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Laplace Transform Initial Value Nonhomogenous, not allowed to use Partial Fractions

  1. Jun 22, 2011 #1
    1. The problem statement, all variables and given/known data
    y''-6y+9y=2 y(0)=y'(0)=0

    *Note* Professor will NOT allow use of partial fractions, so please don't use it.


    2. Relevant equations
    Laplace transform table
    Y=[y'(0)+sy(0)+ay(0)+R]/[(s^2)+as+b]


    3. The attempt at a solution
    Y=L(2)/(s-3)^2
    L(2)=2/s
    Y=(2/s)[1/(s-3)^2]
    Y=2*1/[s^3-6s^2+9s]
    From here I cannot figure out how to continue without using partial fractions since I can't get the roots, from which I would be able to invert and use the Laplace Transform table.
     
  2. jcsd
  3. Jun 22, 2011 #2

    lanedance

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    Re: Laplace Transform Initial Value Nonhomogenous, not allowed to use Partial Fractio

    so you have
    [tex]Y(s) = F(s)G(s) [/tex]
    with
    [tex]F(s)=\frac{2}{s} [/tex]
    [tex]G(s)=\frac{1}{(s-3)^2} [/tex]

    how about consider f(t) and g(t) and using a convolution, if you don't know how to do this have a look at 1.7 in
    http://www.vibrationdata.com/math/Laplace_Transforms.pdf
     
  4. Jun 22, 2011 #3

    vela

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    Re: Laplace Transform Initial Value Nonhomogenous, not allowed to use Partial Fractio

    You could also invert Y(s) using the Bromwich integral
    [tex]y(t)=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} Y(s)e^{st}\,ds[/tex].
     
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