 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/17/d/solution

Take the switch-off time to be

t = 0

. The state remains continuous through the source step, so

y (0) = (1, 1, 1/4)^{T}

. Its eigendirection decomposition is

y (0) = (1, 1, 0)^{T} + \frac{1}{4} (0, 0, 1)^{T} .

The antisymmetric mode has zero amplitude because the first two initial components are equal. Applying the decay factors from part (b),

y (t) = (e^{- t}, e^{- t}, \frac{1}{4} e^{- 4 t})^{T} (t \geq 0) .

The first two responses decay at rate one and the third at rate four, approaching the new zero-source steady state.

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