Green's function for a differential operator

Question

I am having a hard time with the following problem:

$$ y\mapsto Ly=-\frac{d}{dx}(x\frac{d}{dx}y) $$ with boundary conditions $y(1)=0$ and $y^\prime(2)=0$.

There are 2 things to do here:

1. I have to determine a fundamental matrix for the differential equation $Ly=0$.
2. I have to determine the green's function of L for these given boundary conditions and the intervall (1,2).

I have exam in this subject and our professor is bringing stuff to the exam, which we did not have done in the lectures. I would be very grateful if someone could help me out.