Differential Equations | Part IA, 2008

Two cups of hot tea at temperatures T1(t)T_{1}(t) and T2(t)T_{2}(t) cool in a room at ambient constant temperature TT_{\infty}. Initially T1(0)=T2(0)=T0>TT_{1}(0)=T_{2}(0)=T_{0}>T_{\infty}.

Cup 1 has cool milk added instantaneously at t=1t=1; in contrast, cup 2 has cool milk added at a constant rate for 1t21 \leqslant t \leqslant 2. Briefly explain the use of the differential equations

dT1dt=a(T1T)δ(t1)dT2dt=a(T2T)H(t1)+H(t2)\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 δ(t)\delta(t) and H(t)H(t) are the Dirac delta and Heaviside functions respectively, and aa is a positive constant.

(i) Show that for 0t<10 \leqslant t<1

T1(t)=T2(t)=T+(T0T)eatT_{1}(t)=T_{2}(t)=T_{\infty}+\left(T_{0}-T_{\infty}\right) \mathrm{e}^{-a t}

(ii) Determine the jump (discontinuity) condition for T1T_{1} at t=1t=1 and hence find T1(t)T_{1}(t) for t>1t>1.

(iii) Using continuity of T2(t)T_{2}(t) at t=1t=1 show that for 1<t<21<t<2

T2(t)=T1a+eat(T0T+1aea)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 T2(t)T_{2}(t) for t>2t>2 and show that for t>2t>2

T1(t)T2(t)=(1aea11a)e(1t)aT_{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 tt^{*}, after t=1t=1, at which T1=T2T_{1}=T_{2}.

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