 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/2/a/solution

u (x, t) = X (x) T (t)

and divide by

XT

\frac{X ^{''}}{X} = \frac{T ^{''} + 2 λ T ^{'}}{T} = - α^{2} .

X^{''} + α^{2} X = 0, T^{''} + 2 λ T^{'} + α^{2} T = 0 .

X (0) = X (L) = 0

. Nonzero solutions are

X_{n} (x) = sin (nπ x / L)

, with

α_{n} = nπ / L

n \geq 1

. Neither

α = 0

nor a negative spatial eigenvalue produces a nonzero function obeying both endpoint conditions.

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