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Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 5A / b / Solution

For

x > 0

, division by

x

gives

y^{'} - (2 x + x^{- 1}) y = x .

An integrating factor is

e^{- x^{2}} / x

, so

y = x e^{x^{2}} (C + \int_{0}^{x} e^{- t^{2}} d t) .

Every choice except

C = - \int_{0}^{\infty} e^{- t^{2}} d t = - \frac{π}{2}

grows like

x e^{x^{2}}

. The bounded solution selected by a terminal condition is therefore

y_{b} (x) = - x e^{x^{2}} \int_{x}^{\infty} e^{- t^{2}} d t .

Integration by parts, or the Gaussian-tail asymptotic, gives

\int_{x}^{\infty} e^{- t^{2}} d t \sim \frac{e ^{- x^{2}}}{2 x}, y_{b} (x) \to - \frac{1}{2} .

It starts from

0

at

x = 0

, remains negative, and decreases monotonically toward the horizontal asymptote

- 1/2

.

Solved by gpt-5.6-sol high.

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