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Axisymmetric Poisson equation

  1. May 15, 2008 #1
    For a magnetostatics problem I seek the solution to the following equation

    [tex] \frac{1}{x}\frac{d}{dx} \left( x \frac{dy(x)}{dx} \right) = -C^2 y(x) [/tex]

    (C a real constant) or equivalently

    [tex]x \frac{d^2 y(x)}{dx^2} + \frac{dy(x)}{dx} + C^2 x y(x)=0[/tex]

    It seems so simple, but finding a particular solution beats me...is this solvable?
    Last edited: May 15, 2008
  2. jcsd
  3. May 15, 2008 #2


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    If you rescale variables to get rid of the C^2 it looks like you could get it into the form of the differential equation for a zeroth order Bessel function. The general equation for a Bessel function is:

    [tex]x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0[/tex]

    So with alpha = 0, you could divide out an x (or equivalently mutliply your equation by one) and it matches your equation - you just need to scale out the constant. i.e., somehow you want to scale that last term such that [itex]C^2xy \rightarrow xy[/itex] with the other terms remaining unchanged.
  4. May 16, 2008 #3
    That's great! Thank you very much...works like a charm!
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