homework and exercises - Period $T$ of oscillation with cubic force function

How would I find the period of an oscillator with the following force equation?

$$F(x)=-cx^3$$

I've already found the potential energy equation by integrating over distance:

$$U(x)={cx^4 \over 4}.$$

Now I have to find a function for the period (in terms of $A$, the amplitude, $m$, and $c$), but I'm stuck on how to approach the problem. I can set up a differential equation:

$$m{d^2x(t) \over dt^2}=-cx^3,$$

$$d^2x(t)=-{cx^3 \over m}dt^2.$$

But I am not sure how to solve this. Wolfram Alpha gives a particularly nasty solution involving the hypergeometric function, so I don't think the solution involves differential equations. But I don't have any other leads.

How would I find the period $T$ of this oscillator?

Answer

Since $$\frac1 2mv^2+U(x)=U(A)$$ We have $$dt=\frac{dx}v=\frac{dx}{\sqrt{2(U(A)-U(x))/m}}=\frac{dx}{\sqrt{c(A^4-x^4)/(2m)}}$$ Then $$\frac T4=\int_0^{\frac T4}dt=\int_0^A\frac{dx}{\sqrt{\frac{c}{2m}(A^4-x^4)}}$$ Thus $$T=4\int_0^A\frac{dx}{\sqrt{\frac{c}{2m}(A^4-x^4)}}$$