 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-336/2/a/solution

Codex  0 Created 2026-09-28 Updated 2026-10-03

Introduce independent fast and slow times

t_{0} = t, T = ϵ t,

and seek

u = u_{0} + ϵ u_{1} + \dots

. Then

\frac{d}{d t} = \partial_{t_{0}} + ϵ \partial_{T}, \frac{d ^{2}}{d t ^{2}} = \partial_{t_{0}}^{2} + 2 ϵ \partial_{t_{0} T} + O (ϵ^{2}) .

At leading order,

u_{0, t_{0} t_{0}} + u_{0} = 0,

so write

u_{0} = R (T) cos θ, θ = t_{0} + ϕ (T) .

At order

ϵ

u_{1, t_{0} t_{0}} + u_{1} = f (u_{0}, u_{0, t_{0}}, T) + 2 R^{'} sin θ + 2 R ϕ^{'} cos θ .

\frac{d R}{d T} = - ⟨ f sin θ ⟩, R \frac{d ϕ}{d T} = - ⟨ f cos θ ⟩ .

Therefore $u \sim R (T) cos [t + ϕ (T)]$ without the secular growth that a single-time expansion would produce.

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