A second order nonlinear differential equation

Use the transformation $y(x) = \frac{w(x^{n-2})}{x^{n-2}}$, this will transform the ODE into $$ w''(\xi) = \frac{C}{(n-2)^2} \xi^{-2\frac{n-1}{n-2}} w(\xi)^{\frac{n+2}{n-2}}, $$ with $\xi = x^{n-2}$ (I might have made some mistakes, please check). This is the Emden-Fowler equation, and has particular solution $$ \left(\frac{(n-2)^2}{C}\right)^{\frac{n-2}{4}} \xi^{\frac{2-n}{2}}. $$ Hopefully, this will get you a bit further.