Fourier Series and Solving Differential Equations

Question

I am getting stuck on how to use Fourier Series to solve ODE's. Take the problem where

\begin{equation} E(t)=200t(\pi^2-t^2), \end{equation}

for $t$ between $-\pi$ and $\pi$ (period of $2\pi$), $R=100$, $L=10$, $C=10^{-2}$, and

\begin{equation} LI'' +RI'+(1/c)I=E(t). \end{equation}

I know the first step would be to express $E(t)$ as a Fourier Series, then replace $E(t)$ in the equation with that representation, and solve the homogeneous and non-homogeneous solutions. This is where I am confused - what should I be looking to equate?

Any help would be really appreciated! Thank you

Answer 1

Let $I(t)=\sum_{n \in \mathbb{Z}} c_n e^{int}$, and susbstitute so in the equation. Reorganize terms as to have a sum in each side. Equate then the coefficients and you'll get the fourier series of the solution.