 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2005/calculus-linear-algebra-discrete-mathematics-logic-and-proof-and-models-of-computation/4/b/ii/solution

\frac{1}{2} \int_{x - t}^{x + t} 1 d s = t

. Hence, with arbitrary twice continuously differentiable

h

u (x, t) = \frac{h ( x + t ) + h ( x - t )}{2} + t .

The added term has both second partial derivatives zero and has time derivative one, so it supplies precisely the required initial condition while preserving the wave equation.

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