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?