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Math Help - Inverse/Integral operators for DE

  1. #1
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    Inverse/Integral operators for DE

    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.
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  2. #2
    MHF Contributor

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    Re: Inverse/Integral operators for DE

    What you have done so far is good. With g(x) as given, \int_0^\infty g(x,z)f(z)dz= \int_0^x f(z)dz.
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