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Direction of oscillations (2nd order ODE)

  1. May 16, 2016 #1
    1. The problem statement, all variables and given/known data
    [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]


    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

    2. Relevant equations


    3. 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 someting about u + iw vectors, but it's really fuzzy that was the beginning of the semester.
     
    Last edited: May 16, 2016
  2. jcsd
  3. May 16, 2016 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Your DE makes no sense: it equates a vector on one side to a matrix on the other side. Did you mean
    [tex] \frac{d}{dt} \vec{Y}(t) = \begin{bmatrix}
    0 & 2\\
    -2 &-1
    \end{bmatrix} \vec{Y}(t) \: ?
    [/tex]
     
  4. May 16, 2016 #3
    Yes I did, sorry about that forgot my second vector.
     
  5. May 17, 2016 #4
    So, I figure it out eventually and just in case any futuristic internet people find this I figure I can be helpful by saying if you use the point (1,0) and plug it into the original equation and set it equal to the initial condition vector you can draw an arc from the point (1,0) to the next point and consider if the solution is a source or sink.
     
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