# Normal modes / Eigenfrequencies

1.
Homework Statement

Someone studying a dynamical system in another field of science tells you that when they
attempt to model the experiment they’ve been examining they obtain the following set of
coupled ordinary differential equations.

$$\dot{x}$$= -Ax + By
$$\dot{y}$$= -Cx

In what follows you should assume that the material parameters A, B, C are all positive and
non zero. They also tell you that for certain material parameters (that is, for certain A, B, C)
they can sometimes obtain oscillatory behaviour, albeit damped, but sometimes they do not.
Note that x and y are restricted to be real.

(a) Show that this is indeed possible by solving for the normal modes of this system. That is,
find the eigenfrequencies for this system.

## The Attempt at a Solution

So basically my idea of eigenfrequencies are the frequencies at which the system oscillates and all motion of the system is the superposition of these two frequencies/motions. The first thing I'm doing is putting the constants into a matrix:

| -A B | |x| = |0|
| C 0 | |y|= |0|

Then i solve for determinant and get the eigenfrequencies. Basically I just wanted to know if I'm on the right track and this will help me prove that this is solving for the normal modes.

Normal modes imply:

$$x(t) = D e^{i\omega t}$$

So you need to plug in similar functions for x and y. Then you solve for the eigenfrequences.

Shouldn't the function you plug in be of the form of a solution for a damped oscillation?

If $$\omega$$ is imaginary then it will be damped.