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Homework Help: Inverse Laplace Transform

  1. Mar 10, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the inverse laplace transform of [itex]\frac{3s + 7}{s^{2} - 2s + 10}[/itex]

    2. Relevant equations
    completing the square.
    [itex]e^{at}sin(bt) = \frac{b}{(s-a)^{2} + b^{2}}[/itex]
    [itex]e^{at}cos(bt) = \frac{s-a}{(s-a)^{2} + b^{2}}[/itex]

    3. The attempt at a solution
    F(s)= [itex]\frac{3s + 7}{s^{2} - 2s + 10}[/itex]
    F(s) = [itex]\frac{3s + 7}{(s-1)^{2} +9}[/itex]
    F(s) = [itex]\frac{3s}{(s-1)^{2} +9} + \frac{7}{(s-1)^{2} +9} [/itex]

    after this i don't know how to manipulate the first fraction to fit the cosine equation. I know the 3 can be taken up front and a=1 and b=3 im pretty sure when comparing with the cosine equation but there the problem of making s into s-1.
  2. jcsd
  3. Mar 10, 2012 #2


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    Science Advisor
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    Gold Member

    Don't you have the shifting theorems? Like$$
    \mathcal L e^{at}f(t) = \mathcal L(f(t))|_{s \to s-a}$$
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