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Solution to Bessel's Differential Equation

  1. Apr 13, 2013 #1
    Hey everyone. Need some more pairs of eyes for this one:

    "For each positive integer ##n##, the Bessel Function ##J_n(x)## may be defined by:
    [tex]J_n(x) = \frac{x^n}{1\cdot3\cdot5\cdots(2n-1)\pi}\int^1_{-1}(1 - t^2)^{n-\frac{1}{2}}\cos(xt)dt[/tex]

    Prove that ##J_n(x)## satisfies Bessel's differential equation:
    [tex]{J_n}^{\prime\prime} + \frac{1}{x}{J_n}^{\prime} + \Big(1 - \frac{n^2}{x^2}\Big)J_n = 0[/tex]"

    This problem is following a question where I proved differentiating under the integral sign. Thus, the way to do this is to simply calculate the derivatives of ##J_n(x)## and plug back into the equation. However, I can't seem to get all of the terms to cancel. I believe that the only thing that needs "calculus" methods is to calculate the derivative, with respect to x, of all the ##x## terms:
    [tex]\frac{\partial}{\partial{x}}(x^ncos(xt)) = (nx^{n-1}cos(xt) - tx^nsin(xt))[/tex]
    and
    [tex]\frac{\partial^2}{\partial{x^2}}(x^ncost(xt)) = \frac{\partial}{\partial{x}}(nx^{n-1}cos(xt) - tx^nsin(xt) = (n(n-1)x^{n-2}cos(xt) - tnx^{n-1}sin(xt)) - (ntx^{n-1}sin(xt) + t^2x^ncos(xt))[/tex]
    The rest would appear to be algebra. But again, not everything seems to cancel. That ##t^2## term seems to be messing things up for the ##x^n## terms and the ##x^{n-1}## terms seem to just combine. Unless there's an identity I'm overlooking, or my differentiation is wrong, I'm quite lost. Ideas? Thank you.
     
  2. jcsd
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