Paper 2, Section II, A

Differential Equations | Part IA, 2016

(a) Find and sketch the solution of

y+y=δ(xπ/2)y^{\prime \prime}+y=\delta(x-\pi / 2)

where δ\delta is the Dirac delta function, subject to y(0)=1y(0)=1 and y(0)=0y^{\prime}(0)=0.

(b) A bowl of soup, which Sam has just warmed up, cools down at a rate equal to the product of a constant kk and the difference between its temperature T(t)T(t) and the temperature T0T_{0} of its surroundings. Initially the soup is at temperature T(0)=αT0T(0)=\alpha T_{0}, where α>2\alpha>2.

(i) Write down and solve the differential equation satisfied by T(t)T(t).

(ii) At time t1t_{1}, when the temperature reaches half of its initial value, Sam quickly adds some hot water to the soup, so the temperature increases instantaneously by β\beta, where β>αT0/2\beta>\alpha T_{0} / 2. Find t1t_{1} and T(t)T(t) for t>t1t>t_{1}.

(iii) Sketch T(t)T(t) for t>0t>0.

(iv) Sam wants the soup to be at temperature αT0\alpha T_{0} at time t2t_{2}, where t2>t1t_{2}>t_{1}. What value of β\beta should Sam choose to achieve this? Give your answer in terms of α\alpha, k,t2k, t_{2} and T0T_{0}.

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