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Homework Help: Inital value problem, Laplace transform

  1. Sep 5, 2011 #1
    1. The problem statement, all variables and given/known data

    Solve inital value problem: x''+2x'+x=g(t)

    t 0<t<1
    2-t 1<t<2
    0 t>2

    2. Relevant equations

    Second shift theorem, Heaviside function and Laplace transforms. I denote Heaviside,functuon H(t-a), and Laplace transform with L

    3. The attempt at a solution

    I rewrote the problem with Heaviside functions:

    L(t+(2-t)H(t-1))= [itex]\frac{1}{s^{2}}[/itex]+[itex]\frac{2e^{-s}}{s^{2}}[/itex]

    After this I transformed the original diff. equation and plugged in my transform of g(t). After some simplification I got:

    And after partial fraction decomp. I got:


    If I use inverse Laplace transform I get




    According to my book, this is only about half of the answer. I am missing some step and I cant figure out what is it! All help is appreciated

    Thanks in advance!
    Last edited: Sep 5, 2011
  2. jcsd
  3. Sep 5, 2011 #2


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    Science Advisor
    Homework Helper
    Gold Member

    I didn't check your work, but I notice that you haven't mentioned the initial conditions. Unless you were given x(0) = 0 and x'(0) = 0 you have left something out, which might explain why you don't have the whole answer.
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