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Homework Help: Matrix Differential Equation

  1. Feb 17, 2012 #1
    1. The problem statement, all variables and given/known data
    The questions are in the image

    3. The attempt at a solution
    My solutions are
    V1=3*(1 -2)e-2t+ (-2) (1 -3)e-3t
    V2=1*(1 -2)e-2t+ (-1) (1 -3)e-3t

    How do I get the X matrix since my solutions are in exponential still.

    Thank you for all the help
     

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    Last edited: Feb 17, 2012
  2. jcsd
  3. Feb 18, 2012 #2
    According to your solution,
    [tex]X=(v_1\ v_2)=\begin{pmatrix}3e^{-2t}-2e^{-3t}&-6e^{-2t}+6e^{-3t}\\
    e^{-2t}-e^{-3t}&-2e^{-2t}+3e^{-3t}\end{pmatrix}.[/tex]
     
  4. Feb 18, 2012 #3
    That means I just plug in t=0 to prove that x(0)=(1 0
    0 1) and also use the same method to prove that dx/dt =AX
     
    Last edited: Feb 18, 2012
  5. Feb 18, 2012 #4

    vela

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    You have the columns and rows swapped.
     
  6. Feb 18, 2012 #5
    Even after multiplying it I get this respective solutions, what do I do next?
     

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  7. Feb 18, 2012 #6

    vela

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    You can write v1 as a single vector:
    $$\vec{v}_1 = \begin{pmatrix} 3e^{-2t} - 2e^{-3t} \\ -6e^{-2t} + 6e^{-3t}\end{pmatrix}$$Do the same for ##\vec{v}_2##.
     
  8. Feb 18, 2012 #7
    i get that part but after that what do I do to get just numbers in my 2x2 matrix so that I can prove dx/dt =AX and x(0)= \begin{pmatrix}1 0\\ 0 1\end{pmatrix}
     
  9. Feb 18, 2012 #8

    vela

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    Like the problem says, the first column of X is v1. Its second column is v2. It's not going to be just numbers. I'm not sure why you think it has to be.
     
  10. Feb 18, 2012 #9
    Okay but then if I put t=0 into the x equation, I do not get the identity matrix and how would I verify that dx/dt=AX by just differentiating the X matrix?.

    Thank you for all the help
     
  11. Feb 18, 2012 #10

    vela

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    Show us how you're calculating X when t=0.
     
  12. Feb 18, 2012 #11
    Okay I got just did a arithmetic error, but how do I verify that dx/dt=AX
     
  13. Feb 18, 2012 #12

    vela

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    Calculate both sides and show they're equal to each other.
     
  14. Feb 18, 2012 #13
    Okay I will give it a try, thank you very much for all the help.
     
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