- #1
rmiller70015
- 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: