\[ v (v+1) y(x)+y''(x)+\cot (x) y'(x)=0 \] ✓ Mathematica : cpu = 0.149181 (sec), leaf count = 20
\[\left \{\left \{y(x)\to c_1 P_v(\cos (x))+c_2 Q_v(\cos (x))\right \}\right \}\]
✓ Maple : cpu = 0.246 (sec), leaf count = 45
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(-{\frac {v}{2}},{\frac {1}{2}}+{\frac {v}{2}};\,{\frac {1}{2}};\, \left ( \cos \left ( x \right ) \right ) ^{2})}+{\it \_C2}\,\cos \left ( x \right ) {\mbox {$_2$F$_1$}(1+{\frac {v}{2}},{\frac {1}{2}}-{\frac {v}{2}};\,{\frac {3}{2}};\, \left ( \cos \left ( x \right ) \right ) ^{2})} \right \} \]