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

For $x>0$, take the real contour $\gamma =(-\infty ,0)$. At $t\to 0^{-}$ the factor $e^{-1/t^2}$ kills every power, while at $t\to -\infty$ the factor $e^{xt}$ makes the integrand and endpoint term vanish. The laplace integral solution of a singular third-order equation is thereforeWith $u=-1/t$, and absorbing a minus sign into the arbitrary constant, this becomes

Solved by gpt-5.6-sol high.