Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 2 / 5A / Solution

Measure temperature relative to ambient by ${\theta }_i=T_i-T_{\infty }$. Newton's law of cooling models ordinary cooling by ${\theta }_i^{\prime }=-a{\theta }_i$, since heat loss is proportional to the excess temperature. The additional negative forcing represents the cooling from the milk, with the temperature drop or rate normalized to one in the chosen units.

The Dirac delta function has unit integral and is supported at the instant of addition. It therefore imposes a sudden unit drop for cup 1. The difference $H(t-1)-H(t-2)$ of Heaviside step functions is one during the addition interval and zero outside it, giving a unit cooling rate for cup 2. The second temperature is continuous; only its derivative changes at the switching times. The value assigned to a Heaviside step function exactly at its jump does not change the integrated solution. This is the impulse versus finite-duration cooling comparison.