 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017-resit/9/c/solution

The initial vector

(d, 0)

d / 2

in each normalized normal mode. Zero initial velocities remove their sine terms. Therefore

x_{1} (t) = \frac{d}{2} (cos ω_{-} t + cos ω_{+} t), x_{2} (t) = \frac{d}{2} (cos ω_{-} t - cos ω_{+} t) .

At zero these give

(d, 0)

and both derivatives vanish; substitution into the coupled equations confirms the opposite signs of the two higher-frequency components.

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