Ex2. Find the eigenvalues and eigenfunctions for the given BVP:

Solution:

If , the solution is y=Ax+B, with y’=A. The only solution is y=0.

If,the solutions will be of form with

. If y(0)=0, then A+B=0 or A=-B.

Then we would have and forces B=0

Again we have only the solution y=0.

If , the general solution is . The condition y(0)=0 forces A=0.

Therefore we can write , and the other boundary condition forces for some constant C.

This is only possible if .

So the set {1,9,25,…} is the set of eigenvalues with associated eigenfunctions

{sin(x),sin(3x),sin(5x),…}