\[ y(x) \left (-a \sin ^2(x)-(n-1) n\right )+\sin ^2(x) y''(x)=0 \] ✓ Mathematica : cpu = 0.213276 (sec), leaf count = 90
\[\left \{\left \{y(x)\to c_1 \sqrt [4]{\cos ^2(x)-1} P_{\frac {1}{2} i \left (2 \sqrt {a}+i\right )}^{\frac {1}{2} (2 n-1)}(\cos (x))+c_2 \sqrt [4]{\cos ^2(x)-1} Q_{\frac {1}{2} i \left (2 \sqrt {a}+i\right )}^{\frac {1}{2} (2 n-1)}(\cos (x))\right \}\right \}\] ✓ Maple : cpu = 0.322 (sec), leaf count = 120
\[\left \{y \left (x \right ) = \frac {\left (c_{1} \hypergeom \left (\left [\frac {n}{2}+\frac {i \sqrt {a}}{2}, \frac {n}{2}-\frac {i \sqrt {a}}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (\sqrt {\sin }\left (2 x \right )\right )+c_{2} \left (2 \cos \left (2 x \right )+2\right )^{\frac {3}{4}} \left (-2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {n}{2}+\frac {i \sqrt {a}}{2}+\frac {1}{2}, \frac {n}{2}-\frac {i \sqrt {a}}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right ) \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}}}{\sqrt {\sin \left (2 x \right )}}\right \}\]