Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 7D / b

Table of contents
- Solution b

Solution

For

x > 0

, take the real contour

γ = (- \infty, 0)

. At

t \to 0^{-}

the factor

e^{- 1/ t^{2}}

kills every power, while at

t \to - \infty

the factor

e^{x t}

makes the integrand and endpoint term vanish. The laplace integral solution of a singular third-order equation is therefore

y (x) = C \int_{- \infty}^{0} e^{x t - 1/ t^{2}} t^{- 3} d t .

With

u = - 1/ t

, and absorbing a minus sign into the arbitrary constant, this becomes

y (x) = C_{1} \int_{0}^{\infty} u exp (- u^{2} - \frac{x}{u}) d u .

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