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Solving a non-homogeneous ODE with Bessel functions?

  1. Oct 18, 2010 #1
    Hi, I posted this on the homework forum, but I haven't gotten any responses there. I thought there might be a better chance here.

    1. The problem statement, all variables and given/known data

    I have the ODE
    h'' + h'/r + λ2h = 1,
    where h = h(r), and I want to find h(r).

    2. Relevant equations

    The corresponding homogeneous equation is a Bessel equation that has the solution
    hh = c1J0(λr) + c2Y0(λr),
    where J0 and Y0 are Bessel functions.

    Now I was planning on using h(r) = hh + hp,
    where hp is a particular solution of the ODE.

    3. The attempt at a solution

    To find hp, I tried using variation of parameters, but I get to a point where I need to both differentiate and integrate a Bessel function, which turns out to be pretty hard. I'm wondering if I'm going in the wrong direction, or if my logic here is even valid.

  2. jcsd
  3. Oct 18, 2010 #2
    A simple particular solution is easily found by inspection, and it is given by [tex]h_p=1/{\lambda}^2[/tex].
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