\[ y''(x)=-\left (y(x) \csc ^2(x) \left (v (v+1) \sin ^2(x)-n^2\right )\right )-\cot (x) y'(x) \] ✓ Mathematica : cpu = 0.415428 (sec), leaf count = 22
\[\{\{y(x)\to c_1 P_v^n(\cos (x))+c_2 Q_v^n(\cos (x))\}\}\] ✓ Maple : cpu = 0.344 (sec), leaf count = 85
\[\left \{y \left (x \right ) = \left (c_{1} \hypergeom \left (\left [\frac {n}{2}-\frac {v}{2}, \frac {n}{2}+\frac {v}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+c_{2} \sqrt {\cos \left (2 x \right )+1}\, \hypergeom \left (\left [\frac {n}{2}+\frac {v}{2}+1, \frac {n}{2}-\frac {v}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right ) \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}}\right \}\]