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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 12B / iii / Solution

Let

U (s, x) = L_{t} {u (t, x)}

. The initial data turn the wave equation into

- \frac{\partial ^{2} U}{\partial x ^{2}} + s^{2} U = f (x) .

Using the Green function from part (ii) with

m = s

gives

U (s, x) = \frac{1}{2 s} \int_{- \infty}^{\infty} e^{- s ∣ x - y ∣} f (y) d y .

Since

L^{- 1} {\frac{e ^{- a s}}{s}} = H (t - a),

we obtain the D'Alembert formula with initial velocity

u (t, x) = \frac{1}{2} \int_{- \infty}^{\infty} H (t - ∣ x - y ∣) f (y) d y = \frac{1}{2} \int_{x - t}^{x + t} f (y) d y .

Solved by gpt-5.6-sol high.

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