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Substitution in Laplace

  1. Jan 12, 2012 #1
    Hi,

    I'm learning about Laplace and I was wondering what substitution is when using laplace.

    Just say you have to find the laplace transform of

    {x^4 e^4x}

    I know what the individual transforms are = 4!//S^5 and 1/S-3

    but how do you kinda smush them together, apparently thats substitution but I would like to know what it is? and how it's done?

    Any help would be really appreciated!
     
  2. jcsd
  3. Jan 12, 2012 #2

    LCKurtz

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    You use this:$$
    \mathcal L(e^{at}f(t)) = \int_0^\infty e^{at}f(t)e^{-st}\, dt
    =\int_0^\infty e^{-(s-a)t}f(t)\, dt = \mathcal L(f(t))|_{s\rightarrow s-a}$$ What this means is that if you want to take the transform of ##e^{at}## times a function ##f(t)## you can just transform ##f(t)## and replace ##s## by ##s-a## in the answer. Try it in your example.
     
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