 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/3/b/i/solution

For

t < 0

, all forcing points satisfy

τ > t

x (t) = - \frac{1}{4} e^{2 t} \int_{0}^{1} τ e^{- 2 τ} d τ .

\int τ e^{- 2 τ} d τ = - (τ /2 + 1/4) e^{- 2 τ}

. Consequently

x (t) = - \frac{1 - 3 e ^{- 2}}{16} e^{2 t}, t < 0.

The exponential has the required decay as

t \to - \infty

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