\begin {gather*} \boxed {y^{\prime \prime } y^{\prime }-y^{n}=0} \end {gather*} This is solved by reduction of order by using substitution which makes the dependent variable $$y$$ an independent variable. Using $$v(y) = y'(x)$$. Applying this transformation the ode becomes \begin{align*} v \left ( v v' \right )-y^{n} &= 0 \end{align*} Where from now on $$v'$$ is understood as derivative w.r.t. $$y$$. The above is now solved as ﬁrst order ode in $$v \left (y \right )$$. This is separable ﬁrst order ODE. \begin {align*} v^{2}\mathop {\mathrm {d}v}&= y^{n}\mathop {\mathrm {d}y}\\ \int v^{2}\mathop {\mathrm {d}v}&= \int y^{n}\mathop {\mathrm {d}y}\\ \frac {v^{3}}{3} &= \frac {y^{1+n}}{1+n}+c_{1} \end {align*}

Substititing $$v=y'(x)$$ gives the following ode $\frac {\left (y^{\prime }\right )^{3}}{3} = \frac {y^{1+n}}{1+n}+c_{1}$ Solving for $$y^{\prime }$$ gives \begin {align*} y^{\prime }&=\frac {\left (\left (3 c_{2} n +3 y^{1+n}+3 c_{2}\right ) \left (1+n \right )^{2}\right )^{\frac {1}{3}}}{1+n}\tag {1} \\ y^{\prime }&=-\frac {\left (\left (3 c_{2} n +3 y^{1+n}+3 c_{2}\right ) \left (1+n \right )^{2}\right )^{\frac {1}{3}}}{2 \left (1+n \right )}+\frac {i \sqrt {3}\, \left (\left (3 c_{2} n +3 y^{1+n}+3 c_{2}\right ) \left (1+n \right )^{2}\right )^{\frac {1}{3}}}{2+2 n}\tag {2} \\ y^{\prime }&=-\frac {\left (\left (3 c_{2} n +3 y^{1+n}+3 c_{2}\right ) \left (1+n \right )^{2}\right )^{\frac {1}{3}}}{2 \left (1+n \right )}-\frac {i \sqrt {3}\, \left (\left (3 c_{2} n +3 y^{1+n}+3 c_{2}\right ) \left (1+n \right )^{2}\right )^{\frac {1}{3}}}{2 \left (1+n \right )}\tag {3} \end {align*}

Each one of the above ode’s is separable. For example, for the ﬁrst ode above, integrating gives \begin{align*} \int _{0}^{y}\frac {\left (1+n \right ) 3^{\frac {2}{3}}}{3 \left (\left (1+n \right )^{2} \left (c_{2} \left (1+n \right )+\textit {\_a}^{1+n}\right )\right )^{\frac {1}{3}}}d \textit {\_a} -x -c_{1} &= 0 \end{align*} Similarly for the second and third ode’s. At the end, the following are the solutions found \begin{align*} \int _{0}^{y}\frac {\left (1+n \right ) 3^{\frac {2}{3}}}{3 \left (\left (1+n \right )^{2} \left (c_{2} \left (1+n \right )+\textit {\_a}^{1+n}\right )\right )^{\frac {1}{3}}}d \textit {\_a} -x -c_{1} &= 0 \\ \int _{0}^{y}\frac {2+2 n}{\left (\left (1+n \right )^{2} \left (c_{2} \left (1+n \right )+\textit {\_a}^{1+n}\right )\right )^{\frac {1}{3}} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right )}d \textit {\_a} -x -c_{1} &= 0 \\ \int _{0}^{y}-\frac {2+2 n}{\left (\left (1+n \right )^{2} \left (c_{2} \left (1+n \right )+\textit {\_a}^{1+n}\right )\right )^{\frac {1}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right )}d \textit {\_a} -x -c_{1} &= 0 \\ \end{align*}