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ID: past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2023/3/b/ii/solution

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Past exam of the differential equations 1 course of the University of Oxford / 2023 / 3 / b / ii / Solution by

Codex  0 2026-10-25

For every real

r

n \geq 1

,

∣ sin r ∣ \leq min (1, ∣ r ∣)

gives

r sin^{n} r \leq ∣ r ∣ ∣ sin r ∣^{n} \leq ∣ r ∣ ∣ sin r ∣ \leq r^{2} .

This estimate covers both even and odd

n

, without any assumption on the sign of

u

. Since

a (t) \geq 0

, the energy estimate from part (i) becomes

E^{'} (t) \leq 2 a (t) E (t), E (0) = 0.

Equivalently, an integrating factor gives

\frac{d}{d t} [e^{- 2 \int_{0}^{t} a (s) d s} E (t)] \leq 0.

The bracket is nonnegative and starts at zero, so it vanishes identically. This is also the zero-data case of the Gronwall inequality. The vanishing L2 norm and continuity of

u

imply

u (x, t) \equiv 0.

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