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Homework Help: Help With a Laplace Transform

  1. Nov 28, 2015 #1
    1. The problem statement, all variables and given/known data
    [tex] xy''+y'+xy=0, y'(0)=0, y(0)=0 [/tex] Using the method of Laplace transforms, show that the solution is the Bessel function of order zero.

    2. Relevant equations
    [tex] -(d/ds)L{f(x)} [/tex]

    3. The attempt at a solution
    The only thing I got out of this when trying to solve it was y=0. Obviously not the intended answer. In the problem I'm told that the answer is the Bessel function of 0 order.

    I'm pretty sure the parts I'm messing up is the Laplace transform of xy''. I haven't tried to take the Laplace transform of anything like that before and I'm sure it is where I'm messing up. I tried to use the relevant equation up there and I got...

    [tex] -(d/ds)[(s^2Y(s)-s)]+sY(s)-1+-(d/ds)[Y(s)]=0[/tex]

    This gives me [tex] -2sY(s)+1+sY(s)-1=0 [/tex]

    => [tex] -sY(s)=0 [/tex]

    which yeah, just gives me a dumb answer. Again I'm pretty sure it's with my xy'' term. Maybe even more :(

    Thanks anyone for help.
  2. jcsd
  3. Nov 28, 2015 #2
    Oh... as soon as I posted this I thought... Y(s) is a function of s... I need to take that into account when doing derivatives... Dang it... as always it's right after I post I have a decent idea...
    Last edited: Nov 28, 2015
  4. Nov 28, 2015 #3
    Ok so I'm on the right track for sure. My current expression is

    [tex] y = L^-1[1/(s^2+1)^.5] [/tex]

    I know that the inverse of that is the zero order Bessel function. How would I prove with no table though?
  5. Nov 28, 2015 #4


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    Homework Helper

    Well fortunately, there are tables of the Laplace transforms of all kinds of special functions, including Bessel functions. One reference which has such a table is Abramowitz & Stegun, Handbook of Mathematical Functions, p. 1029:


    If you don't trust such a reference, I suppose you could always derive the Laplace transform of your Bessel function from the definition and compare it with your earlier result.
  6. Nov 28, 2015 #5
    Yah I'm supposed to derive it I'm sure.
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