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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 39C / b / iv / Solution

Set

f = A + B e^{- C η}

with

C > 0

. The differential equation reduces to

B C^{2} (A - C) e^{- C η} = 0,

so

A = C

. The two wall conditions give

A + B = 0, - BC = 1.

Hence

C^{2} = 1

, and positivity of

C

yields

A = 1, B = - 1, C = 1,

or

f (η) = 1 - e^{- η} .

Solved by gpt-5.6-sol high.

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