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Solving Transcendental Equations (and Laplace Transforms)

  1. Apr 10, 2008 #1
    1. The problem statement, all variables and given/known data
    Given the equation [tex]H'(t) + u H(t - T) = 0[/tex] u > 0
    Look for solutions of the form [tex]e^{rt}[/tex]

    Show that these solutions are exponentially damped if [tex]e^{-1} > uT > 0[/tex]
    Find uT for which these solutions for r complex are oscillatory with growing, decaying, or constant amplitude.

    The book also hints that a Laplace Transform would be helpful (albeit not necessary), but I'm not sure how to do these.

    2. Relevant equations
    [tex]r = - u e^{-rT}[/tex]
    Let [tex]r = x + yi[/tex]
    [tex]x = -u e^{-xT} cos(yT)[/tex]
    [tex]y = u e^{-xT} sin(yT)[/tex]

    3. The attempt at a solution
    I find found the real solutions.
    Set y = 0, then cos(yT) = 1, so
    [tex]x = -u e^{-xT} [/tex]

    define [tex]F(x) = x + u e^{-xT}[/tex]
    [tex]F(0) = u >0[/tex]
    [tex]F(-1/T) = \frac{-1 + uTe}{T} < \frac{-1 + 1}{T} = 0[/tex]
    when [tex]e^{-1} > uT > 0[/tex]

    For oscillatory, I just set [tex]x = 0[/tex], [tex]so cos(yT) = 0[/tex], which means [tex]sin(yT) = +-1[/tex]
    So [tex]y = +-u[/tex], which means [tex]cos(uT) = 0 sin(uT) = +-1[/tex], which we know happens when uT = pi/2, 3pi/2, etc...

    I'm not sure how to do the rest.

    It says a Laplace Transform would make it faster, well, I took the transform from the DE and got
    [tex]Y(s) = \frac{H(0)}{s + e^{-sT}}[/tex]

    How do I interpret this?
  2. jcsd
  3. Apr 11, 2008 #2
    Although I don't like bumping threads, I want to make sure everyone sees this.

    In particular, I'm really curious to how I would be using Laplace Transforms to solve this problem.
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