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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 12A / c / Solution

The transform factors as

s^{- 2} (s + 2)^{- 2}

, whose inverse factors are

t

and

t e^{- 2 t}

. The Laplace convolution theorem gives

f (t) = \int_{0}^{t} τ (t - τ) e^{- 2 (t - τ)}, d τ,

and direct integration produces

f (t) = \frac{1}{4} (t - 1 + (t + 1) e^{- 2 t}),

confirming part (b).

Solved by gpt-5.6-sol high.

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