Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 11 / b / Solution

Let $\kappa =T/(2ma)$ and seek a normal mode $\mathbf{y}=\mathbf{a}{e}^{i\omega t}$. Its eigenvalue equation isor ${\lambda }^2-5\lambda +4=(\lambda -1)(\lambda -4)=0$, where $\lambda ={\omega }^2/\kappa$. ThusThese are angular frequencies; the corresponding ordinary frequencies are $f_{\pm }={\omega }_{\pm }/(2\pi )$.