Determine? the? eigenfunctions ?and? eigenvalues

Question

Solve the following ?eigenvalue? problem:

        d²y/dx²+λ²y(x)=0       0 < x <1

with? boundary? conditions

y(0) = 0     y'(1) = 0

1. Determine? the? eigenfunctions ?and? eigenvalues

2. Show? that ?the ?eigenfunctions? are? or thogonal

3. Suppose? we ?want? to ?expand? a? given ?smooth? function?  in term of the eigenfunctions determined in (1):

                                        ∞
                              f (x)=  ∑   a_nφ_n (x)
                                      n=0

where? φ_n are? the ?eigenfunctions.?Write? down ?a? formula? for ?the? coefficients? of? a_n