 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-v/5/b/ii/solution

When

α = n

F^{''} = 0

F (0) = 0

F^{'} (0) = 1

give

F (t) = t, y (x, t) = t e^{- n t} sin (n x) .

This is the critically damped normal mode, also obtained by taking

ν \to 0

sin (ν t) / ν

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