\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( {a}^{2}{x}^{2\,n}-1 \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.344544 (sec), leaf count = 26 \[ \text {DSolve}\left [y(x) \left (1-a^2 x^{2 n}\right )+y''(x)=0,y(x),x\right ] \]
Maple: cpu = 0.156 (sec), leaf count = 31 \[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac { {\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) + \left ( -{a}^ {2}{x}^{2\,n}+1 \right ) {\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]