\[ x^{\frac {n}{n+1}} y''(x)-y(x)^{\frac {2 n+1}{n+1}}=0 \] ✗ Mathematica : cpu = 0.0440429 (sec), leaf count = 0
DSolve[-y[x]^((1 + 2*n)/(1 + n)) + x^(n/(1 + n))*Derivative[2][y][x] == 0,y[x],x]
, could not solve
DSolve[-y[x]^((1 + 2*n)/(1 + n)) + x^(n/(1 + n))*Derivative[2][y][x] == 0, y[x], x]
✗ Maple : cpu = 0. (sec), leaf count = 0
dsolve(x^(n/(n+1))*diff(diff(y(x),x),x)-y(x)^((2*n+1)/(n+1))=0,y(x))
, result contains DESol or ODESolStruc
\[y \left (x \right ) = \left (\textit {\_a} \,{\mathrm e}^{-\frac {\left (\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} +c_{1}\right ) \left (n +2\right )}{n}}\right )\:\& \text {where}\:\left [\left \{\frac {d}{d \textit {\_a}}\textit {\_}b\left (\textit {\_a} \right )=-\frac {\left (\textit {\_a}^{\frac {2 n +1}{n +1}} n^{2}-2 \textit {\_a} \,n^{2}-6 \textit {\_a} n -4 \textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )^{3}}{n^{2}}-\frac {\left (3 n +4\right ) \textit {\_}b\left (\textit {\_a} \right )^{2}}{n}\right \}, \left \{\textit {\_a} =y \left (x \right ) x^{\frac {n +2}{n}}, \textit {\_}b\left (\textit {\_a} \right )=\frac {n \,x^{-\frac {n +2}{n}}}{n x \left (\frac {d}{d x}y \left (x \right )\right )+n y \left (x \right )+2 y \left (x \right )}\right \}, \left \{x ={\mathrm e}^{\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} +c_{1}}, y \left (x \right )=\textit {\_a} \,{\mathrm e}^{-\frac {\left (\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} +c_{1}\right ) \left (n +2\right )}{n}}\right \}\right ]\]