\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*}