 Solution

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

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

A a_{j} = λ_{j} a_{j}

(\partial_{t} - A) (\frac{a _{j} e ^{i ω_{j} t}}{i ω _{j} - λ _{j}}) = a_{j} e^{i ω_{j} t} .

With both denominators nonzero, add the two responses:

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

Only the forcing component's own eigenvalue enters its denominator; a frequency matching a different eigenmode does not create resonance in this already diagonal forcing component.

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