\[ \boxed { {x}^{4}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +{a}^{2} \left ( y \left ( x \right ) \right ) ^{n}=0} \]
Mathematica: cpu = 0.032504 (sec), leaf count = 23 \[ \text {DSolve}\left [a^2 y(x)^n+x^4 y''(x)=0,y(x),x\right ] \]
Maple: cpu = 0.764 (sec), leaf count = 160 \[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a}\,{ {\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+ {\it \_C1}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) = \left ( {\frac {{{\it \_a}}^{n}{n}^{2}{a}^{ 2}}{4}}-{\frac {{{\it \_a}}^{n}n{a}^{2}}{2}}+{\frac {{{\it \_a}}^{n}{a }^{2}}{4}}-{\frac {{\it \_a}\,n}{2}}+{\frac {3\,{\it \_a}}{2}} \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{3}+ \left ( -{\frac {n}{2}}+{\frac {5}{2}} \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2} \right \} , \left \{ {\it \_a}=y \left ( x \right ) {x}^{-2\, \left ( n-1 \right ) ^{-1}},{\it \_b} \left ( {\it \_a} \right ) =-2\,{\frac {1}{-nx{\frac {\rm d}{{\rm d}x}} y \left ( x \right ) +x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +2\,y \left ( x \right ) } \left ( {x}^{-2\, \left ( n-1 \right ) ^{-1}} \right ) ^{-1}} \right \} , \left \{ x={{\rm e}^{{\frac { \left ( \int \! {\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) n}{2}}-{\frac {\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}}{2}}-{\frac {{\it \_C1}}{2}}}},y \left ( x \right ) = {\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right \} ] \right ) \right \} \]