\[ y(x) \left (-a \sin ^2(x)-(n-1) n\right )+\sin ^2(x) y''(x)=0 \] ✓ Mathematica : cpu = 0.190951 (sec), leaf count = 90
DSolve[(-((-1 + n)*n) - a*Sin[x]^2)*y[x] + Sin[x]^2*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \sqrt [4]{\cos ^2(x)-1} P_{\frac {1}{2} i \left (2 \sqrt {a}+i\right )}^{\frac {1}{2} (2 n-1)}(\cos (x))+c_2 \sqrt [4]{\cos ^2(x)-1} Q_{\frac {1}{2} i \left (2 \sqrt {a}+i\right )}^{\frac {1}{2} (2 n-1)}(\cos (x))\right \}\right \}\] ✓ Maple : cpu = 0.258 (sec), leaf count = 120
dsolve(sin(x)^2*diff(diff(y(x),x),x)-(a*sin(x)^2+n*(n-1))*y(x),y(x))
\[y \left (x \right ) = \frac {\left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {n}{2}} \left (\left (-2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {1}{2}+\frac {i \sqrt {a}}{2}+\frac {n}{2}, \frac {1}{2}-\frac {i \sqrt {a}}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (2 \cos \left (2 x \right )+2\right )^{\frac {3}{4}} c_{2}+\left (\sqrt {\sin }\left (2 x \right )\right ) \hypergeom \left (\left [\frac {n}{2}+\frac {i \sqrt {a}}{2}, \frac {n}{2}-\frac {i \sqrt {a}}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1}\right )}{\sqrt {\sin \left (2 x \right )}}\]