 Solution

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

When

α > n

μ = α^{2} - n^{2} > 0

F^{''} - μ^{2} F = 0

. Enforcing the same initial conditions gives

F (t) = \frac{sinh ( μ t )}{μ}, μ = α^{2} - n^{2} .

The corresponding overdamped normal mode can also be written

y (x, t) = \frac{e ^{(- α + μ) t} - e ^{(- α - μ) t}}{2 μ} sin (n x) .

Both rates in the exponential functions are negative because

0 < μ < α

, so the hyperbolic sine factor does not prevent decay.

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