I'm having trouble understanding how I'm supposed to show the following for this question.

Let be an arbitraryfunctiondefined for such that . Consider the ordinary differential operator which assigns to each such function the new continuous function . Show that the inverse operator, say B, assigns to each continuous function , defined for , the function

, where

Consequently, the solution of the problem with boundary condition , is given in terms of the integral operator B with Green's function g(x,z).

I don't really understand what any of this means. So, I don't think my "work" can be considered that.

My thought is that we have

Which is the same thing as

By using the inverse operator B on both sides, I get

Which gives

So,

.

But the way the question is posed, makes me think I'm supposed to somehow derive this integral. Am I making this overly complicated? Could someone explain how they got this integral?

Any help is appreciated. I'm just trying to understand the material.