 Solution

ID: 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/solution

∣ 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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