1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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,

    4...3
    -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

    (a
    -2a)

    and

    (b
    -b)

    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

    HallsofIvy

    User Avatar
    Staff Emeritus
    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.
    Then
    [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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook