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Linear systems question

  1. Oct 21, 2007 #1
    Hi all,
    I have a system of linear DE 's as under.

    [tex]\dot{x}_1(t)=-tx_1+x_2[/tex]

    and

    [tex]\dot{x}_2(t)=-tx_2+x_1[/tex]

    Now how do we find the solution


    [tex]x(t)^{T}=[x_1, x_2][/tex]


    I tried to find a similar post but could not. Any help would be highly appreciated. I am in the midst of writing a code wherein I need to solve this system of linear DE's.

    Thanks in advance.


    Gaganaut.
     
  2. jcsd
  3. Oct 21, 2007 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I can think of about a dozen different ways to do that problem. Unfortunately, since you have not shown how you would attempt such a probem, I have no idea which of them is appropriate for you. Do you see my dilemma?
     
  4. Oct 21, 2007 #3
    The most straightforward way seems to be that you could subtract the first from the second to obtain one (separable) equation for their difference.
     
  5. Oct 21, 2007 #4
    Linear differential equations: Solution so far

    First of all thanks for getting back. So far I have brought the system in a matrix form as under.

    [tex]\[ \left[ \begin{array}{c}
    \dot{x}_1\\
    \dot{x}_2 \end{array} \right]=\left[ \begin{array}{cc}
    -t & 1\\
    1 & -t \end{array} \right] \[ \left[ \begin{array}{c}
    x_1\\
    x_2 \end{array} \right]\]
    [/tex]

    I am a bit skeptical about the further steps that I did and that's when I decided to get help on this. I have written a formula for integrating the system as under.

    [tex]\underline{x}(t) = exp\left(\int_{t_0}^t A(\tau)\,d\tau \right)\underline{x}_0(t)[/tex]

    where [tex]A(\tau)=\left[ \begin{array}{cc}
    -\tau & 1\\
    1 & -\tau \end{array} \right][/tex]

    I cannot get any further. I definitely want to take the matrix approach as it is easier to code in for me. Also, I might be wrong with the integral method, so I would appreciate a better method preserving the matrix and vector form.

    Thank you.

    Gaganaut
     
    Last edited: Oct 21, 2007
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