Link to actual problem [9743] \[ \boxed {y^{\prime \prime }+\frac {\left (1+2 n \right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}+\left (v +n +1\right ) \left (v -n \right ) y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {\sin \left (x \right )^{n} y}{\operatorname {LegendreP}\left (v , n , \cos \left (x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {\sin \left (x \right )^{n} y}{\operatorname {LegendreQ}\left (v , n , \cos \left (x \right )\right )}\right ] \\ \end{align*}