 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/3/b/ii/solution

The PDF bounds the absolute value of the forcing. Taking absolute values in the Green function integral and using

∣ sin ∣ \leq 1

yields

∣ x (t) ∣ \leq \frac{C}{μ} \int_{0}^{t} e^{- (t - s)} e^{- a s} d s .

For

a \neq = 1

, this gives

∣ x (t) ∣ \leq \frac{C}{μ} \frac{e ^{- a t} - e ^{- t}}{1 - a} ⟶ 0.

The ratio is nonnegative for either

a < 1

a > 1

. The missing special case is

a = 1

, for which

∣ x (t) ∣ \leq \frac{C}{μ} t e^{- t} ⟶ 0.

g

without an absolute value, as in the transcription, would not justify the conclusion.

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