\[ y''(x)=-a y(x) \csc ^2(x) \] ✓ Mathematica : cpu = 0.0562062 (sec), leaf count = 70
DSolve[Derivative[2][y][x] == -(a*Csc[x]^2*y[x]),y[x],x]
\[\left \{\left \{y(x)\to c_1 \sqrt [4]{\cos ^2(x)-1} P_{-\frac {1}{2}}^{\frac {1}{2} \sqrt {1-4 a}}(\cos (x))+c_2 \sqrt [4]{\cos ^2(x)-1} Q_{-\frac {1}{2}}^{\frac {1}{2} \sqrt {1-4 a}}(\cos (x))\right \}\right \}\] ✓ Maple : cpu = 0.25 (sec), leaf count = 132
dsolve(diff(diff(y(x),x),x) = -a/sin(x)^2*y(x),y(x))
\[y \left (x \right ) = \frac {\sqrt {-2 \cos \left (2 x \right )+2}\, \left (2 \cos \left (2 x \right )+2\right )^{\frac {1}{4}} \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {\sqrt {1-4 a}}{4}} \left (\hypergeom \left (\left [\frac {\sqrt {1-4 a}}{4}+\frac {3}{4}, \frac {\sqrt {1-4 a}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {2 \cos \left (2 x \right )+2}\, c_{2}+\hypergeom \left (\left [\frac {\sqrt {1-4 a}}{4}+\frac {1}{4}, \frac {\sqrt {1-4 a}}{4}+\frac {1}{4}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1}\right )}{\sqrt {\sin \left (2 x \right )}}\]