Two cups of hot tea at temperatures $T_{1}(t)$ and $T_{2}(t)$ cool in a room at ambient constant temperature $T_{\infty}$. Initially $T_{1}(0)=T_{2}(0)=T_{0}>T_{\infty}$.

Cup 1 has cool milk added instantaneously at $t=1$; in contrast, cup 2 has cool milk added at a constant rate for $1 \leqslant t \leqslant 2$. Briefly explain the use of the differential equations

$$\begin{aligned} &\frac{d T_{1}}{d t}=-a\left(T_{1}-T_{\infty}\right)-\delta(t-1) \\ &\frac{d T_{2}}{d t}=-a\left(T_{2}-T_{\infty}\right)-H(t-1)+H(t-2) \end{aligned}$$

where $\delta(t)$ and $H(t)$ are the Dirac delta and Heaviside functions respectively, and $a$ is a positive constant.

(i) Show that for $0 \leqslant t<1$

$$T_{1}(t)=T_{2}(t)=T_{\infty}+\left(T_{0}-T_{\infty}\right) \mathrm{e}^{-a t}$$

(ii) Determine the jump (discontinuity) condition for $T_{1}$ at $t=1$ and hence find $T_{1}(t)$ for $t>1$.

(iii) Using continuity of $T_{2}(t)$ at $t=1$ show that for $1<t<2$

$$T_{2}(t)=T_{\infty}-\frac{1}{a}+\mathrm{e}^{-a t}\left(T_{0}-T_{\infty}+\frac{1}{a} \mathrm{e}^{a}\right)$$

(iv) Compute $T_{2}(t)$ for $t>2$ and show that for $t>2$

$$T_{1}(t)-T_{2}(t)=\left(\frac{1}{a} \mathrm{e}^{a}-1-\frac{1}{a}\right) \mathrm{e}^{(1-t) a}$$

(v) Find the time $t^{*}$, after $t=1$, at which $T_{1}=T_{2}$.