 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/15/b/ii/solution

m^{2} + 2 m + 2 = 0

, with roots

- 1 \pm i

. Therefore

u (t) = e^{- t} (A cos t + B sin t) .

If the second component is also wanted, the relation

v = u^{'} + 3 u

gives

v (t) = e^{- t} [(2 A + B) cos t + (2 B - A) sin t] .

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