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

1. ## 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 .

2. 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?

3. sure i know linearization is a sulution. however i think there should be a general solution somewhere since it is "known" system , isnt it?

4. 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?