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DE question, how can I use straight line trajectories to come up with

  1. Jun 24, 2013 #1
    Ok, so this is a differential equation question.
    How can I use the eigenvectors/eigenvalues to find the formulas for straight line trajectories and from those formulas that I come up with, how can I alter them so as to start at any given point that I would like them to (like with starter data)?

    I know that I can come up with the straight line trajectories starting at the tips of the eigenvectors using:

    {x1[t_], y1[t_]} = eigenvector[1] E^(eigenvalue[1] t)
    {x2[t_], y2[t_]} = eigenvector[2] E^(eigenvalue[2] t)

    Using the above, then if I want my solution plots to obey certain starter data, say x[0] = A, y[0] = B,
    What do I have to do to the above to make the by solution plots start at those values?
    Any suggested reading or videos on the subject to get a good understanding of it?

    Thanks for any help,
    -James
     
    Last edited by a moderator: Jun 24, 2013
  2. jcsd
  3. Jun 25, 2013 #2
    Ok, so I figured it out:

    If you have a DE system that is linear, then if we know eigenvalue and eigenvectors of a coefficient matrix A, we know that trajectory plots starting at the ends of the eigenvectors will be:

    {x1[t_], y1[t_]} = eigenvector[1] E^(eigenvalue[1] t)
    {x2[t_], y2[t_]} = eigenvector[2] E^(eigenvalue[2] t)

    These will go with the flow. However, if we would like to start at any point, then we can say:

    {x[t],y[t]} = C1 {x1[t], y1[t]} + C2 {x2[t], y2[t]}
    where C1 and C2 are chosen, based off of the coefficients of {x1[t], y1[t]} and {x2[t], y2[t]} such that the resulting coefficients will be where you want the trajectory to start.
    This is all possible because the system is linear.
     
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