\[ y''(x)-x^{n-2} h\left (x^{-n} y(x),x^{1-n} y'(x)\right )=0 \] ✗ Mathematica : cpu = 4.43017 (sec), leaf count = 0 , could not solve
DSolve[-(x^(-2 + n)*h[y[x]/x^n, x^(1 - n)*Derivative[1][y][x]]) + Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 0.86 (sec), leaf count = 125
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\frac {{\it \_a}}{{{\rm e}^{- \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) n}}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) = \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2} \left ( -{\it \_b} \left ( {\it \_a} \right ) h \left ( {\it \_a},{\frac {{\it \_b} \left ( {\it \_a} \right ) {\it \_a}\,n+1}{{\it \_b} \left ( {\it \_a} \right ) }} \right ) +n{\it \_a}\, \left ( n-1 \right ) {\it \_b} \left ( {\it \_a} \right ) +2\,n-1 \right ) \right \} , \left \{ {\it \_a}=y \left ( x \right ) {x}^{-n},{\it \_b} \left ( {\it \_a} \right ) ={\frac {1}{{x}^{-n} \left ( x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -ny \left ( x \right ) \right ) }} \right \} , \left \{ x={{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},y \left ( x \right ) ={\frac {{\it \_a}}{{{\rm e}^{- \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right ) n}}}} \right \} ] \right ) \right \} \]