-2 m x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (-2 \left (a +n \right ) x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}\left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x \left (a \,x^{2}+b x +1\right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y^{\prime }+\left (n x +m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x \left (x -1\right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }-\left (-\lambda ^{2}+x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\left (\alpha \gamma +\beta \right ) x +\beta \lambda \right ) y^{\prime }-\left (\alpha x +\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-\lambda x +x^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \left (2-k \right ) x^{2}+b \left (1-k \right ) x -c k \right ) y^{\prime }+\lambda \,x^{k +1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{4} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{4} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a \,x^{n -1}+a b \,x^{n -2}+b^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}a \,x^{2} \left (x -1\right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }-\left (\nu \left (\nu +1\right ) \left (x^{2}-1\right )+n^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right )^{2} y^{\prime \prime }-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (\nu \left (\nu +1\right ) \left (-x^{2}+1\right )-\mu ^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}a \left (x^{2}-1\right )^{2} y^{\prime \prime }+b x \left (x^{2}-1\right ) y^{\prime }+\left (c \,x^{2}+d x +e \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+a \right )^{2} y^{\prime \prime }+b \,x^{n} \left (x^{2}+a \right ) y^{\prime }-\left (b \,x^{n +1}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (x^{2}+a \right )^{2} y^{\prime \prime }+b \,x^{n} \left (x^{2}+a \right ) y^{\prime }-m \left (b \,x^{n +1}+\left (m -1\right ) x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }+\left (\left (x^{2}-1\right ) \left (a^{2} x^{2}-\lambda \right )-m^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+\left (\left (x^{2}+1\right ) \left (a^{2} x^{2}-\lambda \right )+m^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{n} y^{\prime \prime }+c \left (a x +b \right )^{n -4} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{n} y^{\prime \prime }+a x y^{\prime }-\left (b^{2} x^{n}+2 b \,x^{n -1}+a b x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (a \,x^{n -1}+b x \right ) y^{\prime }+\left (a -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (2 x^{n -1}+a \,x^{2}+b x \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{n}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (a \,x^{n}-x^{n -1}+a b x +b \right ) y^{\prime }+a^{2} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (a \,x^{n +m}+1\right ) y^{\prime }+a \,x^{m} \left (1+m \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (a \,x^{n}+b \right ) y^{\prime \prime }+\left (c \,x^{n}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{n}+d -b \lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (a -b \right ) x^{n}+a -n \right ) y^{\prime }+b \left (1-a \right ) x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a^{2} \left (n +1\right ) x^{2 n}+n -1\right ) y^{\prime }-\nu \left (\nu +1\right ) a^{2} n^{2} x^{2 n} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a p \,x^{n}+q \right ) y^{\prime }+\left (a r \,x^{n}+s \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}\left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{n -2} \left (\left (b -1\right ) x^{n}+a \left (n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}\left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) \left (c \,x^{n}+d \right ) y^{\prime }+n \left (-a d +b c \right ) x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{n}+a \right )^{2} y^{\prime \prime }+b \,x^{m} \left (x^{n}+a \right ) y^{\prime }-x^{n -2} \left (b \,x^{m +1}+a n -a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+c \,x^{m} \left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{m}-a n \,x^{n -1}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (n +1\right ) x \left (a^{2} x^{2 n}-b^{2}\right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (a \,x^{n +1}+b \,x^{n}+c \right )^{2} y^{\prime \prime }+\left (\alpha \,x^{n}+\beta \,x^{n -1}+\gamma \right ) y^{\prime }+\left (n \left (-a n -a +\alpha \right ) x^{n -1}+\left (n -1\right ) \left (-b n +\beta \right ) x^{n -2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda ^{2}-x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}2 \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+a n \,x^{n -1} b m \,x^{m -1} y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+a \left (\lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{2 \lambda x}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+c \,{\mathrm e}^{2 \lambda x}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{\mu x} \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+\mu \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+2 k \,{\mathrm e}^{\mu x} y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+k^{2} {\mathrm e}^{2 \mu x}+k \mu \,{\mathrm e}^{\mu x}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+c \left (a \,{\mathrm e}^{\lambda x}+b -c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a +b \,{\mathrm e}^{2 \lambda x}\right ) y^{\prime }+\lambda \left (a -\lambda -b \,{\mathrm e}^{2 \lambda x}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (b -\lambda \right ) {\mathrm e}^{2 \lambda x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+c \,{\mathrm e}^{\mu x}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b -\lambda \right ) y^{\prime }+\left (c \,{\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+b^{2}-b \lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +b c -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{2 \mu x}+c \,{\mathrm e}^{\mu x}+k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b -\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \mu x}+d \,{\mathrm e}^{\mu x}+k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}\right ) y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\mu x}+\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{2 \mu x}+b \right ) y^{\prime }+\mu \left ({\mathrm e}^{\lambda x} \left (b -a \,{\mathrm e}^{2 \mu x}\right )-\mu \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+a c \,{\mathrm e}^{\lambda x}+b \mu \,{\mathrm e}^{\mu x}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right ) = \left (x^{2}+y^{2}+x \right ) \left (y^{\prime } x -y\right )
\] |
[_rational] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{4}+x^{2} y^{3}+x y^{2}+y+\left (x^{4} y^{3}-x^{3} y^{2}-x^{3} y+x \right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
✗ |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }-2 n x \left (x +1\right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{4} y^{\prime \prime }+2 x^{3} \left (x +1\right ) y^{\prime }+n^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}\left (x^{2}-2 x +2\right ) y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+2 y^{\prime } x -2 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime } x +y = -x^{2}+1
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{3}-x \right ) y^{\prime \prime \prime }+\left (8 x^{2}-3\right ) y^{\prime \prime }+14 y^{\prime } x +4 y = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
✓ |
|
|
\[
{}2 x^{3} y y^{\prime \prime \prime }+6 x^{3} y^{\prime } y^{\prime \prime }+18 x^{2} y y^{\prime \prime }+18 x^{2} {y^{\prime }}^{2}+36 x y y^{\prime }+6 y^{2} = 0
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} \left (-x^{3}+1\right ) y^{\prime \prime }-x^{3} y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-5 x y^{\prime \prime }+\left (4 x^{4}+5\right ) y^{\prime }-8 x^{3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{2} y y^{\prime \prime }+\left (y^{\prime } x -y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }-\left (y^{\prime } x -y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{2} \ln \left (y\right )-x^{2} y^{2}
\] |
[[_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}\sin \left (x \right )^{2} y^{\prime \prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{3}+1\right ) y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+18 y^{\prime } x +6 y = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
✓ |
|
|
\[
{}x x^{\prime } = 1-x t
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}3 x^{2} y+2-\left (x^{3}+y\right ) y^{\prime } = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x +x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\left (x^{4}-2 x^{3}+x^{2}\right ) y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }+x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}\left (x^{5}+x^{4}-6 x^{3}\right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (x -2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = x y^{3}+x^{2}
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}{y^{\prime \prime \prime }}^{2}+{y^{\prime \prime }}^{2} = 1
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
✓ |
|
|
\[
{}m x^{\prime \prime } = f \left (x\right )
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=y \left (t \right ) \\ y^{\prime }\left (t \right )=\frac {y \left (t \right )^{2}}{x \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \sin \left (y x \right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = t \ln \left (y^{2 t}\right )+t^{2}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✓ |
|
|
\[
{}y^{\prime } = \ln \left (y x \right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+y x = \sin \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+y y^{\prime } = 1
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+y y^{\prime \prime \prime \prime } = 1
\] |
