Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 2 / 5A / i

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- Solution i

Solution

Before time one, both forcing terms vanish. The separable differential equation

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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