Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 2 / 7C (Differential Equations)

Put $\Delta =T_0-T_{\infty }$. Before either forcing term acts, Newton's law of cooling gives

Integrating the equation through $t=1$ gives the jump condition for an impulseFor $1<t<2$ the equation is again homogeneous, so propagation of the post-jump temperature givesTherefore

For $t>2$, the Heaviside step function contributes a constant unit forcing. Solving the resulting first-order linear differential equation from the value at $t=2$ gives

The cups have equal temperatures up to the intervention at $t=1$. For the later recrossing, subtract the formula for $T_2$ from that for $T_1$ and set the result to zero:Writing $z=e^{-a(t-2)}$ gives $z=(1+a{e}^{-a}{)}^{-1}$. Hence the nontrivial meeting time is

Using the Taylor expansionwe obtainWhen cooling is negligible, the milk lowers the temperature by one unit at $t=1$. The hot-water term begins at $t=2$ and raises it at one unit per unit time, so one further unit of time is needed to erase the deficit.