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Simplifying a solution that has complex eigenvalues

  1. Apr 27, 2010 #1
    1. The problem statement, all variables and given/known data

    I'll give an example.

    Ex: x'=[-1/2 1; -1 -1/2]x.

    2. Relevant equations

    Assume a solution of the form x=$ert for these type of problems.

    Euler's formula: ebi = cosb + isinb

    3. The attempt at a solution

    |A-rI|=0

    ---> r= -1/2 +/- i

    ---> x= e-t/2 ( C1(cost + isint)(1 i)T + C2(cos(-t) +isin(-t))(1 -i)T )

    I understand that I can simplify a little with the fact that sin(-t)=sin(t) and cos(-t)=-cos(t), but I don't understand how to simplify it all the way to

    C1e-t/2 (cost -sint)T + C2e-t/2(sint cost)T,

    which is the answer in the book.

    So, explain.
     
  2. jcsd
  3. Apr 27, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    so basically your 2 linearly independent solutions are

    [tex] \textbf{x}_1 = e^{-t/2}(cos(t) + i.sin(t))(\begin{matrix} 1 \\ i \end{matrix}) [/tex]
    [tex] \textbf{x}_2 = e^{-t/2} (cos(-t) + i.sin(-t))(\begin{matrix} 1 \\ -i \end{matrix}) [/tex]

    note that any linear combination of these will also be a solution, so perhaps you could try taking 2 linear combinations that simplify things... making sure they are still linearly independent
     
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