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Multiple eigenvalue solutions

  1. Nov 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Solve the system.

    dx/dt=[1 -4; 4 -7]*x with x(0)=[3; 2]


    2. Relevant equations



    3. The attempt at a solution
    I am apparently not getting this at all. Can someone walk me through it? I konw I have to first find the eigenvalues and eigenvectors:

    (1-λ)(-7-λ)+16=0
    λ2+6λ+9=0
    λ=-3,-3

    So, (A-3I)C1 = 0
    (A-3I)= [4 -4; 4 -4]

    So, eigenvector = [1; 1]

    (A-3I)C2=C1

    Eigenvector = [1; 0]


    And, x1= [1; 1] e-3t
    x2 = ([1; 1]t + [1; 0])e-3t

    So, using fundamental matrices...

    F = [ e-3t (t+3) e-3t; e-3t t e-3t]
    F(0) = [1 3; 1 0]
    (F(0))'= [0 1; 1/3 -1/3]

    So,
    x(t)=F*(F(0))'*X0 = [X1 ; X2]

    Is there anything wrong with my method?
    The homework asks for two answers: X1 and X2 and I'm not exactly sure what that is asking for. Thanks! Any help is appreciated.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 8, 2009 #2

    lanedance

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    Homework Helper

  4. Nov 9, 2009 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The initial value problem has, of course, a single solution. Perhaps the "X1" and "X2" are the two independent solutions to the equation without the initial values.

     
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