Linear differential equation; Green's function

[capandbells]

I have this problem:

Consider the differential equation
y'' + P(x) y' + Q(x) y = 0
on the interval a ≤ x ≤ b. Suppose we know two solutions y₁(x), y₂(x) such that
y₁(a) = 0, y₁(b) ≠ 0
y₂(a) ≠ 0, y₂(b) = 0

Give the solution of the equation
y'' + P(x) y' + Q(x) y = f(x)
which obeys the conditions y(a) = y(b) = 0 in the form
[PLAIN]http://mathbin.net/equations/57612_0.png

where G(x,x') involves only the solutions y₁, y₂ and assumes different functional forms for x' < x and x' > x.

Ok, I don't really know where to begin here. My initial reaction is that I need to use variation of parameters, because it's the only method I know of that's going to give me a solution in terms of an integral, but I'm not sure if that makes sense. Anyway, if I do that, I get an expression like

[PLAIN]http://mathbin.net/equations/57613_0.png

which I don't think is right because it doesn't obey the y(a) = y(b) = 0 condition. I also don't get that "assumes different functional forms..." part. I really have no idea what I'm doing here, I'm mostly flailing aimlessly.

 

[LCKurtz]

This problem sounds a lot like the example towards the bottom of this article:

http://en.wikipedia.org/wiki/Green's_function

Look at that and see if it doesn't help you.