A second order differential operator usually needs two initial conditions. I'm guessing that f(0) and f'(1) should both be 0?

In that case, if T is the differential operator given by , and f is an eigenfunction for the operator T, with eigenvalue , then (just as in linear algebra). So you need to solve the differential equation , together with the initial conditions f(0) = f'(1) = 0.

The general solution is , where . The initial condition f(0)=0 tells you that A=0. The condition f'(1)=0 then tells you that . So the equation will only have a nonzero solution if or . In terms of , that implies that or , for some integer k. So those are the eigenvalues (an infinite sequence of them, unlike in linear algebra where you only have finitely many eigenvalues for a matrix).