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Homework Help: Using Eigenvalues and Eigenvectors to solve Differential Equations

  1. Aug 18, 2009 #1
    1. The problem statement, all variables and given/known data

    x1(t) and x2(t) are functions of t which are solutions of the system of differential equations

    x(dot)1 = 4x1 + 3x2
    x(dot)2 = -6x1 - 5x2

    Express x1(t) and x2(t) in terms of the exponential function, given that x1(0) = 1 and x2(0) = 0

    3. The attempt at a solution

    I've already put this into matrix form,

    -6 -5

    and then

    (4-λ) 3
    -6 (-5-λ)

    To find that the eigenvalues are -2 and 1, and then that the eigenvectors are




    but after that I'm completely stuck. I know you get into constants and such but I just can't remember how to continue.
  2. jcsd
  3. Aug 18, 2009 #2


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    Science Advisor

    You can write (a, -2a)= a(1, -2) and (b, -b) as b(1, -1). Use those vectors as columns in a matrix:
    [tex]A= \begin{bmatrix}1 & 1\\ -2 & -1\end{bmatrix}[/tex]
    and write [begin]A^{-1}[/itex] as its inverse matrix.
    [tex]A^{-1}\begin{bmatrix}1 & 1 \\ -2 & -1\end{bmatrix}A= \begin{bmatrix}-2 & 0\\ 0 & 1\end{bmatrix}[/tex]

    Writing your differential equation as differential equation as dY/dt= BY, We can multiply on both sides by [itex]A^{-1}[/itex] and write Y as [itex]Y= AX[/itex] so that [math]X= A^{-1}Y[/math] and the differential equation becomes [itex]d(A^{-1}Y)/dt= dX/dt= A^{-1}BY= (A^{-1}BA)[/itex] which is just
    [tex]\frac{dX}{dt}= \begin{bmatrix}-2 & 0 \\ 0 & 1\end{bmatrix}X[/tex]
    which is easy to solve. Once you have X, then, of course, Y= AX.
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