Link to actual problem [9752] \[ \boxed {y^{\prime \prime }+\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}}=0} \]
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 {\left (2 a \cos \left (x \right )-\cos \left (x \right )-2\right ) \sqrt {\cos \left (\frac {x}{2}\right )}\, \sin \left (x \right )^{a -\frac {1}{2}}}{\sin \left (\frac {x}{2}\right )^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sin \left (x \right )^{-a} \sqrt {\sin \left (x \right )}\, \sin \left (\frac {x}{2}\right )^{\frac {3}{2}} y}{\left (-2+\left (2 a -1\right ) \cos \left (x \right )\right ) \sqrt {\cos \left (\frac {x}{2}\right )}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {a}{2}-\frac {3}{4}} \left (\frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )^{\frac {3}{4}-\frac {a}{2}} \operatorname {hypergeom}\left (\left [-a -\frac {1}{2}, a -\frac {1}{2}\right ], \left [\frac {3}{2}-a \right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )}{\sqrt {\sin \left (x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{-\frac {a}{2}} \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {3}{4}} \left (\frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )^{\frac {a}{2}} \sqrt {\sin \left (x \right )}\, y}{\left (\frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-a -\frac {1}{2}, a -\frac {1}{2}\right ], \left [\frac {3}{2}-a \right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}