Find the eigenvalues and eigenfunctions of the operator

Question

Find the eigenvalues and eigenfunctions of the operator $$Ly=\frac{d^2y}{dx^2},-\pi\le x \le \pi,$$ which operates on even-2$\pi$ periodic functions.

I am unsure of where to start. Any help would be appreciated.

Answer 1

Let $\lambda \in \mathbb{R}$.

Consider the following homogeneous differential equation with constant coefficients: $$ (E_\lambda): \frac{d^2y}{dx^2}=\lambda y $$ Solving for eigenvalues comes down to finding the values of $\lambda$ so that $(E_\lambda)$ has even $2\pi$-periodic solutions.

Solving for eigenfunctions comes down to finding those solutions.