The "Green's function" for a given differential operator, D(f), is a function, G(x,x'), satisfying D(G)= 0 as long as and such that .
Functions satisfying are all of the form y(x)= C cos(kx)+ D sin(kx). So we must have for some constants (with respect to x, they may depend on x') for and for . So the problem is to determine , , , and .
If x= 0 then because x' must also be between 0 and L so we must have .
Similarly of x= L then so we must have . That gives
So far, we have if and .
At x= x' those must both be true: .
The derivative, from the left, at x= x', is and, from the right, so we must have .
That gives two equation to solve for and in terms of x'.