Exercise 105

Let $f(t) = t + \sum_{n=1}^\infty \frac{1}{2^n} \sin(nt)$. Solve $x'' + \pi x = f(t)$ for the Dirichlet conditions $x(0) = 0$ and $x(\pi) = 1$. Hint: Note that $\frac{t}{\pi}$ satisfies the given Dirichlet conditions.