What you have done so far is good. With g(x) as given, .
I'm having trouble understanding how I'm supposed to show the following for this question.
Let be an arbitrary function defined 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.