Link to actual problem [9747] \[ \boxed {\cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (-1+n \right )\right ) y=0} \]
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 &= \frac {\sin \left (x \right )^{\frac {3}{2}} \cos \left (x \right )^{n +\frac {1}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i \sqrt {a}}{2}+\frac {n}{2}, \frac {1}{2}-\frac {i \sqrt {a}}{2}+\frac {n}{2}\right ], \left [n +\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )}{\sqrt {\sin \left (2 x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\cos \left (x \right )^{-n} \sqrt {\sin \left (2 x \right )}\, y}{\sin \left (x \right )^{\frac {3}{2}} \sqrt {\cos \left (x \right )}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i \sqrt {a}}{2}+\frac {n}{2}, \frac {1}{2}-\frac {i \sqrt {a}}{2}+\frac {n}{2}\right ], \left [n +\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sin \left (x \right )^{\frac {3}{2}} \left (\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{\frac {3}{4}-\frac {n}{2}} \operatorname {hypergeom}\left (\left [1+\frac {i \sqrt {a}}{2}-\frac {n}{2}, 1-\frac {i \sqrt {a}}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}-n \right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )}{\sqrt {\sin \left (2 x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{\frac {n}{2}} \sqrt {\sin \left (2 x \right )}\, y}{\sin \left (x \right )^{\frac {3}{2}} \left (\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [1+\frac {i \sqrt {a}}{2}-\frac {n}{2}, 1-\frac {i \sqrt {a}}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}-n \right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}