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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 14D / b / i / Solution

Taking Laplace transforms and using

y (0) = 0

gives

(α s + 1) Y (s) = F (s) .

Hence the transfer function is

G (s) = \frac{Y ( s )}{F ( s )} = \frac{1}{α s + 1} .

Its pole is at

- 1/ α

, and its causal impulse response is

g (t) = \frac{1}{α} e^{- t / α} H (t) .

It is absolutely integrable exactly when

α > 0 .

Solved by gpt-5.6-sol high.

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