Link to actual problem [9645] \[ \boxed {x \left (x^{2}-1\right ) y^{\prime \prime }+\left (x^{2} a +b \right ) y^{\prime }+c x y=0} \]
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 &= \operatorname {hypergeom}\left (\left [-\frac {1}{4}+\frac {a}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, -\frac {1}{4}+\frac {a}{4}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}\right ], \left [\frac {1}{2}-\frac {b}{2}\right ], x^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {1}{4}+\frac {a}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, -\frac {1}{4}+\frac {a}{4}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}\right ], \left [\frac {1}{2}-\frac {b}{2}\right ], x^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{b +1} \operatorname {hypergeom}\left (\left [\frac {1}{4}+\frac {a}{4}+\frac {b}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, \frac {1}{4}+\frac {a}{4}+\frac {b}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}\right ], \left [\frac {3}{2}+\frac {b}{2}\right ], x^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-b} y}{x \operatorname {hypergeom}\left (\left [\frac {1}{4}+\frac {a}{4}+\frac {b}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, \frac {1}{4}+\frac {a}{4}+\frac {b}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}\right ], \left [\frac {3}{2}+\frac {b}{2}\right ], x^{2}\right )}\right ] \\ \end{align*}