Hello,

I am trying to solve what I thought was going to be a simple differential equation.

$\displaystyle m\ddot{x}=-kx^{2n+1}:n \in \mathbb{Z}-\{0\} and k,m \in \Re$.

So I set

$\displaystyle y=\frac{dx}{dt}\Rightarrow \frac{dy}{dt}=-\frac{k}{m}x^{2n+1}$

$\displaystyle

\frac{dy}{dt}\frac{dt}{dx}=-\frac{k}{m}x^{2n+1}\frac{1}{y}=-\frac{k}{m}\frac{x^{2n+1}}{y}=\frac{dy}{dx}$

Then separating variables and integrating I got

$\displaystyle y^2+\frac{k}{m(n+1)} x^{2n+2}=C$

Solving for y

$\displaystyle y=\underline{+}\sqrt{C-\frac{k}{m(n+1)} x^{2n+2}$

remembering $\displaystyle y=\frac{dx}{dt} $ we can again separate variables

$\displaystyle \frac{dx}{\underline{+}\sqrt{C-\frac{k}{m(n+1)} x^{2n+2}}}=dt$

This is where I get stuck. I know this should be integrable to a sinusoid, but I don't know how. The actual question asks you to prove that the solution oscillates with a period proportional to $\displaystyle A^{-n}$ where A is the amplitude.

Any help would be appreciated.

Brad