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Solve the system of linear differential equations using eigenvectors and eigenvalues

  1. Dec 7, 2010 #1
    1. The problem statement, all variables and given/known data
    solve the system of first-order linear differential equations:
    (y1)' = (y1) - 4(y2)
    (y2)' = 2(y2)

    using the equation:
    (λI -A)x = 0

    2. Relevant equations
    using eigenvectors and eigenvalues
    in the book 'Elementary Linear Algebra' by Larson and Falvo - Section 7.4 #19

    3. The attempt at a solution
    (y1)' = (y1) - 4(y2)
    (y2)' = 2(y2)

    makes the matrix:
    [1 -4]
    [0 2]

    (λI-A)x = 0
    [λ-1 4]
    [0 λ-2]

    which gives the eiganvalues:
    (λ-1)(λ-2)
    λ = 1
    λ = 2

    then input it back into the original matrix A:
    λ = 1
    (1I - A) =
    [0 4]
    [0 -1]

    which reduces to:
    [0 0]
    [0 1]

    and creates the eiganvector:
    [0]
    [1]

    then continued onto:
    λ = 2
    (2I - A) =
    [1 4]
    [0 0]

    which creates the eigenvector:
    [1]
    [-4]

    so then I put the two eigenvectors together and create P:
    [1 1]
    [0 -4]

    then I need to find P^-1 so that I may use PAP^-1 to find the differential equation:
    I set P equal to the identity matrix and solve (I think this is where things start to go wrong...)
    [1 1/4]
    [0 -1/4]

    then I solve for PAP^-1
    [1 1] [1 -4] [1 1/4]
    [0 -4] [0 2] [0 -1/4]

    I take AP^-1 first in the multiplication and get:
    [1 5/4]
    [0 1/2]

    and then add on P and get:
    [1 7/4]
    [0 2]

    and that just looks wrong, I stopped there in fear of continuing further, but the answer IN THE BOOK reads:
    (y1) = (C1)e^t - 4(C2)e^2t
    (y2) = (C2)e^2t

    and it looks like if I were to continue it would not even come close to that answer...Where did I go wrong?
     
  2. jcsd
  3. Dec 7, 2010 #2

    vela

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    Re: solve the system of linear differential equations using eigenvectors and eigenval

    You calculated the eigenvectors incorrectly.
     
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