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Solveing differential equations system using diagonal matrix

  1. Dec 10, 2012 #1
    1. The problem statement, all variables and given/known data
    Solve this system of differential equations

    \begin{equation}
    x'_1=5x_1 + 2 x_2 - x_3 \\
    x'_2=-2x_1 + x_2 - 2x_3 \\
    x'_3=-6x_1 - 6 x_2
    \end{equation}


    2. Relevant equations
    3. The attempt at a solution
    This is my first time solving a problem like this and I just wanted to make sure if what I did was correct.

    Using the eigenvalues and eigenvectors, I found the matrix [itex]P[/itex] such that [itex]P^{-1} AP[/itex] is a diagonal matrix where [itex]A[/itex] is the coefficient matrix of the system above.

    [itex]P=[/itex]
    \begin{pmatrix}
    0&-1&-1\\
    1&1&0\\
    2&0&1
    \end{pmatrix}

    [itex]P^{-1} AP=[/itex]
    \begin{pmatrix}
    -3&0&0\\
    0&3&0\\
    0&0&6
    \end{pmatrix}


    ----------

    I am sure everything above is correct. So assuming it is correct, is the following process correct?

    [itex]U'=(P^{-1} AP)U[/itex]

    \begin{equation}
    u'_1=-3u_1 \\
    u'_2=3u_2 \\
    u'_3=6u_3
    \end{equation}

    \begin{equation}
    u_1=c_1e^{-3x} \\
    u_2=c_2e^{3x} \\
    u_3=c_3e^{6x}
    \end{equation}

    [itex]U=[/itex]
    \begin{pmatrix}
    c_1e^{-3x} \\
    c_2e^{3x} \\
    c_3e^{6x}
    \end{pmatrix}

    [itex]X=PU[/itex]

    I understand this is pretty long. I'd appreciate if someone can just give a quick look at the process and the substitutions I made (with U and U'). Thanks.
     
  2. jcsd
  3. Dec 10, 2012 #2
    Never mind. I just realized I can verify by seeing if my solution satisfies the given system. And it does. So I guess that means my soultion is correct. :)
     
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