Link to actual problem [9754] \[ \boxed {y^{\prime \prime }+\frac {\left (-\left (b^{2} a^{2}-\left (1+a \right )^{2}\right ) \sin \left (x \right )^{2}-a \left (1+a \right ) b \sin \left (2 x \right )-a \left (-1+a \right )\right ) y}{\sin \left (x \right )^{2}}=0} \]
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 &= \left (\cot \left (x \right )+i\right )^{-\frac {1}{2}-\frac {1}{2} i a b -\frac {1}{2} a} \left (b +\cot \left (x \right )\right ) \left (\cot \left (x \right )-i\right )^{-\frac {1}{2}+\frac {1}{2} i a b -\frac {1}{2} a}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {\cot \left (x \right )+i}\, \left (\cot \left (x \right )+i\right )^{\frac {i a b}{2}} \left (\cot \left (x \right )+i\right )^{\frac {a}{2}} \sqrt {\cot \left (x \right )-i}\, \left (\cot \left (x \right )-i\right )^{-\frac {i a b}{2}} \left (\cot \left (x \right )-i\right )^{\frac {a}{2}} y}{b +\cot \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\cot \left (x \right )+i\right )^{\frac {1}{2}+\frac {1}{2} a +\frac {1}{2} i a b} \left (\cot \left (x \right )-i\right )^{-\frac {1}{2}+\frac {1}{2} i a b -\frac {1}{2} a} \operatorname {hypergeom}\left (\left [a \left (i b +1\right ), i a b -a +1\right ], \left [i a b +a +2\right ], \frac {1}{2}-\frac {i \cot \left (x \right )}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\cot \left (x \right )+i\right )^{-\frac {a}{2}} \left (\cot \left (x \right )+i\right )^{-\frac {i a b}{2}} \sqrt {\cot \left (x \right )-i}\, \left (\cot \left (x \right )-i\right )^{-\frac {i a b}{2}} \left (\cot \left (x \right )-i\right )^{\frac {a}{2}} y}{\sqrt {\cot \left (x \right )+i}\, \operatorname {hypergeom}\left (\left [a \left (i b +1\right ), i a b -a +1\right ], \left [i a b +a +2\right ], \frac {1}{2}-\frac {i \cot \left (x \right )}{2}\right )}\right ] \\ \end{align*}