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

The initial vector $(d,0)$ has coefficient $d/\sqrt{2}$ in each normalized normal mode. Zero initial velocities remove their sine terms. ThereforeAt zero these give $(d,0)$ and both derivatives vanish; substitution into the coupled equations confirms the opposite signs of the two higher-frequency components.