Link to actual problem [11051] \[ \boxed {a \left (x^{2}-1\right )^{2} y^{\prime \prime }+b x \left (x^{2}-1\right ) y^{\prime }+\left (c \,x^{2}+d x +e \right ) y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\frac {x}{2}-\frac {1}{2}\right )^{\frac {1}{2}+\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}} \left (x^{2}-1\right )^{-\frac {b}{4 a}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}-2 a b -4 a c +b^{2}}-\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}, \frac {1}{2}-\frac {-\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}-2 a b -4 a c +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}\right ], \left [1-\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {x}{2}+\frac {1}{2}\right ) \left (\frac {x}{2}+\frac {1}{2}\right )^{\frac {1}{2}-\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {4 \left (\frac {x}{2}-\frac {1}{2}\right )^{-\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}} \left (x^{2}-1\right )^{\frac {b}{4 a}} \left (\frac {x}{2}+\frac {1}{2}\right )^{\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}} y}{\sqrt {2 x -2}\, \operatorname {hypergeom}\left (\left [\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, -\frac {-\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}-2 a}{4 a}\right ], \left [-\frac {-2 a +\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {x}{2}+\frac {1}{2}\right ) \sqrt {2 x +2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\frac {x}{2}-\frac {1}{2}\right )^{\frac {1}{2}+\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}} \left (x^{2}-1\right )^{-\frac {b}{4 a}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}-2 a b -4 a c +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}, \frac {1}{2}-\frac {-\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}-2 a b -4 a c +b^{2}}-\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}\right ], \left [1+\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {x}{2}+\frac {1}{2}\right ) \left (\frac {x}{2}+\frac {1}{2}\right )^{\frac {1}{2}+\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {4 \left (\frac {x}{2}-\frac {1}{2}\right )^{-\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}} \left (x^{2}-1\right )^{\frac {b}{4 a}} \left (\frac {x}{2}+\frac {1}{2}\right )^{-\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}} y}{\sqrt {2 x -2}\, \operatorname {hypergeom}\left (\left [\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}-2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [\frac {2 a +\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {x}{2}+\frac {1}{2}\right ) \sqrt {2 x +2}}\right ] \\ \end{align*}