\[ a x^r y(x)^n+y''(x)=0 \] ✗ Mathematica : cpu = 0.0338408 (sec), leaf count = 0
, could not solve
DSolve[a*x^r*y[x]^n + Derivative[2][y][x] == 0, y[x], x]
✗ Maple : cpu = 0. (sec), leaf count = 0
, result contains DESol or ODESolStruc
\[y \relax (x ) = \left (\textit {\_a} \,{\mathrm e}^{\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} +c_{1}}\right )\boldsymbol {\mathrm {where}}\left [\left \{\frac {d}{d \textit {\_a}}\mathrm {\_}\mathrm {b}\left (\textit {\_a} \right )=\frac {\left (\textit {\_a}^{n} a \,n^{2}-2 \textit {\_a}^{n} a n +\textit {\_a} r n +\textit {\_a} \,r^{2}+\textit {\_a}^{n} a +2 \textit {\_a} n +3 \textit {\_a} r +2 \textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )^{3}}{\left (r +2\right )^{2}}+\frac {\left (2 r +n +3\right ) \textit {\_}b\left (\textit {\_a} \right )^{2}}{r +2}\right \}, \left \{\textit {\_a} =y \relax (x ) x^{\frac {r +2}{n -1}}, \textit {\_}b\left (\textit {\_a} \right )=-\frac {\left (r +2\right ) x^{-\frac {r +2}{n -1}}}{n x \left (\frac {d}{d x}y \relax (x )\right )-x \left (\frac {d}{d x}y \relax (x )\right )+y \relax (x ) r +2 y \relax (x )}\right \}, \left \{x ={\mathrm e}^{-\frac {\left (\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} +c_{1}\right ) \left (n -1\right )}{r +2}}, y \relax (x )=\textit {\_a} \,{\mathrm e}^{\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} +c_{1}}\right \}\right ]\]