# Lecture 1: Zero-Input Response

When there is no input, we can write

(What happened for continuous time?) Similar. Let's try a simply case to get started:

Try a solution . (Comment on the difference from earlier case.) Substitute in and show that it works.

or

Write as

is thus the characteristic polynomial, and we look at its roots. The roots are , .

As before, we take all possible solutions in a linear combination:

Repeated roots: If

So how do things behave as a function of root location. (0.8, -0.8, -.5, 1.2,
(0.8),
1, etc.) Same old, same old.

Complex roots: With roots , then

with and arbitrary constants (same steps as before).