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System of ODEs

  1. Nov 18, 2013 #1
    1. The problem statement, all variables and given/known data

    Given the matrix [itex]b=\begin{pmatrix}-1&0&-1\\-4&3&-1\\0&0&-2\end{pmatrix}[/itex] decide if the system of ODEs, [itex]\frac{dx}{dt}=Bx[/itex] is decoupled. If yes find the general solution x=xh(t)

    2. Relevant equations



    3. The attempt at a solution
    I would say the matrix is decoupled since the second equation involving [itex]2x[/itex]2(t) can be solved without the other two equations. Then the third equation can be solved without knowing [itex]x[/itex]1(t). We have:

    x1(t)-x2(t)-3x3(t)=x'1
    2x2(t)=x'2
    x2(t)+4x3(t)=x'3

    Im not sure where to go from here.
     
  2. jcsd
  3. Nov 18, 2013 #2

    pasmith

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    Homework Helper

    How did you get that? You have
    [tex]
    \begin{pmatrix} \dot x_1 \\ \dot x_2 \\ \dot x_3 \end{pmatrix}
    = \begin{pmatrix}-1&0&-1\\-4&3&-1\\0&0&-2\end{pmatrix}
    \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}
    [/tex]
    Hence
    [tex]
    \dot x_1 = -x_1 - x_3 \\
    \dot x_2 = -4x_1 + 3x_2 - x_3 \\
    \dot x_3 = -2x_3
    [/tex]
     
  4. Nov 19, 2013 #3
    yes thats what I wrote down on paper. There was a bit of a mistranslation when trying to write it in latex.
     
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