\[ y''(x)-k x^a y(x)^b y'(x)^c=0 \] ✗ Mathematica : cpu = 0.0645049 (sec), leaf count = 0
, could not solve
DSolve[-(k*x^a*y[x]^b*Derivative[1][y][x]^c) + 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}^{b} \left (-\frac {\left (a -c +2\right ) \left (\textit {\_a} \textit {\_}b\left (\textit {\_a} \right )+1\right )}{\textit {\_}b\left (\textit {\_a} \right ) \left (b +c -1\right )}\right )^{c} b^{2} k +2 \textit {\_a}^{b} \left (-\frac {\left (a -c +2\right ) \left (\textit {\_a} \textit {\_}b\left (\textit {\_a} \right )+1\right )}{\textit {\_}b\left (\textit {\_a} \right ) \left (b +c -1\right )}\right )^{c} b c k +\textit {\_a}^{b} \left (-\frac {\left (a -c +2\right ) \left (\textit {\_a} \textit {\_}b\left (\textit {\_a} \right )+1\right )}{\textit {\_}b\left (\textit {\_a} \right ) \left (b +c -1\right )}\right )^{c} c^{2} k -2 \textit {\_a}^{b} \left (-\frac {\left (a -c +2\right ) \left (\textit {\_a} \textit {\_}b\left (\textit {\_a} \right )+1\right )}{\textit {\_}b\left (\textit {\_a} \right ) \left (b +c -1\right )}\right )^{c} b k -2 \textit {\_a}^{b} \left (-\frac {\left (a -c +2\right ) \left (\textit {\_a} \textit {\_}b\left (\textit {\_a} \right )+1\right )}{\textit {\_}b\left (\textit {\_a} \right ) \left (b +c -1\right )}\right )^{c} c k +\textit {\_a}^{b} \left (-\frac {\left (a -c +2\right ) \left (\textit {\_a} \textit {\_}b\left (\textit {\_a} \right )+1\right )}{\textit {\_}b\left (\textit {\_a} \right ) \left (b +c -1\right )}\right )^{c} k -\textit {\_a} \,a^{2}-\textit {\_a} a b +\textit {\_a} a c +\textit {\_a} c b -3 a \textit {\_a} -2 b \textit {\_a} +c \textit {\_a} -2 \textit {\_a} \right ) \textit {\_}b\left (\textit {\_a} \right )^{3}}{\left (a -c +2\right )^{2}}+\frac {\left (2 a +b -c +3\right ) \textit {\_}b\left (\textit {\_a} \right )^{2}}{a -c +2}\right \}, \left \{\textit {\_a} =y \relax (x ) x^{\frac {a -c +2}{b +c -1}}, \textit {\_}b\left (\textit {\_a} \right )=-\frac {\left (a -c +2\right ) x^{-\frac {a -c +2}{b +c -1}}}{b x \left (\frac {d}{d x}y \relax (x )\right )+x \left (\frac {d}{d x}y \relax (x )\right ) c +a y \relax (x )-c y \relax (x )-x \left (\frac {d}{d x}y \relax (x )\right )+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 (b +c -1\right )}{a -c +2}}, y \relax (x )=\textit {\_a} \,{\mathrm e}^{\int \textit {\_}b\left (\textit {\_a} \right )d \textit {\_a} +c_{1}}\right \}\right ]\]