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Homework Help: Test questions

  1. Dec 9, 2005 #1
    I'm going over old exam questions for the final. I'm not sure what the departament will put on the exams so I'm trying to go over as much as possible, but I having problems figuring certain problems out:

    1)
    y^(5) + 3y^(4) - 5y''' - 15'' + 4y' + 12y = 0

    How do you find the five solutions to the equation and then put it into the 5 dimensional system of first order equations.

    2)
    Let A be a square matrix, and let Y be a fundamental matrix for the homogeneous linear system x' = Ax.

    a) Verify by substituion that e^tA and (YY_0)^-1 are both solutions to the matrix IVP E(t)' = AE(t), E(0) = I, where Y_0 = Y(0).

    All I can intepret from this is the e^tA = (YY_0)^-1 so if I can find Y I can slove this...but nothing else...

    b) Find e^-A

    A = [[1 2],[2,1]]

    Any help would be apprecaited, thank you
     
  2. jcsd
  3. Dec 9, 2005 #2
    For (1), the standard trick of assuming y=A*ekt gives you a solution basis fast, as all the roots of the characteristic equation are integers.
     
  4. Dec 9, 2005 #3

    George Jones

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    Staff Emeritus
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    Gold Member

    For 2 b), use eigenvectors and eigenvalues to write -A = S D S^(-1), where D is a diagonal matrix. Note that e^(-A) = e^(S D S^(-1)) = S (e^D) S^(-1).

    Regards,
    George
     
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