 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-2/13/iii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

- 1

, matching the exponential forcing. Put

y = e^{- x} w

; then

y^{''} + 2 y^{'} + y = e^{- x} w^{''}

, so

w^{''} = 8

w (0) = w^{'} (0) = 0

, hence

y = 4 x^{2} e^{- x} .

The extra quadratic factor is the resonance associated with the double root.

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