 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ia/paper-2/7b/i/b/solution

t > 0

y (t) = \frac{1}{α} \int_{0}^{t} e^{- (t - s) / α} d s = 1 - e^{- t / α} .

Before zero it vanishes. Therefore

y (t) = (1 - e^{- t / α}) H (t) .

This causal response of a first-order relaxation equation is continuous at zero and approaches the new constant equilibrium exponentially.

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