Link to actual problem [9755] \[ \boxed {y^{\prime \prime }+\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}}=0} \]
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 {\sqrt {\cos \left (x \right )}\, \left (-\frac {1}{2}+\frac {\cos \left (2 x \right )}{2}\right )^{\frac {1}{2}+\frac {\sqrt {-4 a +1-4 c}}{4}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {-4 a +1-4 c}}{4}-\frac {\sqrt {-a +b}}{2}+\frac {1}{4}, \frac {\sqrt {-4 a +1-4 c}}{4}+\frac {\sqrt {-a +b}}{2}+\frac {1}{4}\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 {\sqrt {-4 a +1-4 c}}{4}} \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 {\sqrt {-4 a +1-4 c}}{4}-\frac {\sqrt {-a +b}}{2}+\frac {1}{4}, \frac {\sqrt {-4 a +1-4 c}}{4}+\frac {\sqrt {-a +b}}{2}+\frac {1}{4}\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 {1}{2}+\frac {\sqrt {-4 a +1-4 c}}{4}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {-4 a +1-4 c}}{4}+\frac {\sqrt {-a +b}}{2}+\frac {3}{4}, \frac {\sqrt {-4 a +1-4 c}}{4}-\frac {\sqrt {-a +b}}{2}+\frac {3}{4}\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 {\sqrt {-4 a +1-4 c}}{4}} \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 [\frac {\sqrt {-4 a +1-4 c}}{4}+\frac {\sqrt {-a +b}}{2}+\frac {3}{4}, \frac {\sqrt {-4 a +1-4 c}}{4}-\frac {\sqrt {-a +b}}{2}+\frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}