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Math Help - System of second order linear homogenous differential coupled equations

  1. #1
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    System of second order linear homogenous differential coupled equations

    my question is: what is the general solution of this system of coupled diff. equations:

    f ''i = Cijfj

    Cij is a matrix, fj(z) are functions dependent of z. indexes i and j go from 0 to N .
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  2. #2
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    Hi. This is what I think it is without looking so I'm not sure ok? You have N second-order DEs which I can convert to 2N first-order DEs just by adding N more variables. I'd then have a system of 2N first-order autonomous DEs which I can then in principle, compute the eigenvalues and vectors and then claim, the solution is:

    \textbf{F}_{2N}=\sum_{n=1}^{2N} k_n e^{\lambda_n z} \textbf{V}_n

    where \{\lambda_n\} is the set of eigenvalues and \{\textbf{V}_n\}, the set of eigenvectors.

    I don't know. Is that right guys?
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  3. #3
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    sure i know linearization is a sulution. however i think there should be a general solution somewhere since it is "known" system , isnt it?
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  4. #4
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    okay, i found a possible solution:

    f_j=G_je^{i\sqrt{c_j}z}+H_je^{-i\sqrt{c_j}z}

    where G_j and H_j are integrating constants.. cj are eigenvalues of C and are complex..
    if so, one question remains... how are the indexes assigned to eigenvalues? i mean which eigenvalue will be c1....?

    is this correct?
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  5. #5
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    anyone can answer me ? please?
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