System of homogeneous second order differential equations

Question

I'm trying to solve a system of three homogeneous second order differential equations, that is I want $x_1(t)$, $x_2(t)$ and $x_3(t)$ so that I can plot them in Matlab.

I am given \begin{align} \frac{d^2}{dt^2}x_1+\omega_0^2x_1-\omega_0^2x_2=0\\ \frac{d^2}{dt^2}x_1+2\omega_0^2x_2-\omega_0^2x_3=0\\ \frac{d^2}{dt^2}x_3+\omega_0^2x_3-\omega_0^2x_2=0\\ \end{align}

From the lecture I had I understand that this system can be solved by taking the Fourier transform and in the frequency domain express the problem as a system of linear equations and solve for the eigenvalues and eigenvectors \begin{align} (4\pi^2\nu^2+\omega_0^2)x_1(\nu)-\omega_0^2(\nu)=0\\ -\omega_0^2x_1(\nu)+(4\pi^2\nu^2+2\omega_0^2)x_2(\nu)-\omega_0^2x_3(\nu)=0\\ -\omega_0^2x_2(\nu)+(4\pi^2\nu^2+\omega_0^2)x_3(\nu)=0 \end{align} Or in matrix form with $4\pi^2\nu^2=-\lambda$ $$ \begin{bmatrix} -\omega_0^2 - \lambda & -\omega_0^2 & 0 \\ -\omega_0^2 & 2\omega_0^2 - \lambda & -\omega_0^2 \\ 0 & -\omega_0^2 & \omega_0^2 - \lambda \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} $$ This is an eigenvalue problem which I solve in Matlab with [V,D]=eig(X) with X being the matrix $$ X=\omega_0^2 \begin{bmatrix} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{bmatrix}\\ , D= \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0\\ 0 & 0 & \lambda_3 \end{bmatrix} ,V= \begin{bmatrix} V_{11} & V_{12} & V_{13} \\ V_{21} & V_{22} & V_{23} \\ V_{31} & V_{32} & V_{33} \\ \end{bmatrix} $$ Where the eigenvalues are confined in the diagonal of D, and the eigenvectors are the columns of V with the first column being the first eigenvector, the second column the second eigenvector and so on. Now I want to write the solutions to the system of the differential equation specified in the beginning. I want something like $$ \begin{bmatrix} x_1(t)\\ x_2(t)\\ x_3(t) \end{bmatrix} = C1\cos(\sqrt{\lambda_1}t) \begin{bmatrix} V_{11} \\ V_{21} \\ V_{31} \end{bmatrix} + C2\sin(\sqrt{\lambda_1}t) \begin{bmatrix} V_{11} \\ V_{21} \\ V_{31} \end{bmatrix} + C3\cos(\sqrt{\lambda_2}t) \begin{bmatrix} V_{12} \\ V_{22} \\ V_{32} \end{bmatrix} + C4\sin(\sqrt{\lambda_2}t) \begin{bmatrix} V_{12} \\ V_{22} \\ V_{32} \end{bmatrix} + C5\cos(\sqrt{\lambda_3}t) \begin{bmatrix} V_{13} \\ V_{23} \\ V_{33} \end{bmatrix} + C6\sin(\sqrt{\lambda_3}t) \begin{bmatrix} V_{13} \\ V_{23} \\ V_{33} \end{bmatrix} $$ With $C1,C2,C3,C4,C5,C6$ being constants that can be found from initial conditions, I don't know if imaginary or real. Any suggestions for how the general solution should look, is it correct written in the trigonometric functions or should there be some exponential also that would than add damping to the oscillation?

Answer 1

It looks like you don't really have a system of ODE the way you have it written because you have two relationships for $\frac{d^2x_1}{dt^2}$. The type of system you have is really called a $Differential$ $Algebraic$ $System$, mostly because it can be written as

$\begin{cases} x''_1 + 2\omega^2x_2 - \omega^2x_3=0 \\ x''_3 - \omega^2x_2 + \omega^2 x_3=0 \\ \omega^2x_1 - 3\omega^2x_2 + \omega^2x_3 =0 \end{cases}$

Unless, of course, you wrote a typo in your post as I noted in my comments. It is possible to turn this system into a system of ODE by repeatedly differentiating the third equation, but because of my sneaking suspicion that the original post was made in error, I'm going to refrain from continuing the problem until OP either acknowledges the mistake or says that this problem was stated correctly.

Lastly, Fourier Transforms are absolutely not necessary and completely going to be missing the point of this problem. In fact, they won't do you a lot of good. Fourier Transforms allow us to turn our problem into a a linear problem of the form $Ax = 0$, as you have written, but that has $nothing$ to do with eigenvalues or eigenvectors.