\[ \left (n^2-a^2\right ) y(x)+2 n \cot (x) y'(x)+y''(x)=0 \] ✓ Mathematica : cpu = 0.14574 (sec), leaf count = 114
\[\left \{\left \{y(x)\to c_1 \left (\cos ^2(x)-1\right )^{\frac {1}{4} (1-2 n)} P_{\frac {1}{2} \left (2 \sqrt {2 n^2-a^2}-1\right )}^{\frac {1}{2} (2 n-1)}(\cos (x))+c_2 \left (\cos ^2(x)-1\right )^{\frac {1}{4} (1-2 n)} Q_{\frac {1}{2} \left (2 \sqrt {2 n^2-a^2}-1\right )}^{\frac {1}{2} (2 n-1)}(\cos (x))\right \}\right \}\] ✓ Maple : cpu = 0.977 (sec), leaf count = 60
\[ \left \{ y \left ( x \right ) = \left ( \sin \left ( x \right ) \right ) ^{-n+{\frac {1}{2}}} \left ( {\it LegendreQ} \left ( -{\frac {1}{2}}+\sqrt {-{a}^{2}+2\,{n}^{2}},n-{\frac {1}{2}},\cos \left ( x \right ) \right ) {\it \_C2}+{\it LegendreP} \left ( -{\frac {1}{2}}+\sqrt {-{a}^{2}+2\,{n}^{2}},n-{\frac {1}{2}},\cos \left ( x \right ) \right ) {\it \_C1} \right ) \right \} \]