\[ v (v+1) y(x)+\cot (x) y'(x)+y''(x)=0 \] ✓ Mathematica : cpu = 0.114882 (sec), leaf count = 20
\[\{\{y(x)\to c_1 P_v(\cos (x))+c_2 Q_v(\cos (x))\}\}\] ✓ Maple : cpu = 1.13 (sec), leaf count = 45
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(-{\frac {v}{2}},{\frac {1}{2}}+{\frac {v}{2}};\,{\frac {1}{2}};\, \left ( \cos \left ( x \right ) \right ) ^{2})}+{\it \_C2}\,\cos \left ( x \right ) {\mbox {$_2$F$_1$}(1+{\frac {v}{2}},{\frac {1}{2}}-{\frac {v}{2}};\,{\frac {3}{2}};\, \left ( \cos \left ( x \right ) \right ) ^{2})} \right \} \]