# A second order nonlinear differential equation

How should I start to solve the following differential equation? $$xy''+(n-1)y'-Cxy^\frac{n+2}{n-2}=0,$$ where $$x>0$$, and $$C$$ is some constant.

I have very little knowledge in differential equations, tried some substitutions that did not work. I guess if the zero order term was not raised to the power, than it would be a more standard task, because we could re-write this to get a system of ODEs.

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.

Hint:

Case $$1$$: $$n=1$$ and $$C\neq0$$

Then $$xy''-Cxy^{-3}=0$$

$$\dfrac{d^2y}{dx^2}=Cy^{-3}$$

This reduces to an autonomous ODE.

Case $$2$$: $$n\neq1,2$$ and $$C\neq0$$

Then $$xy''+(n-1)y'-Cxy^\frac{n+2}{n-2}=0$$

$$xy''-(1-n)y'=Cxy^\frac{n+2}{n-2}$$

This belongs to a modified Emden–Fowler equation according to this.

Let $$r=x^{2-n}$$ ,

Then $$\dfrac{dy}{dx}=\dfrac{dy}{dr}\dfrac{dr}{dx}=(2-n)x^{1-n}\dfrac{dy}{dr}$$

$$\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}\left((2-n)x^{1-n}\dfrac{dy}{dr}\right)=(2-n)x^{1-n}\dfrac{d}{dx}\left(\dfrac{dy}{dr}\right)+(2-n)(1-n)x^{-n}\dfrac{dy}{dr}=(2-n)x^{1-n}\dfrac{d}{dr}\left(\dfrac{dy}{dr}\right)\dfrac{dr}{dx}+(2-n)(1-n)x^{-n}\dfrac{dy}{dr}=(2-n)x^{1-n}\dfrac{d^2y}{dr^2}(2-n)x^{1-n}+(2-n)(1-n)x^{-n}\dfrac{dy}{dr}=(2-n)^2x^{2-2n}\dfrac{d^2y}{dr^2}+(2-n)(1-n)x^{-n}\dfrac{dy}{dr}$$

$$\therefore(2-n)^2x^{3-2n}\dfrac{d^2y}{dr^2}+(2-n)(1-n)x^{1-n}\dfrac{dy}{dr}-(1-n)(2-n)x^{1-n}\dfrac{dy}{dr}=Cxy^\frac{n+2}{n-2}$$

$$(n-2)^2x^{3-2n}\dfrac{d^2y}{dr^2}=Cxy^\frac{n+2}{n-2}$$

$$\dfrac{d^2y}{dr^2}=\dfrac{Cx^{2n-2}y^\frac{n+2}{n-2}}{(n-2)^2}$$

$$\dfrac{d^2y}{dr^2}=\dfrac{Cr^{-\frac{2n-2}{n-2}}y^\frac{n+2}{n-2}}{(n-2)^2}$$

Which reduces to an Emden–Fowler equation.

Let $$\begin{cases}y=\dfrac{u}{s}\\r=\dfrac{1}{s}\end{cases}$$ ,

Then $$\dfrac{d^2u}{ds^2}=\dfrac{Cr^{-\frac{2}{n-2}-2}u^\frac{n+2}{n-2}}{(n-2)^2}$$