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Laplace Transform equation help

  1. May 29, 2008 #1
    [SOLVED] Laplace Transform

    1. The problem statement, all variables and given/known data
    Suppose F(s) = [tex]\displaystyle\mathcal{L}(f(t)) [/tex]
    Show that [tex]\displaystyle\mathcal{L}(f(ct)) = 1/c F(s/c) [/tex]


    2. Relevant equations



    3. The attempt at a solution

    [tex]\displaystyle\mathcal{L}(f(t)) = \int_0^{inf} e^{-st} f(t) dt[/tex]
    [tex]\displaystyle\mathcal{L}(f(ct)) = \int_0^{inf} e^{-st} f(ct) dt[/tex]
    I'm not quite sure what to do after this...

    I could play around with integration by parts, but in this case I don't think it yields anything useful
    [tex]\frac{F(ct)}{c} e^{-st} - \int \frac{F(ct)}{c} (-s e^{-st} ) dt[/tex]

    [tex]{F(t)e^{-st} + \int F(t) se^{-st} dt}{}[/tex]
     
    Last edited: May 29, 2008
  2. jcsd
  3. May 29, 2008 #2
    Suppose [tex]u=ct [/tex] then we get:

    [tex] \int_0^{ \infty} e^{-st} f(ct)\ \mbox{d}t = \frac{1}{c} \int_0^{\infty}\ e^{\frac{-s}{c} u}\ f(u)\ \mbox{d}u = \frac{1}{c}\ F \left( \frac{s}{c} \right) [/tex]
     
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