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Inverse Laplace Transforms

  1. Jul 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Use partial fraction decomposition to find the inverse Laplace Transform.

    F(s)= 1/[(s+1)(s^2 + 1)]

    2. Relevant equations



    3. The attempt at a solution
    1/[(s+1)(s^2 + 1)] = A/(s+1) + (Bs + C)/(s^2 + 1)

    1 = A(s^2 + 1) + (Bs + C)(s+1)

    I do not know how to solve for A and B or C
     
  2. jcsd
  3. Jul 25, 2009 #2

    jgens

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    Gold Member

    To solve for A, let s = -1 and go from there.

    To solve for B and C, note that: [As2 + (A - 1)]/(s + 1) = Bs + C
     
  4. Jul 25, 2009 #3
    s = -1
    1 = A(1 + 1) + B(-1)^2 + B(-1) + C(-1) + C
    A = 1/2

    I dont understand your next step
    do you mean
    [(1/2)(-1)^2 + (1/2 - 1)]/(-1 + 1) = B(-1) + C
     
  5. Jul 25, 2009 #4

    jgens

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    No, I don't mean that. We have that A = 1/2. This means that, (1/2 - s2)/(s + 1) = Bs + C.

    Edit: Fixed algebra errors. Wow, really bad algebra on my part!
     
    Last edited: Jul 26, 2009
  6. Jul 26, 2009 #5

    HallsofIvy

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    Or: choose any 3 values for s to get 3 equations in A, B, and C.

    For example, choosing, arbitrarily, s= 1, 2, 3 gives:
    s=1 2A+ 2B+ 2C= 1
    s=2 5A+ 6B+ 3C= 1
    s=3 10A+ 12B+ 4C= 1

    Or: multiply out the right side and set corresponding coefficients equal.

    1 = A(s^2 + 1) + (Bs + C)(s+1)= As^2+ A+ Bs^2+ Bs+ Cs+ C
    = (A+ B)s^2+ (B+ C)s+ (A+ C)
    0x^2+ 0x+ 1= (A+ B)s^2+ (B+C)s+ (A+ C) so

    A+ B= 0, B+ C= 0, A+ C= 1.

    You have three unknown numbers, A, B, and C. Any way you can get three equations to solve for them is valid.
     
  7. Jul 26, 2009 #6
    Thanks for the help guys. I appreciate it!
     
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