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Exp Matrix to solve an initial value problem

  1. Sep 23, 2008 #1
    Exp Matrix to solve an initial value problem(urgent)

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

    Given the matrix [tex]A = \left[\[\begin{array}{ccc} 2 & 2 \\ 1& 3 \end{array}\right][/tex]

    a) Find the [tex]e^{tA}[/tex]

    2) Solve the [tex]x' = Ax + (1,0) \begin{array}{c} \end{array}[/tex] where x(0) = (0,0)

    3. The attempt at a solution

    a) [tex]e^{tA} = P_{A} \cdot e^{Dt} \cdot P'[/tex]

    Which in my book gives

    [tex]e^{tA} = \left[\[\begin{array}{ccc} -2 & 1 \\ 1& 1 \end{array}\right] \cdot \left[\[\begin{array}{ccc} e^{t} & 0 \\ 0& e^{16t} \end{array}\right] \cdot \left[\[\begin{array}{ccc} -\frac{1}{3} & \frac{1}{3} \\ \frac{4}{3}& \frac{8}{3} \end{array}\right] [/tex]

    [tex]e^{tA} = \left[\[\begin{array}{ccc} \frac{2}{3}\cdot e^{t} + \frac{4}{3}\cdot e^{16t} & \frac{-2}{3}\cdot e^{t} + \frac{8}{3}\cdot e^{16t} \\ \frac{-1}{3}\cdot e^{t} + \frac{4}{3}\cdot e^{16t} & \frac{1}{3}\cdot e^{t} + \frac{8}{3}\cdot e^{16t} \end{array} \right][/tex]

    Doesn't that look okay??

    b) From what I remember the solution for x' can be written as [tex]X = e^{tA} \cdot C[/tex]

    Which in my case gives [tex]X = e^{tA} \cdot \left[\begin{array}{c} 0 \\ 0 \end{array} \right][/tex]

    This is how my textbook argues how solve such eqn, but if I fry to x' I totally different result. What am I doing wrong??? Or could somebody please be so kind to lead me on the right path/track?? :)

    Last edited: Sep 23, 2008
  2. jcsd
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