 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/ia/paper-2/7d/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

The second component is resonant. Differentiating

t e^{λ_{2} t}

supplies the term missing from the exponential ansatz:

(\partial_{t} - A) (t a_{2} e^{λ_{2} t}) = a_{2} e^{λ_{2} t} .

x_{p} = \frac{a _{1} e ^{i ω_{1} t}}{i ω _{1} - λ _{1}} + t a_{2} e^{λ_{2} t} .

The linearly growing oscillatory envelope is forced resonance in a differential equation; it does not contradict bounded oscillation of the complementary function. The worked initial-value solution in the question's root Solution uses this same resonant mechanism.

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