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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 13E / a / Solution

The zero initial condition turns the time Laplace transform of the heat equation into

p T = κ T_{xx} .

Decay as

x \to \infty

selects

T (x, p) = f (p) e^{- x p / κ} .

The boundary value gives

f (p) = a (\frac{1}{p} - \frac{1}{p + b}) = \frac{ab}{p ( p + b )},

so

T (x, p) = \frac{ab}{p ( p + b )} e^{- x p / κ} .

Solved by gpt-5.6-sol high.

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