How to find the eigenfunctions of a differential operator.

Question

Consider a linear differential operator $$L=\frac{d^2}{dx^2}.$$ How would one determine that the normalised eigenfunctions of $L$ are $$\phi_n(x)=\sqrt{2}\sin{(n\pi x)}?$$

Answer 1

To find its eigenfunction $f$, it is equivalent to solve $Lf=\lambda f$, that is, $$\frac{d^2f}{dx^2}=\lambda f.$$ This is an second order ODE with constant coefficient, which can be solved. After finding all the possible solutions for $f$, we can consider the normalized condition and initial conditions to find the specify $f$.