Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 9 / a / Solution

The central spring extension is $x_2-x_1$. Applying Hooke's law and Newton's second law to each mass givesIn matrix form $\ddot{x}+(k/m)\left(\begin{array}{cc}1+ϵ& -ϵ\\ -ϵ& 1+ϵ\end{array}\right)x=0$. The positive off-diagonal coupling in the force equations means that displacing one mass pulls the other in the same direction.