 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-2/6/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Expanding gives

u = 2 x^{2} + 2 c^{2} t^{2}

, so

u_{tt} = 4 c^{2}

and

u_{xx} = 4

. Hence

u_{tt} = c^{2} u_{xx}

, proving that it solves the one-dimensional wave equation. Equivalently its two summands are oppositely travelling profiles, as in D'Alembert's formula.

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