first order linear differential equation

Question

I'm currently studying the Dirichlet and Neumann Laplacians, How can I satisfy both boundary conditions, one at each end.

i.e. if I had $$-f''(x)=\lambda f(x), \qquad x \in [-a,a]$$ where $f(-a)=f'(a)=0$

Answer 1

Using the usual prescription: Solve the equation $-f''(x)=\lambda f(x)$ for each value of $\lambda$ (separate into zero, positive and negative) and substitute your conditions $f(-a)=f'(a)=0$ to check whether that particular value of $\lambda$ gives a nonzero solution.