\[ y''(x)=y(x) \csc ^2(x) \left (-\left (v (v+1) \sin ^2(x)-n^2\right )\right )-\cot (x) y'(x) \] ✓ Mathematica : cpu = 0.360065 (sec), leaf count = 22
\[\{\{y(x)\to c_1 P_v^n(\cos (x))+c_2 Q_v^n(\cos (x))\}\}\] ✓ Maple : cpu = 1.214 (sec), leaf count = 85
\[ \left \{ y \left ( x \right ) = \left ( {\frac {\cos \left ( 2\,x \right ) }{2}}-{\frac {1}{2}} \right ) ^{{\frac {n}{2}}} \left ( \sqrt {\cos \left ( 2\,x \right ) +1}{\mbox {$_2$F$_1$}(1+{\frac {v}{2}}+{\frac {n}{2}},{\frac {1}{2}}-{\frac {v}{2}}+{\frac {n}{2}};\,{\frac {3}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})}{\it \_C2}+{\mbox {$_2$F$_1$}(-{\frac {v}{2}}+{\frac {n}{2}},{\frac {1}{2}}+{\frac {v}{2}}+{\frac {n}{2}};\,{\frac {1}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})}{\it \_C1} \right ) \right \} \]