1. (I) First to find the formal eigenfunction series expansion for
the solution y(x) of -y''-μy=f(x), y(0)=0,y'(1)=0, 0< x<1 where f is a given continuous function on [0, 1].
(II) What happens if the parameter μ is an eigenvalue?
2. (I)(20 points) Find the eigenvalues and normalized eigenfunctions of the
Sturm-Liouville system -(x^2)(y'*x^2) = μ*y, y'(1)=0, y'(2)=0, 1≤x≤2. What are the orthogonality relations for the eigenfunctions?
Does anyone can give some tips?