\[ \left (n^2-a^2\right ) y(x)+2 n \cot (x) y'(x)+y''(x)=0 \] ✓ Mathematica : cpu = 0.158271 (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 = 0.271 (sec), leaf count = 60
\[\left \{y \left (x \right ) = \left (c_{1} \LegendreP \left (-\frac {1}{2}+\sqrt {-a^{2}+2 n^{2}}, n -\frac {1}{2}, \cos \left (x \right )\right )+c_{2} \LegendreQ \left (-\frac {1}{2}+\sqrt {-a^{2}+2 n^{2}}, n -\frac {1}{2}, \cos \left (x \right )\right )\right ) \left (\sin ^{-n +\frac {1}{2}}\left (x \right )\right )\right \}\]