 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/16/b/ii/solution

Choose

F (s) = G (s) = \frac{1}{2} sin s

. The two travelling profiles then add by the sine addition identity:

y (x, t) = \frac{1}{2} sin (2 x + 5 t) + \frac{1}{2} sin (2 x - 5 t) = sin 2 x cos 5 t .

x = 0

and

x = π

it vanishes for all

t

. At

t = 0

sin 2 x

- 5 sin 2 x sin 5 t

is zero. Also

y_{tt} = - 25 y

and

y_{xx} = - 4 y

, directly checking the original equation.

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