\[ y''(x)=-\frac {1}{4} y(x) \csc ^2(x) \left (-4 n^2+4 v (v+1) \sin ^2(x)-\cos ^2(x)+2\right ) \] ✓ Mathematica : cpu = 0.548315 (sec), leaf count = 33
\[\left \{\left \{y(x)\to \sqrt [4]{-\sin ^2(x)} \left (c_1 P_v^n(\cos (x))+c_2 Q_v^n(\cos (x))\right )\right \}\right \}\]
✓ Maple : cpu = 0.281 (sec), leaf count = 113
\[ \left \{ y \left ( x \right ) ={1\sqrt {-2\,\cos \left ( 2\,x \right ) +2}\sqrt [4]{2\,\cos \left ( 2\,x \right ) +2} \left ( {\frac {\cos \left ( 2\,x \right ) }{2}}-{\frac {1}{2}} \right ) ^{{\frac {n}{2}}} \left ( {\mbox {$_2$F$_1$}(1+{\frac {v}{2}}+{\frac {n}{2}},{\frac {1}{2}}-{\frac {v}{2}}+{\frac {n}{2}};\,{\frac {3}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})}\sqrt {2\,\cos \left ( 2\,x \right ) +2}{\it \_C2}+{\mbox {$_2$F$_1$}(-{\frac {v}{2}}+{\frac {n}{2}},{\frac {1}{2}}+{\frac {v}{2}}+{\frac {n}{2}};\,{\frac {1}{2}};\,{\frac {\cos \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})}{\it \_C1} \right ) {\frac {1}{\sqrt {\sin \left ( 2\,x \right ) }}}} \right \} \]