\[ y''(x)=-a^2 n y(x) \sec ^2(a x) \left ((n-1) \sin ^2(a x)+\cos ^2(a x)\right )-a (n-1) \sin (2 a x) \sec ^2(a x) y'(x) \] ✓ Mathematica : cpu = 0.162152 (sec), leaf count = 81
\[\left \{\left \{y(x)\to c_1 e^{-i a x} \cos ^{n-1}(a x)-\frac {i c_2 e^{2 i a x} \left (\frac {1}{2} e^{-i a x}+\frac {1}{2} e^{i a x}\right )^n}{a \left (1+e^{2 i a x}\right )}\right \}\right \}\] ✓ Maple : cpu = 0.127 (sec), leaf count = 27
\[\{y \left (x \right ) = c_{2} \left (\cos ^{n -1}\left (a x \right )\right ) \sin \left (a x \right )+c_{1} \left (\cos ^{n}\left (a x \right )\right )\}\]