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System of differential equations, Maple tells me I'm wrong

  1. Sep 10, 2010 #1
    Hi there,

    I have a system of differential equations which I set up in a matrix like this:

    [tex]\left[ \begin {array}{ccc} 3/2&-1&-1/2\\ \noalign{\medskip}-1/2&2&1/2
    \\ \noalign{\medskip}1/2&1&5/2\end {array} \right] = \left[
    \begin {array}{c} {\frac {d}{dt}}x \left( t \right)
    \\ \noalign{\medskip}{\frac {d}{dt}}y \left( t \right)
    \\ \noalign{\medskip}{\frac {d}{dt}}z \left( t \right) \end {array}

    Now I need to find the real solution to the system, so I find the eigenvectors and eigenvalues
    Eigenvalues <3,1,2> and Eigenvectors <-1,1,1>,<-1,-1,1> and <1,-1,1>

    and set up my solution like this


    I even get a nice matrix out saying
    \left[ \begin {array}{c} {\frac {d}{dt}}x \left( t \right)
    \\ \noalign{\medskip}{\frac {d}{dt}}y \left( t \right)
    \\ \noalign{\medskip}{\frac {d}{dt}}z \left( t \right) \end {array}
    \right] = \left[ \begin {array}{c} -{\it c\_2}\,{{\rm e}^{3\,t}}-{
    \it c\_3}\,{{\rm e}^{t}}+{\it c\_1}\,{{\rm e}^{2\,t}}
    \\ \noalign{\medskip}{\it c\_2}\,{{\rm e}^{3\,t}}-{\it c\_3}\,{{\rm e}
    ^{t}}-{\it c\_1}\,{{\rm e}^{2\,t}}\\ \noalign{\medskip}{\it c\_2}\,{
    {\rm e}^{3\,t}}+{\it c\_3}\,{{\rm e}^{t}}+{\it c\_1}\,{{\rm e}^{2\,t}}
    \end {array} \right]

    But Maple keeps giving me this solution instead:

    \left\{ x \left( t \right) ={\it \_C1}\,{{\rm e}^{2\,t}}+{\it \_C2}\,
    {{\rm e}^{3\,t}}+{\it \_C3}\,{{\rm e}^{t}},y \left( t \right) =-{\it
    \_C1}\,{{\rm e}^{2\,t}}-{\it \_C2}\,{{\rm e}^{3\,t}}+{\it \_C3}\,{
    {\rm e}^{t}},z \left( t \right) ={\it \_C1}\,{{\rm e}^{2\,t}}-{\it
    \_C2}\,{{\rm e}^{3\,t}}-{\it \_C3}\,{{\rm e}^{t}} \right\}

    I really can't figure out the problem here - is there something wrong with my eigenvectors or is it the solution I'm doing all wrong? Hope you can help :)
  2. jcsd
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