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Homework Help: Stability of a solution

  1. Sep 6, 2007 #1
    1. The problem statement, all variables and given/known data
    Let a(t), b(t) and c(t) be continuous functions of t over the interval [tex][0,\infty)[/tex]. Assume (x,y) = [tex](\phi(t), \psi(t))[/tex] is a solution of the system
    [tex]\dot{x} = -a^2(t)y + b(t), \dot{y} = a^2(t)x + c(t)[/tex]

    Show that this solution is stable.

    3. The attempt at a solution
    I rearranged the system to get

    [tex]\frac{d}{dt}\binom{x}{y} = \binom{-a^2(t)y + b(t)}{a^2(t)x + c(t)}
    = \left( \begin{array}{cc} 0 & -a^2(t) \\ a^2(t) & 0 \end{array} \right)\binom{x}{y} + \binom{b(t)}{c(t)}[/tex]

    I've only dealt with constant coefficient linear systems before, so I'm having trouble with this question.

    Let [tex]A = \left( \begin{array}{cc} 0 & -a^2(t) \\ a^2(t) & 0 \end{array} \right) [/tex].

    I'm not sure if I can treat A like a constant matrix, i.e. take the determinant of A to be [tex]a^4(t)[/tex] and trace(A) = 0 by treating t as constant. Also, what role does the [tex]\binom{b(t)}{c(t)}[/tex] play here?

    Can someone please help me?

    Thank you.

    Last edited: Sep 6, 2007
  2. jcsd
  3. Sep 6, 2007 #2


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  4. Sep 7, 2007 #3
    Do I need to find a Lyapunov function?
  5. Sep 7, 2007 #4
    CouldnĀ“t you just find an integrating factor h integrate the whole thing and find an integral expression for the whole thing.
    Then you got the time T operator differentiate ( take integral sign away) and show that your solution is an eigenvector with Eigenvalue(floquet multiplier) = 1 for the system?
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