 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-2/17/b/solution

\overset{x}{¨}_{i} = - ω^{2} a_{i} cos ω t

. Matching the coefficients of the common time factor in the three equations gives

m ω^{2} a_{1} m ω^{2} a_{2} m ω^{2} a_{3} = k (2 a_{1} - a_{2}), = k (- a_{1} + 2 a_{2} - a_{3}), = k (- a_{2} + 2 a_{3}) .

These algebraic relations hold for the entire oscillation, including the isolated times when the common cosine vanishes.

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