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Math Help - Using eigenvectors and eigenvalues to solve DE in matrix form

  1. #1
    Newbie
    Joined
    Apr 2009
    Posts
    9

    Using eigenvectors and eigenvalues to solve DE in matrix form

    Hi, my friend asked for help with a question and I'm wondering if there's a better method than mine by the question wording.

    We have the matrix
    A =\left( \begin{array}{cc}<br />
2 & 1 \\<br />
1 & 2<br />
\end{array}\right)
    and have to find the eigen values and vectors which are
    3 with \left( \begin{array}{c}<br />
1 \\<br />
1<br />
\end{array} \right)
    and
    1 with \left( \begin{array}{c}<br />
-1 \\<br />
1<br />
\end{array}\right)

    The next part is to solve the set of equations

    u_1'' = 2 u_1' + u_2'
    u_2'' = u_1' + 2 u_2'
    u_1(0) = 0
    u_1'(0) = 1
    u_2(0) = 1
    u_2'(0) = 0
    using the eigenvalues and eigenvectors.

    As this is of the matrix form I solve this by writing characteristic equations with the eigenvalues as the exponentials
    u_1 = A e^{t} + B e^{3t} + C
    u_2 = D e^{t} + E e^{3t} + F

    and substituting into the above equations to get six equations in A,B,C,D,E and F thus leading to
    u_1 = \frac{1}{6}\left( 3e^{t} + e^{3t} -4 \right)
     <br />
u_2 = \frac{1}{6}\left( -3e^{t} + e^{3t} +8 \right)<br />
    which is right!

    However, I haven't used the eigenvectors only the eigenvalues, is there another (better ? ) method that I should be using?

    Thanks!
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  2. #2
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Hi

    You can write the set of equations as
    U'' = AU' where U = \left( \begin{array}{c}<br />
u_1 \\<br />
u_2<br />
\end{array} \right)

    A is diagonalisable into the diagonal matrix D =\left( \begin{array}{cc}<br />
3 & 0 \\<br />
0 & 1<br />
\end{array}\right) through a transition matrix P such that P^{-1}AP = D

    Let V = \left( \begin{array}{c}<br />
 v_1 \\<br />
 v_2<br />
 \end{array} \right) = P^{-1}U \Rightarrow U = PV

    U'' = AU' is transposed into PV'' = APV' \Rightarrow V'' = P^{-1}APV' = DV'

    Since D is diagonal V'' = DV' is easily solved.

    Then U = PV is used to find u_1 and u_2
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