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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 14A / c / Solution

For

λ = 2

, the Green representation of the boundary value problem is

y (x) = \int_{0}^{\infty} G (x; ξ) [- (ξ + 3) e^{- ξ}], d ξ .

Splitting the integral at

ξ = x

gives

y (x) = e^{- x} \int_{0}^{x} (ξ + 2), d ξ + (x + 2) \int_{x}^{\infty} e^{- ξ}, d ξ .

Therefore

y (x) = e^{- x} (\frac{x ^{2}}{2} + 3 x + 2) .

It satisfies

y (0) = 2 y^{'} (0)

and tends to zero as

x \to \infty

.

Solved by gpt-5.6-sol high.

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