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Elementary ODEs matrix integration help

  1. Mar 28, 2008 #1
    1. The problem statement, all variables and given/known data
    I'm trying to understand the Variation of Parameters in ODEs and I came up to this following expression which i cannot solve:

    [tex]{2\,{e}^{-t}{e}^{-3\,t}\choose {e}^{-t}{e}^{-3\,t}} \int {\,{e}^{t} {e}^{\,t}\choose {e}^{3t}{2e}^{-3\,t}} {10\,\cos \left( t \right) \choose 2\,{e}^{-t}}[/tex]

    Can I just integrate each individual component or must I use matrix multiplication first? If anyone could help me I would appreciate it greatly. I'm not sure how to even start on this so I don't have any work to show.
  2. jcsd
  3. Mar 28, 2008 #2


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    It's not at all clear what those mean. Do you mean
    [tex]\left(\begin{array}{cc}2e^{-t} & e^{-3t} \\ e^{-t} & e^{-3t}\end{array}\right)\int \left(\begin{array}{cc}e^t & e^t \\ e^{3t} & 2e^{-3t}\end{array}\right)\left(\begin{array}{cc}10 & cos(t) \\ 2 & e^{-t}\end{array}\right) dt[/tex]
    (Click on the LaTex to see the code I used.)

    To answer your question, yes, you must multiply before integrating: [itex]\int f(x)g(x) dx[/itex] is NOT [itex](\int f(x)dx)(\int g(x)dx)[/itex].
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