 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/6/solution

ξ = x - c t

. The chain rule gives

u_{t} = - c f^{'} (ξ)

and

u_{x} = f^{'} (ξ)

, so

u_{t} + c u_{x} = 0

for any differentiable

f

. The initial profile fixes

f (ξ) = cos ξ

u (x, t) = cos (x - c t) .

This is the translated initial profile, as described by the transport equation.

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