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Differential Equations

  1. Mar 22, 2005 #1
    For this linear system with complex eigenvalues
    a) find the eigenvalues
    b) determine whether the origin is a spiral source, sink or center
    c) Determine the direction of oscillations, clockwise or anticlockwise

    [tex] \frac{dY}{dt} = \left(\begin{array}{cc}0&2\\-2&0\end{array}\right) Y [/tex] with initial conditions [tex] Y_{0} = (1,0) [/tex]

    i foudn the eigenvalues to be
    [tex] \lambda = \pm i \sqrt{2} [/tex] which would make it a center
    also the eigenvectors
    [tex] \left(\begin{array}{cc}0&2\\-2&0\end{array}\right) \left(\begin{array}{cc}x\\y\end{array}\right) = \pm i \sqrt{2} \left(\begin{array}{cc}x\\y\end{array}\right) [/tex] i computed to be
    [tex] V_{1} = \left(\begin{array}{cc}i\sqrt{2}\\1\end{array}\right) [/tex]
    and [tex] V_{1} = -V_{2} [/tex]

    i feel i made a mistake in finding the eigenvectors
    also what would be the direction of the oscillations then?? Do i solve the Initial value problem to get hte direction of the oscillations??
     
  2. jcsd
  3. Mar 22, 2005 #2

    HallsofIvy

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    You have V1 correct but V2 is NOT -V1.
    [tex] V_{2} = \left(\begin{array}{cc}-i\sqrt{2}\\1\end{array}\right) [/tex]

    In order to determine the direction of rotation, look what happens to (1, 0):
    dx/dt= 2y= 0 but dy/dt= -2 so the "motion" is downward and the rotation is clearly clockwise.
     
  4. Mar 22, 2005 #3
    thank you very much :smile: didnt realize that the dx/dt and dy/dt were the directions of the vector. But what if the matrix was i nthe form
    a b
    c d then would i have to reduce this till i get zeroes in the 1x2 and 2x1 spots?
     
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