 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2003/calculus-probability-linear-algebra-discrete-mathematics-logic-and-proof-and-models-of-computation/2/b/solution

Write

z (x, y) = Z (r, s)

in the new coordinates. Since

\partial_{x} = \partial_{r} - \partial_{s}, \partial_{y} = \partial_{r} + \partial_{s},

z_{xx} - z_{yy} = - 4 Z_{rs} = 0.

For a twice continuously differentiable solution of a differential equation,

Z_{r}

is independent of

s

r

and then in

s

gives

z (x, y) = ϕ (y + x) + ψ (y - x) .

This is the decomposition of the one-dimensional wave equation into waves traveling in opposite directions.

ϕ (y) + ψ (y) = cos y

ϕ^{'} (y) - ψ^{'} (y) = 0

. Differentiating the first relation and combining the two relations yields

ϕ^{'} = p s i^{'} = - \frac{1}{2} sin y

. Thus

ϕ (y) = \frac{1}{2} cos y + C

and

ψ (y) = \frac{1}{2} cos y - C

z (x, y) = \frac{cos ( y + x ) + cos ( y - x )}{2} = cos x cos y .

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