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 Incoming links: Decay of a damped oscillator under exponentially decaying forcing

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 2026-10-18

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

or

a > 1

. The missing special case is

a = 1

, for which

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

This proves decay of a damped oscillator under exponentially decaying forcing. A one-sided upper bound on

g

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

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