 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-1/8/a/solution

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

The Green's function is nonpositive, so

f \geq 0

gives

y \leq 0

. If also

f \leq m

, then

\int_{0}^{x} ζ f (ζ) d ζ \leq \frac{m x ^{2}}{2}, \int_{x}^{\infty} \frac{f ( ζ )}{ζ ^{2}} d ζ \leq \frac{m}{x} .

The integral solution therefore gives

- \frac{m}{2} = - \frac{1}{3} (\frac{m}{2} + m) \leq y (x) \leq 0.

Both bounds are attained, by the constant forcings

f = m

and

f = 0

, respectively. In particular, boundedness of the solution does not require it to approach zero.

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