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Homework Help: Repeated Eigenvalues problem

  1. Apr 24, 2010 #1
    This problem, and all the others on this homework assignment, are making me angry.

    1. The problem statement, all variables and given/known data

    Find the general solution of the system of equations.

    .....

    x'=[-3 5/2; -5/2 2]x

    2. Relevant equations

    Just watch me solve

    3. The attempt at a solution

    Assume there's a solution x= $ert, where I'm denoting a vector with constant entries $.

    ----> (A-rI)=$
    ----> (A-rI) is singular
    ----> det(A-rI)=0
    ---->(-3-r)(2-r)-(-5/2)(5/2)=0
    ----> r= 1/2
    ---->(A-(1/2)I)$=(0 0)T

    But then I have a problem because the only solution is $=(0 0)T.

    I'd know how to proceed, were it not for this dilemma. Next I would Assume there's a second solution x=$tert + #ert, where # is another vector with constant entries, and then solve.
     
  2. jcsd
  3. Apr 24, 2010 #2

    Dick

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    The solution to (-3-r)*(2-r)-(-5/2)*(5/2)=0 is r=(-1/2), isn't it? Not r=1/2.
     
  4. Apr 24, 2010 #3
    r2 + 3r - 2r -6 + 25/4 = 0
    r2 + r + 1/4 = 0
    (r+1/2)2=0
    r= -1/2

    D'oh!

    (More than likely, I will some more questions about this assignment. Keep checking in to this thread.)
     
  5. Apr 24, 2010 #4
    Alright, Dick. Here is that other question I promised you.

    I have a problem in the form x'=P(t)x + g(t), and P(t) just happens to be a singular matrix with constant entries. Therefor I can't do a transformation to the form y'= Dy + h(t).

    Grrrrr! Now what?
     
  6. Apr 24, 2010 #5

    Dick

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    If P were diagonal, then you could split it into two separate equations. The usual way to do this is to find a basis of R^2 where P is diagonal, then express x and g in terms of that basis and solve. You aren't being very specific here.
     
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