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Thread: 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 $\displaystyle N$ second-order DEs which I can convert to $\displaystyle 2N$ first-order DEs just by adding $\displaystyle N$ more variables. I'd then have a system of $\displaystyle 2N$ first-order autonomous DEs which I can then in principle, compute the eigenvalues and vectors and then claim, the solution is:

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

    where $\displaystyle \{\lambda_n\}$ is the set of eigenvalues and $\displaystyle \{\textbf{V}_n\}$, 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:

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

    where $\displaystyle G_j and H_j$ 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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