 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014/9/b/solution

x = a cos (ω t + φ)

. The two normal modes satisfy

[(2 - 1 - 1 1) - s I] a = 0, s = \frac{m ω ^{2}}{k} .

(2 - s) (1 - s) - 1 = s^{2} - 3 s + 1 = 0

. Hence

a_{\pm} = \frac{3 \pm 5}{2}, ω_{\pm}^{2} = \frac{k}{m} \frac{3 \pm 5}{2} .

Here

a_{\pm}

denotes the scalar coefficient requested in the source, whereas bold

a

denotes a mode-amplitude vector. Both roots are positive.

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