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Eigenvectors/values and diff equations

  1. Mar 27, 2007 #1
    1. The problem statement, all variables and given/known data
    the higher order equation y'' + y = 0 can be written as a first order system by introducing y' as another unknown besides y:

    2. Relevant equations
    d[y y']t/dt = [y' y'']t = [y' -y]t

    if this is du/dt = Au what is 2x2 matrix A?
    3. The attempt at a solution
    I think it's
    [0 1]
    [-1 0]
    but then I get complex eigenvalues so I am a bit suspicious... could someone verify or help me out?
    thanks.
     
  2. jcsd
  3. Mar 27, 2007 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You are exactly correct. If you let x= dy/dt, then d2y/dt2= dx/dt+ y= 0 or dx/dt= -y so your two equations are dy/dt= x and dx/dt= -1. Letting Y(t)= (y(t) x(t))t, yes, your differential equation is
    [tex]Y(t)= \left[\begin{array}{cc} 0 & 1 \\ -1 & 0\end{array}\right]Y(t)[/tex]
    and, yes, that has complex eigenvalues. Why does that bother you?
    I notice that the problem only asks you to write the matrix equation, not to find the eigenvalues or solve the equation so I guess you will learn what complex eigenvalues mean in this situation later.
     
  4. Nov 6, 2011 #3
    I'm trying to finish the very same problem. In addition to what is stated above, the problem says: "Find its eigenvectors, and compute the solution that starts from y(0), y'(0)=0."

    I found the eigenvalues to be i and -i, and the eigenvectors to be
    [1 -i]^T and [1 i]^T.

    I wrote the solution to start as y(t)=(1/2)*e^it*[1 i]^T + (1/2)*e^-it*[1 -i]^T.

    From there I don't know how to proceed. Any suggestions?
     
  5. Nov 6, 2011 #4

    Mark44

    Staff: Mentor

    eit = cos(t) + isin(t)

    Note that (1/2)(eit + e-it) = cos(t)
    and that (1/(2i))(eit - e-it) = sin(t)
     
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