[[_high_order, _missing_x], [_high_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+y x = \cosh \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+y x = \cosh \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}2 y^{\prime \prime }+3 y^{\prime }+4 x^{2} y = 1
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime }+x y^{\prime \prime }-y^{2} = \sin \left (x \right )
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}\sin \left (y^{\prime \prime }\right )+y y^{\prime \prime \prime \prime } = 1
\] |
[[_high_order, _missing_x], [_high_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\sinh \left (x \right ) {y^{\prime }}^{2}+y^{\prime \prime } = y x
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}{y^{\prime \prime \prime }}^{2}+\sqrt {y} = \sin \left (x \right )
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}\left (x -3\right ) y^{\prime \prime }+\ln \left (x \right ) y = x^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cot \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+\left (x -1\right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x y^{\prime \prime }+2 x^{2} y^{\prime }+y \sin \left (x \right ) = \sinh \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
|
|
\[
{}\sin \left (x \right ) y^{\prime \prime }+y^{\prime } x +7 y = 1
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }-\left (x -1\right ) y^{\prime }+x^{2} y = \tan \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\frac {k x}{y^{4}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+2 \left (1-x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}\ln \left (x^{2}+1\right ) y^{\prime \prime }+\frac {4 x y^{\prime }}{x^{2}+1}+\frac {\left (-x^{2}+1\right ) y}{\left (x^{2}+1\right )^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }+\left (6 x y^{2}+1\right ) y^{\prime }+2 y^{3}+1 = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✗ |
|
|
\[
{}\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime \prime }+\left (7 \sin \left (x \right )+4 \cos \left (x \right )\right ) y^{\prime }+10 y \cos \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+\frac {\left (x -1\right ) y^{\prime }}{x}+\frac {y}{x^{3}} = \frac {{\mathrm e}^{-\frac {1}{x}}}{x^{3}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
|
|
\[
{}t^{2} y^{\prime \prime }-6 t y^{\prime }+\sin \left (2 t \right ) y = \ln \left (t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+3 y^{\prime }+\frac {y}{t} = t
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+t y^{\prime }-y \ln \left (t \right ) = \cos \left (2 t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
|
|
\[
{}t^{3} y^{\prime \prime }-2 t y^{\prime }+y = t^{4}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✗ |
✓ |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime }-2 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✗ |
✓ |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\[
{}y^{\prime } \left (x^{2} y^{3}+y x \right ) = 1
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
|
|
\[
{}x^{\prime \prime }+x^{\prime }+x-x^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}x^{\prime \prime }+x^{\prime }+x+x^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime \prime }-4 y^{\prime } x +6 y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = x^{3}+y^{3}
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \frac {1}{\sqrt {15-x^{2}-y^{2}}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}\sqrt {1-x}\, y^{\prime \prime }-4 y = \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
|
|
\[
{}\left (x^{2}-4\right ) y^{\prime \prime }+\ln \left (x \right ) y = x \,{\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=\frac {2 y_{1} \left (x \right )}{x}-\frac {y_{2} \left (x \right )}{x^{2}}-3+\frac {1}{x}-\frac {1}{x^{2}} \\ y_{2}^{\prime }\left (x \right )=2 y_{1} \left (x \right )+1-6 x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=\frac {5 y_{1} \left (x \right )}{x}+\frac {4 y_{2} \left (x \right )}{x}-2 x \\ y_{2}^{\prime }\left (x \right )=-\frac {6 y_{1} \left (x \right )}{x}-\frac {5 y_{2} \left (x \right )}{x}+5 x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=\sin \left (x \right ) y_{1} \left (x \right )+\sqrt {x}\, y_{2} \left (x \right )+\ln \left (x \right ) \\ y_{2}^{\prime }\left (x \right )=\tan \left (x \right ) y_{1} \left (x \right )-{\mathrm e}^{x} y_{2} \left (x \right )+1 \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=\sin \left (x \right ) y_{1} \left (x \right )+\sqrt {x}\, y_{2} \left (x \right )+\ln \left (x \right ) \\ y_{2}^{\prime }\left (x \right )=\tan \left (x \right ) y_{1} \left (x \right )-{\mathrm e}^{x} y_{2} \left (x \right )+1 \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )={\mathrm e}^{-x} y_{1} \left (x \right )-\sqrt {x +1}\, y_{2} \left (x \right )+x^{2} \\ y_{2}^{\prime }\left (x \right )=\frac {y_{1} \left (x \right )}{\left (x -2\right )^{2}} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )={\mathrm e}^{-x} y_{1} \left (x \right )-\sqrt {x +1}\, y_{2} \left (x \right )+x^{2} \\ y_{2}^{\prime }\left (x \right )=\frac {y_{1} \left (x \right )}{\left (x -2\right )^{2}} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=2 x y_{1} \left (x \right )-x^{2} y_{2} \left (x \right )+4 x \\ y_{2}^{\prime }\left (x \right )={\mathrm e}^{x} y_{1} \left (x \right )+3 \,{\mathrm e}^{-x} y_{2} \left (x \right )-\cos \left (3 x \right ) \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=\frac {5 y_{1} \left (x \right )}{x}+\frac {4 y_{2} \left (x \right )}{x} \\ y_{2}^{\prime }\left (x \right )=-\frac {6 y_{1} \left (x \right )}{x}-\frac {5 y_{2} \left (x \right )}{x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} y_{1}^{\prime }\left (x \right )=\frac {5 y_{1} \left (x \right )}{x}+\frac {4 y_{2} \left (x \right )}{x}-2 x \\ y_{2}^{\prime }\left (x \right )=-\frac {6 y_{1} \left (x \right )}{x}-\frac {5 y_{2} \left (x \right )}{x}+5 x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}y^{\prime } = 2 y^{3}+t^{2}
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \left (y-3\right ) \left (\sin \left (y\right ) \sin \left (t \right )+\cos \left (t \right )+1\right )
\] |
[‘x=_G(y,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \left (y-1\right ) \left (y-2\right ) \left (y-{\mathrm e}^{\frac {t}{2}}\right )
\] |
[_Abel] |
✗ |
✗ |
|
|
\[
{}y^{2} y^{\prime \prime } = 8 x^{2}
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}\sin \left (x +y\right )-y y^{\prime } = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{2} y^{\prime }+3 x^{2} y = \sin \left (x \right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}x y^{\prime \prime }-{y^{\prime }}^{2} = 6 x^{5}
\] |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+x^{2} y^{\prime }-4 y = x^{3}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+x^{2} y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+x^{2} y^{\prime } = 4 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime }+x^{2} y^{\prime }+4 y = y^{3}
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}y y^{\prime \prime \prime }+6 y^{\prime \prime }+3 y^{\prime } = y
\] |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{3} y^{\prime \prime \prime }-4 y^{\prime \prime }+10 y^{\prime }-12 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}t y^{\prime \prime }+y^{\prime }+t y = 0
\] |
[_Lienard] |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} t x^{\prime }\left (t \right )+2 x \left (t \right )=15 y \left (t \right ) \\ t y^{\prime }\left (t \right )=x \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right ) y \left (t \right )-6 y \left (t \right ) \\ y^{\prime }\left (t \right )=x \left (t \right )-y \left (t \right )-5 \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}y y^{\prime }+y^{4} = \sin \left (x \right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✗ |
|
|
\[
{}x {y^{\prime \prime }}^{2}+2 y = 2 x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
|
|
\[
{}x^{\prime \prime }+2 \sin \left (x\right ) = \sin \left (2 t \right )
\] |
[NONE] |
✗ |
✗ |
|
|
\[
{}4 x \left (x^{2}+y^{2}\right )-5 y+4 y \left (x^{2}+y^{2}-5 x \right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
✗ |
|
|
\[
{}y^{\prime }+t^{2} = \frac {1}{y^{2}}
\] |
[_rational] |
✗ |
✗ |
|
|
\[
{}1-y^{2} \cos \left (t y\right )+\left (t y \cos \left (t y\right )+\sin \left (t y\right )\right ) y^{\prime } = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime }-2 y = t^{2} \sqrt {y}
\] |
[_Bernoulli] |
✓ |
✓ |
|
|
\[
{}y^{4}+\left (t^{4}-t y^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✗ |
|
|
\[
{}{y^{\prime \prime }}^{2}-5 y^{\prime \prime } y^{\prime }+4 y^{2} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✗ |
|
|
\[
{}{y^{\prime \prime }}^{2}-2 y^{\prime \prime } y^{\prime }+y^{2} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✗ |
|
|
\[
{}x \left (x +1\right ) y^{\prime \prime }+\frac {y^{\prime }}{x^{2}}+5 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
|
|
\[
{}x^{\prime \prime }+x = \left \{\begin {array}{cc} \cos \left (t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right .
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right )^{2} \\ y^{\prime }\left (t \right )={\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}y^{\prime } = \sin \left (y\right )-\cos \left (x \right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
✗ |
|
|
\[
{}y^{\prime } = \sin \left (y x \right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}x^{2} y^{\prime } \cos \left (y\right )+1 = 0
\] |
[_separable] |
✓ |
✗ |
|
|
\[
{}x^{2} y^{\prime }+\cos \left (2 y\right ) = 1
\] |
[_separable] |
✓ |
✗ |
|
|
\[
{}x^{3} y^{\prime }-\sin \left (y\right ) = 1
\] |
[_separable] |
✗ |
✗ |
|
|
\[
{}x^{2} y^{\prime }+\sin \left (2 y\right ) = 1
\] |
[_separable] |
✓ |
✗ |
|
|
\[
{}y^{\prime }-2 y \,{\mathrm e}^{x} = 2 \sqrt {y \,{\mathrm e}^{x}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y^{\prime } = y \left ({\mathrm e}^{x}+\ln \left (y\right )\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\[
