Link to actual problem [9742] \[ \boxed {y^{\prime \prime }+\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}+\left (-a^{2}+n^{2}\right ) y=0} \]
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 &= \sinh \left (x \right )^{-n +\frac {1}{2}} \operatorname {LegendreP}\left (a -\frac {1}{2}, n -\frac {1}{2}, \cosh \left (x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sinh \left (x \right )^{n} y}{\sqrt {\sinh \left (x \right )}\, \operatorname {LegendreP}\left (a -\frac {1}{2}, n -\frac {1}{2}, \cosh \left (x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sinh \left (x \right )^{-n +\frac {1}{2}} \operatorname {LegendreQ}\left (a -\frac {1}{2}, n -\frac {1}{2}, \cosh \left (x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sinh \left (x \right )^{n} y}{\sqrt {\sinh \left (x \right )}\, \operatorname {LegendreQ}\left (a -\frac {1}{2}, n -\frac {1}{2}, \cosh \left (x \right )\right )}\right ] \\ \end{align*}