- #1

- 110

- 1

## Homework Statement

[tex] \frac{d\vec{Y}}{dt} =

\begin{bmatrix}

0 & 2\\

-2 &-1

\end{bmatrix}\vec{Y}

[/tex]

With an initial condition of [tex] \vec{Y_0} = (-1,1)[/tex][/B]

a) Find the eigenvalues

b) Determine if the origin is a spiral sink, source, or center

c) Determine the natural period and frequency of the oscillations

d) Determine the direction of oscillations in the phase plane

e) Graph it

## Homework Equations

## The Attempt at a Solution

I have the solutions, the eigens are [tex] \frac{-1}{2} \pm \frac{\sqrt{15}i}{2}[/tex]

Because λ is negative and the roots of the characteristic polynomial are complex the solutions oscillate and it is a sink.

The frequency and periods can be found with [tex] f = \frac{2π}{μ}[/tex]

But the directions of oscillation is where I have a problem. Again I have the answer to this problem (graphing it with mathematica gives the directions), but my question is how do I typically find the direction of oscillations for equations of this form using algebra or some other relationship that exists?

Edit: I remember there being something about u + iw vectors, but it's really fuzzy that was the beginning of the semester.

Last edited: