How should I show that solutions can be expressed as a Fourier series?

[Math100]

Homework Statement

Consider the equation $\ddot{x}+x+\epsilon(\alpha x^2\operatorname{sgn}(x)+\beta x^3)=0, 0<\epsilon<<1, \alpha>0, \beta>0$, where $\alpha$ and $\beta$ are constants, and the signum function, $\operatorname{sgn}(x)$, is defined by $\operatorname{sgn}(x)=1$ for $x>0$, $\operatorname{sgn}(x)=0$ for $x=0$ and $\operatorname{sgn}(x)=-1$ for $x<0$. With initial condition $x(0)=0$, show that solutions with angular frequency $\omega$ can be expressed as a Fourier series of the form $x(t)=\sum_{k=1}^{\infty}B_{2k-1}\sin((2k-1)\omega t)=B_{1}\sin\omega t+B_{3}\sin 3\omega t+\cdots$.

Relevant Equations

None.

Proof:

Let $\epsilon=0$.
Then the unperturbed equation is $\ddot{x}+x=0$ and the general solution is
$x(t)=A\sin\omega t+B\cos\omega t$ where $\omega=1$ is the angular frequency
with the constants $A$ and $B$.
With the initial condition $x(0)=0$, we have that $B=0$.
This gives $x(t)=A\sin\omega t$ where $A$ is a constant.
Note that the function of $x(t)=A\sin\omega t$ is periodic and odd.

From the work/proof shown above, how should I show that solutions with angular frequency $\omega$
can be expressed as a Fourier series of the given form on the problem? What needs to be done?

 

[renormalize]

From the work/proof shown above, how should I show that solutions with angular frequency $\omega$
can be expressed as a Fourier series of the given form on the problem? What needs to be done?

Can you show the work you've done on this problem so far?

 

[Math100]

Can you show the work you've done on this problem so far?

That's all the work I have so far.

 

[renormalize]

That's all the work I have so far.

OK, can you substitute the form $x(t)=\sum_{k=1}^{\infty}B_{2k-1}\sin((2k-1)\omega t)=B_{1}\sin\omega t+B_{3}\sin 3\omega t+\cdots$ into the differential equation to see what you can learn about $\omega$ and the coefficients $B_{2k-1}$?