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What did I do wrong in this Bessel equation?

  1. Dec 21, 2009 #1
    I need to convert[tex]x^{2}y''+2xy'+[kx^{2}-n(n+1)]y=0[/tex] using [tex]y=x^{-\frac{1}{2}}w[/tex] to a normal Modified Bessel Equation and I cannot get to that. I check many times and I must be having a blind spot!!!!

    This is my work:

    [tex]y=x^{-\frac{1}{2}} w \Rightarrow y'=-\frac{1}{2}x^{-\frac{3}{2}}w+x^{-\frac{1}{2}}w' \Rightarrow y''=\frac{3}{4}x^{-\frac{5}{2}}w-\frac{1}{2}x^{-\frac{3}{2}}w'+x^{-\frac{1}{2}}w''-\frac{1}{2}x^{-\frac{3}{2}}w'[/tex]

    [tex]x^{2}y''+2xy'+(kx^{2}-n(n+1))y=0 \Rightarrow[/tex]

    [tex] [\frac{3}{4}x^{-\frac{1}{2}}w-\frac{1}{2}x^{\frac{1}{2}}w'+x^{\frac{3}{2}}w''-\frac{1}{2}x^{\frac{1}{2}}w']+[2x^{\frac{1}{2}}w'-x^{-\frac{1}{2}}w]+[kx^{\frac{3}{2}}w-n(n+1)x^{-\frac{1}{2}})w]=0[/tex]


    Grouping w'', w' and w terms:

    [tex]x^{\frac{3}{2}}w''+(-x^{\frac{1}{2}}+2x^{\frac{1}{2}})w'+(\frac{3}{4}x^{-\frac{1}{2}}-x^{-\frac{1}{2}}+kx^{\frac{3}{2}}-n(n+1)x^{-\frac{1}{2}})w=0[/tex]

    Multiply by [tex]x^{\frac{1}{2}}[/tex] This give:

    [tex]x^{2}w''+xw'+(\frac{3}{4}-1+kx^{2}-n(n+1))w=0[/tex]

    I can't get the standard Modified Bessel equation because the 3/4-1 in the w term cannot be cancel out.
    Please tell me what I did wrong.

    Thanks
    Alan
     
    Last edited: Dec 21, 2009
  2. jcsd
  3. Dec 21, 2009 #2
    Your arithmetic is correct (except that in the first line you have an exponent [tex]-\frac23[/tex] that should be [tex]-\frac32[/tex], but later steps are correct).

    Is there any reason why the [tex]n[/tex] (order) in the equation you end up with is required to be the same you began with? I notice that your problematic constant term is [tex]-n^2 - n - \frac14[/tex], which factors nicely.
     
  4. Dec 21, 2009 #3
    You are right, I was blind!!! The final equation is with [tex](n+\frac{1}{2})^{2}[/tex]!!!

    It must be old age!!! I read the final equation wrong and I just kept concentrate on what I did and never look at the final equation!!!! Wasted almost an hour on this!!! The only excuse I can come up is I am 56!!!! Too old!!!:rolleyes: BUT I am not old enough to conceed to you guys yet!!!!

    Thanks a million.

    Alan
     
    Last edited: Dec 21, 2009
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