Link to actual problem [9763] \[ \boxed {y^{\prime \prime }+\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}}=0} \]
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 {\sqrt {\cos \left (x \right )}\, \left (-\frac {1}{2}+\frac {\cos \left (2 x \right )}{2}\right )^{\frac {n}{2}+\frac {1}{2}} \operatorname {hypergeom}\left (\left [-\frac {v}{2}+\frac {n}{2}, \frac {1}{2}+\frac {v}{2}+\frac {n}{2}\right ], \left [\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 {2 \left (-\frac {1}{2}+\frac {\cos \left (2 x \right )}{2}\right )^{-\frac {n}{2}} \sqrt {\sin \left (2 x \right )}\, y}{\sqrt {\cos \left (x \right )}\, \sqrt {2 \cos \left (2 x \right )-2}\, \operatorname {hypergeom}\left (\left [-\frac {v}{2}+\frac {n}{2}, \frac {1}{2}+\frac {v}{2}+\frac {n}{2}\right ], \left [\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 {\cos \left (x \right )^{\frac {3}{2}} \left (-\frac {1}{2}+\frac {\cos \left (2 x \right )}{2}\right )^{\frac {n}{2}+\frac {1}{2}} \operatorname {hypergeom}\left (\left [1+\frac {v}{2}+\frac {n}{2}, \frac {1}{2}-\frac {v}{2}+\frac {n}{2}\right ], \left [\frac {3}{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 {2 \left (-\frac {1}{2}+\frac {\cos \left (2 x \right )}{2}\right )^{-\frac {n}{2}} \sqrt {\sin \left (2 x \right )}\, y}{\cos \left (x \right )^{\frac {3}{2}} \sqrt {2 \cos \left (2 x \right )-2}\, \operatorname {hypergeom}\left (\left [1+\frac {v}{2}+\frac {n}{2}, \frac {1}{2}-\frac {v}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}