# Solving Transcendental Equations (and Laplace Transforms)

1. Homework Statement
Given the equation $$H'(t) + u H(t - T) = 0$$ u > 0
Look for solutions of the form $$e^{rt}$$

Show that these solutions are exponentially damped if $$e^{-1} > uT > 0$$
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
$$r = - u e^{-rT}$$
Let $$r = x + yi$$
$$x = -u e^{-xT} cos(yT)$$
$$y = u e^{-xT} sin(yT)$$

3. The Attempt at a Solution
I find found the real solutions.
Set y = 0, then cos(yT) = 1, so
$$x = -u e^{-xT}$$

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

For oscillatory, I just set $$x = 0$$, $$so cos(yT) = 0$$, which means $$sin(yT) = +-1$$
So $$y = +-u$$, which means $$cos(uT) = 0 sin(uT) = +-1$$, 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
$$Y(s) = \frac{H(0)}{s + e^{-sT}}$$

How do I interpret this?