I'm trying to solve this exercise, however to be quite frank, I don't really have any idea on where to begin and proceed from there. It goes as follows:
Let $f$ be a real periodic function, $f: [0,2\pi] \to \mathbb{R}$ with $f(0) = f(2\pi)$. Let $\gamma,\beta \in \mathbb{R}$ with $\beta \neq 0$. The following ODE is given:
$$(1): \frac{d^2u}{dt^2}(t)+\beta\frac{du}{dt}(t)+\gamma^2u(t) = f(t)$$
(a): Assume that $f(t) = \sum_{i=1}^\infty a_k$sin$(kt)$ , and let $(u_k)_{k=1}^\infty$ denote solutions of $$(2): \frac{d^2u_k}{dt^2}(t)+\beta\frac{du_k}{dt}(t)+\gamma^2u_k(t) = \sin(kt) , \qquad k = 1,2,3\dots$$
Using the fact that $u_k(t) = A_k(t)\cos(kt) + B_k\sin(kt), \quad A_k,B_k \in \mathbb{R}$, solve the ODE $(2)$ for every $k$ by finding $A_k, B_k$ and consequently, $u_k(t)$
(b): By superposition (linearity) we then have $u(t) = \sum_{i=1}^\infty a_ku_k(t)$ is the solution to $(1)$. Write the complete solution $u(t)$
(c): Find $N \in \mathbb{N}$ such that $$ \max_{0\leq j\leq1000}\lvert (f_1^N-f)(\frac{j}{1000}2\pi)\rvert \lt \varepsilon$$ for $\epsilon= 10^{-\ell} , \ell \in \{1,2,3\}$ and
$f(t) = \begin{cases} 1, & \text{if $0\lt t \lt \pi$,} \\ 0, & \text{if $t \in \{0,\pi,2\pi\}$}\\ -1, & \text{if $\pi \lt t \lt 2\pi$} \end{cases}$
Now I even struggle to come up with anything really. How would I start solving an exercise like this?