Two cups of 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 $t=1$ and then hot water added at a constant rate after $t=2$ which is modelled as follows

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

whereas cup 2 is left undisturbed and evolves as follows

$$\frac{d T_{2}}{d t}=-a\left(T_{2}-T_{\infty}\right)$$

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

(a) Derive expressions for $T_{1}(t)$ when $0<t \leqslant 1$ and for $T_{2}(t)$ when $t>0$.

(b) Show for $1<t<2$ that

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

(c) Derive an expression for $T_{1}(t)$ for $t>2$.

(d) At what time $t^{*}$ is $T_{1}=T_{2}$ ?

(e) Find how $t^{*}$ behaves for $a \rightarrow 0$ and explain your result.