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Laplace tranform problem

  1. Nov 25, 2007 #1
    1. The problem statement, all variables and given/known data
    find solution using laplace transforms


    y'' + 4y = 8



    alright, so i did the laplace transform of both sides and i get

    (s^2 + 4)L(y) - 11s - 5 = 8/s

    so i isolate L(y) and i get this expression:

    L(y) = (11s^2 + 5s + 8)/(s*(s^2 + 4))

    however, the textbook says the answer is:

    L(y) = 2/s + (9s + 5)/(s^2 + 4)

    And i dont know how to get from my expression to the book's.
    I'm pretty good at doing inverse laplace transforms, its just that i cant seem to do the algebra right.

    CAn someone help me see how to get to the right expression for L(y)?
    thanks!!
     
  2. jcsd
  3. Nov 26, 2007 #2

    benorin

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    Homework Helper

    Use partial fraction decomposition:

    Start off by setting

    [tex] \frac{11s^2 + 5s + 8}{s(s^2 + 4)} = \frac{A}{s}+\frac{ Bs + C}{s^2 + 4}[/tex]

    multiply through by the common demoninator [tex]s(s^2 + 4)[/tex] and plug-in 3 different values of [itex]s[/itex] to generate 3 equations involving [tex]A,B, \mbox{ and }C[/tex]. Nice values of [itex]s[/itex] here include [itex]s=0[/itex] (this will give the value of [tex]A[/tex]), and either [itex]s=\pm 1[/itex] (which gives 2 equations in [tex]A\mbox{ and }B[/tex]) or [itex]s=2i[/itex] (which gives the values of [tex]A\mbox{ and }B[/tex] by equating real and imaginary parts). Enjoy :).

    Dear Moderator: I know this post goes beyond what we by rule give in guidance in solving a problem, yet I offer this apology: I could not quickly find a web page that gave instructions for the above easy method of partial fraction decomposition to my satisfaction: hence my post.
     
  4. Nov 26, 2007 #3
    well dont worry i found out a way anyways.

    so i got:

    L(y) = 8/s(s^2 + 4) + (11s + 5)/(s^2 + 4)

    i wont type it all out cause it's annoying, but what i did was i partial fraction decomposed 8/s(s^2 + 4) and i expanded all the terms and equated coeficcients cause i hate dealing with complex numbers when using partial fractions. So i find the values of A B C and combine all the terms in it comes out to the expression i was looking for.
     
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