Link to actual problem [9741] \[ \boxed {y^{\prime \prime }+\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (-1+n \right )\right ) y}{\sinh \left (x \right )^{2}}=0} \]
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 {\cosh \left (x \right )}\, \sinh \left (x \right )^{n +\frac {1}{2}} \operatorname {hypergeom}\left (\left [-\frac {a}{2}+\frac {n}{2}, \frac {a}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right )}{\sqrt {\sinh \left (2 x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sinh \left (x \right )^{-n} \sqrt {\sinh \left (2 x \right )}\, y}{\sqrt {\cosh \left (x \right )}\, \sqrt {\sinh \left (x \right )}\, \operatorname {hypergeom}\left (\left [-\frac {a}{2}+\frac {n}{2}, \frac {a}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cosh \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 {\cosh \left (x \right )^{\frac {3}{2}} \sinh \left (x \right )^{n +\frac {1}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {a}{2}+\frac {n}{2}, \frac {1}{2}+\frac {a}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right )}{\sqrt {\sinh \left (2 x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sinh \left (x \right )^{-n} \sqrt {\sinh \left (2 x \right )}\, y}{\cosh \left (x \right )^{\frac {3}{2}} \sqrt {\sinh \left (x \right )}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {a}{2}+\frac {n}{2}, \frac {1}{2}+\frac {a}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}