 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-1/3/i/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

The factor satisfying the left condition is

x

, while

x^{- 1}

satisfies decay at infinity. Matching the value and unit derivative jump gives

G_{1} (x, ξ) = {- x /2, - ξ^{2} / (2 x), x \leq ξ, x \geq ξ .

For the specified compactly supported forcing, integrating over

0 < ξ < 1

gives

y_{1} (x) = {x^{2} /3 - x /2, - 1/ (6 x), 0 < x \leq 1, x \geq 1.

For example, on

0 < x < 1

, the two pieces contribute

- x^{2} /6

and

- x (1 - x) /2

. Both

y_{1}

and

y_{1}^{'}

match at one, and the function tends to zero at both ends. A homogeneous difference

a x + b / x

with those two limits must vanish, establishing uniqueness.

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