 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016-resit/11/b/solution

Let

κ = T / (2 ma)

y = a e^{iω t}

det (3 - ω^{2} / κ - 1 - 2 2 - ω^{2} / κ) = 0,

λ^{2} - 5 λ + 4 = (λ - 1) (λ - 4) = 0

, where

λ = ω^{2} / κ

. Thus

ω_{-} = \frac{T}{2 ma}, ω_{+} = 2 \frac{T}{2 ma} .

These are angular frequencies; the corresponding ordinary frequencies are

f_{\pm} = ω_{\pm} / (2 π)

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