Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 3 / iii

Table of contents
- Solution iii

Solution

The original PDF has the absolute-value hypothesis

∣ g (ξ) ∣ < C e^{- s ξ}

, which is missing from the local OCR. From the Green's function formula and

∣ sin ∣ \leq 1

,

∣ y (x) ∣ \leq \frac{C}{b} \int_{0}^{x} e^{- a (x - ξ)} e^{- s ξ} d ξ = \frac{C}{b} e^{- a x} \int_{0}^{x} e^{- (s - a) ξ} d ξ = \frac{C}{b ( s - a )} e^{- a x} [1 - e^{- (s - a) x}] \leq \frac{C}{b ( s - a )} e^{- a x} .

Because

s > a > 0

, the upper bound tends to zero. The squeeze theorem proves

y (x) \to 0 as x \to \infty

.

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