\[ y''(x)=-a^2 n y(x) \sec ^2(a x) \left ((n-1) \sin ^2(a x)+\cos ^2(a x)\right )-a (n-1) \sin (2 a x) \sec ^2(a x) y'(x) \] ✓ Mathematica : cpu = 0.140855 (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 = 0.149 (sec), leaf count = 27
\[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( \cos \left ( ax \right ) \right ) ^{n}+{\it \_C2}\, \left ( \cos \left ( ax \right ) \right ) ^{n-1}\sin \left ( ax \right ) \right \} \]