Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 4 / 12C (Complex Methods)

$Y(s)={\int }_0^{\infty }{e}^{-st}y(t)dt$, with transforms $sY-y(0)$ and $s^{2}Y-sy(0)-\dot{y}(0)$; $e^{\lambda t}$ transforms to $(s-\lambda {)}^{-1}$ and $\delta$ to $1$. Algebra gives $Y=GH$ with $H=F+(s+2\gamma )y(0)+\dot{y}(0)$, so $y=g\ast h$. Writing $a=\sqrt{|{\gamma }^2-{\omega }^2|}$ gives $g=e^{-\gamma t}\sinh (at)/a$ for $\gamma >\omega$, $e^{-\gamma t}\sin (at)/a$ for $\gamma <\omega$, and $t{e}^{-\gamma t}$ at equality. If $y(0)=0$, $h=f+\dot{y}(0)\delta$, so the initial velocity contributes $\dot{y}(0)g(t)$.