\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -{x}^{n-2}h \left ( {\frac {y \left ( x \right ) }{{x}^{n}}},{\frac {{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{{x}^{n-1}}} \right ) =0} \]
Mathematica: cpu = 3.899495 (sec), leaf count = 39 \[ \text {DSolve}\left [y''(x)-x^{n-2} h\left (x^{-n} y(x),x^{1-n} y'(x)\right )=0,y(x),x\right ] \]
Maple: cpu = 0.640 (sec), leaf count = 132 \[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\frac {{\it \_a }}{{{\rm e}^{- \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) n}}}},[ \left \{ {\frac {\rm d}{ {\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) = \left ( {\it \_a }\,{n}^{2}-{\it \_a}\,n-h \left ( {\it \_a},{\frac {{\it \_b} \left ( { \it \_a} \right ) {\it \_a}\,n+1}{{\it \_b} \left ( {\it \_a} \right ) }} \right ) \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{3}+ \left ( 2\,n-1 \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2} \right \} , \left \{ {\it \_a}=y \left ( x \right ) {x}^{-n},{\it \_b} \left ( {\it \_a} \right ) =-{\frac {1}{{x}^ {-n} \left ( ny \left ( x \right ) -x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) }} \right \} , \left \{ x={{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},y \left ( x \right ) ={\frac {{\it \_a}}{{{\rm e}^{- \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) n}}}} \right \} ] \right ) \right \} \]