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Homework Help: Laplace Integral Equation

  1. Sep 20, 2012 #1
    1. The problem statement, all variables and given/known data
    By taking the Laplace transform and using the convolution theorem, obtain the solution of the integral equation

    2. Relevant equations
    f(t) = sin t + ∫e^(t-u)*f(u) du
    integral is from 0 to t

    3. The attempt at a solution
    I used the following site as a reference for how to construct the problem

    I rewrote the equation using the convolution theorem to be this
    f(t) = sin t + e^t*f(t)
    Letting y = L{f(t)} this becomes
    y = 1/s^2 + y/s-1

    The website that i referenced you too somehow removes the y and gets the RHS purely in terms of s. I cannot reproduce the simplication the site used on their problem nor can i apply it to my own. I get
    y = y(s^2+1)+(s-2)/[(s^2+1)(s-2)]

    Hopefully I am just missing something obvious but I am unsure what to do from here. I will continue to play around with it but hopefully someone can nudge me in the right direction.
  2. jcsd
  3. Sep 20, 2012 #2


    User Avatar
    Science Advisor

    What you are missing is basic algebra!

    Solve y = 1/s^2 + y/(s-1) for y and apply the inverse transform.
  4. Sep 20, 2012 #3
    Finally got it. Took me hours to work through that but I just couldn't see a solution until you gave me a push. Cheers

    Solve for y then solve using partial fractions before being able to invert
    Final answer
    F(t) = 1/5*e^2t - 1/5cos(t) + 3/5sin(t)

    Thanks again!
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