Fourier Series and differential equation with epsilon

Question

Happy New Year!

I am stuck for days on expressing the solution of a differential equation using Fourier series.

The question is:

Consider the equation: $$\ddot{x}+x+\epsilon\left(\alpha x^2\text{sgn}(x)+\beta x^3\right)=0,\ 0<\epsilon \ll1,\ \alpha>0,\ \beta>0$$ where $\alpha$ and $\beta$ are constants

With initial condition $x(0)=0$, show that solution with angular frequency $w$ can be expresed as Fourier series: $$x(t)=\sum_{k=1}^{\infty}B_{2k-1}\sin\left((2k-1)wt\right)=B_1\sin(wt)+B_3\sin(3wt)+...$$

Assuming $w$ and $B_n$ can be expanded as power series in $\epsilon$, show that as $\epsilon\to0$, $w=1+O(\epsilon)$, $B_1=O(1)$, $B_n=O(\epsilon)$ for $n=3,5,7,...$

Appreciate for some lights!

Answer 1

Cheers, I managed to solve it.