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**1. Homework Statement**

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. Homework 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?