 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ia/paper-2/6c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For

k \neq = m

, the required Euler-equation solution is

x (t) = \frac{t ^{k} - t ^{m}}{k ^{2} - m ^{2}} .

For

k = m > 0

, the limiting resonant form is

x (t) = \frac{t ^{m} l o g t}{2 m} .

Both extend continuously to zero and satisfy the two boundary conditions. If

k = m = 0

, integration gives

x = \frac{1}{2} (lo g t)^{2} + C lo g t + D,

which cannot have a finite zero limit at

t = 0

, so no such solution exists.

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