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Dirac Delta function

  1. Jul 3, 2011 #1
    The problem statement, all variables and given/known data
    Solve the given symbolic initial value problem: [tex]y''-2y'-3y=2\delta (t-1)-\delta (t-3) ;y(0)=2,y'(0)=2[/tex]


    The attempt at a solution

    Let Y(s):= L{y(t)}(s)

    Taking laplace transform of both sides:


    [tex][s^{2}Y(s)-2s-2]-2[sY(s)-2]-3Y(s)=2e^{-s}-e^{-3s}[/tex]
    [tex]s^{2}Y(s)-2sY(s)-3Y(s)=2e^{-s}-e^{-3s}+2s-2[/tex]
    [tex]Y(s)=\frac{2e^{-s}-e^{-3s}+2s-2}{s^{2}-2s-3}[/tex]
    [tex]Y(s)=\frac{2e^{-s}}{s^{2}-2s-3}- \frac{e^{-3s}}{s^{2}-2s-3}+\frac{2s-2}{s^{2}-2s-3}[/tex]
    [tex]y(t)=e^{-(t-1)}\frac{1}{2}e^{-(t-1)}(e^{4(t-1)}-1)u(t-1)-e^{-(t-3)}e^{-(t-3)}(e^{4(t-3)}-1)u(t-3)+e^{-t}+e^{3t}[/tex]

    And my final answer:


    [tex]y(t)=e^{1-t}(e^{4t-4}-1)u(t-1)-2e^{3-t}(e^{4t-12}-1)u(t-3)+e^{-t}+e^{3t}[/tex]

    Is this correct?
     
  2. jcsd
  3. Jul 3, 2011 #2

    hunt_mat

    User Avatar
    Homework Helper

    It looks to be right sort of thing to be doing.
     
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