1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    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:

    http://people.math.sfu.ca/~cbm/aands/

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Help With a Laplace Transform
  1. Laplace Transform Help (Replies: 0)

Loading...