{}y y^{\prime }+1 = \left (x -1\right ) {\mathrm e}^{-\frac {y^{2}}{2}}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime }+x \sin \left (2 y\right ) = 2 x \,{\mathrm e}^{-x^{2}} \cos \left (y\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\[
{}y^{\prime \prime \prime } = 3 y y^{\prime }
\] |
[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
✗ |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime }+4 y = 2 \,{\mathrm e}^{x} \left (\sin \left (x \right )+7 \cos \left (x \right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
|
|
\[
{}\left (1-x \right ) y^{\prime \prime }+y^{\prime } x -y = \left (x -1\right )^{2} {\mathrm e}^{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\[
{}x^{\prime \prime }-2 {x^{\prime }}^{2}+x^{\prime }-2 x = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}x^{\prime \prime }+{\mathrm e}^{-x^{\prime }}-x = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}x^{\prime \prime }-x^{\prime }+x-x^{2} = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
|
|
\[
{}y^{\prime \prime \prime }+x \sin \left (y\right ) = 0
\] |
[NONE] |
✓ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }\left (t \right )=-2 t x_{1} \left (t \right )^{2} \\ x_{2}^{\prime }\left (t \right )=\frac {x_{2} \left (t \right )+t}{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }\left (t \right )={\mathrm e}^{t -x_{1} \left (t \right )} \\ x_{2}^{\prime }\left (t \right )=2 \,{\mathrm e}^{x_{1} \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=y \left (t \right ) \\ y^{\prime }\left (t \right )=\frac {y \left (t \right )^{2}}{x \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }\left (t \right )=\frac {x_{1} \left (t \right )^{2}}{x_{2} \left (t \right )} \\ x_{2}^{\prime }\left (t \right )=x_{2} \left (t \right )-x_{1} \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=\frac {{\mathrm e}^{-x \left (t \right )}}{t} \\ y^{\prime }\left (t \right )=\frac {x \left (t \right ) {\mathrm e}^{-y \left (t \right )}}{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=\frac {y \left (t \right )+t}{x \left (t \right )+y \left (t \right )} \\ y^{\prime }\left (t \right )=\frac {x \left (t \right )-t}{x \left (t \right )+y \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=\frac {t -y \left (t \right )}{-x \left (t \right )+y \left (t \right )} \\ y^{\prime }\left (t \right )=\frac {x \left (t \right )-t}{-x \left (t \right )+y \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=\frac {y \left (t \right )+t}{x \left (t \right )+y \left (t \right )} \\ y^{\prime }\left (t \right )=\frac {t +x \left (t \right )}{x \left (t \right )+y \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=y \left (t \right ) \\ y^{\prime \prime }\left (t \right )=x \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )+y^{\prime }\left (t \right )+x \left (t \right )=0 \\ x^{\prime }\left (t \right )+y^{\prime \prime }\left (t \right )=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=3 x \left (t \right )+y \left (t \right ) \\ y^{\prime }\left (t \right )=-2 x \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }\left (t \right )=x \left (t \right )^{2}+y \left (t \right ) \\ y^{\prime }\left (t \right )=-2 x \left (t \right ) x^{\prime }\left (t \right )+x \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=x \left (t \right )^{2}+y \left (t \right )^{2} \\ y^{\prime }\left (t \right )=2 x \left (t \right ) y \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=-\frac {1}{y \left (t \right )} \\ y^{\prime }\left (t \right )=\frac {1}{x \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=\frac {x \left (t \right )}{y \left (t \right )} \\ y^{\prime }\left (t \right )=\frac {y \left (t \right )}{x \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=\frac {y \left (t \right )}{x \left (t \right )-y \left (t \right )} \\ y^{\prime }\left (t \right )=\frac {x \left (t \right )}{x \left (t \right )-y \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=\sin \left (x \left (t \right )\right ) \cos \left (y \left (t \right )\right ) \\ y^{\prime }\left (t \right )=\cos \left (x \left (t \right )\right ) \sin \left (y \left (t \right )\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} {\mathrm e}^{t} x^{\prime }\left (t \right )=\frac {1}{y \left (t \right )} \\ {\mathrm e}^{t} y^{\prime }\left (t \right )=\frac {1}{x \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=\cos \left (x \left (t \right )\right )^{2} \cos \left (y \left (t \right )\right )^{2}+\sin \left (x \left (t \right )\right )^{2} \cos \left (y \left (t \right )\right )^{2} \\ y^{\prime }\left (t \right )=-\frac {\sin \left (2 x \left (t \right )\right ) \sin \left (2 y \left (t \right )\right )}{2} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
✗ |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }\left (t \right )=-4 x \left (t \right )-2 y \left (t \right )+\frac {2}{{\mathrm e}^{t}-1} \\ y^{\prime }\left (t \right )=6 x \left (t \right )+3 y \left (t \right )-\frac {3}{{\mathrm e}^{t}-1} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|