 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ia/paper-2/5a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

θ_{i}^{'} = - a θ_{i}

gives

θ_{i} (t) = C_{i} e^{- a t}

. The initial values give

C_{1} = C_{2} = T_{0} - T_{\infty}

, and hence

T_{1} (t) = T_{2} (t) = T_{\infty} + (T_{0} - T_{\infty}) e^{- a t}, 0 \leq t < 1.

In particular the common left-hand limit at time one is

T_{\infty} + (T_{0} - T_{\infty}) e^{- a}

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