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Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 6C / a / Solution

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.

Solved by gpt-5.6-sol high.

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