\[ y''(x)-x^{n-2} h\left (x^{-n} y(x),x^{1-n} y'(x)\right )=0 \] ✗ Mathematica : cpu = 3.13394 (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 = 0.821 (sec), leaf count = 125
\[\left \{y \left (x \right ) = \mathit {ODESolStruc} \left (\textit {\_a} \,{\mathrm e}^{\left (c_{1}+\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} \right ) n}, \left [\left \{\frac {d}{d \textit {\_a}}\mathrm {\_}\mathrm {b}\left (\textit {\_a} \right )=\left (\left (n -1\right ) \textit {\_a} n \textit {\_}b\left (\textit {\_a} \right )-\textit {\_}b\left (\textit {\_a} \right ) h \left (\textit {\_a} , \frac {\textit {\_a} n \textit {\_}b\left (\textit {\_a} \right )+1}{\textit {\_}b\left (\textit {\_a} \right )}\right )+2 n -1\right ) \textit {\_}b\left (\textit {\_a} \right )^{2}\right \}, \left \{\textit {\_a} =x^{-n} y \left (x \right ), \textit {\_}b\left (\textit {\_a} \right )=\frac {x^{n}}{-n y \left (x \right )+x \left (\frac {d}{d x}y \left (x \right )\right )}\right \}, \left \{x ={\mathrm e}^{c_{1}+\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a}}, y \left (x \right )=\textit {\_a} \,{\mathrm e}^{\left (c_{1}+\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} \right ) n}\right \}\right ]\right )\right \}\]