 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ia/paper-2/2a/a/solution

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

Since

1 + cos 2 y = 2 cos^{2} y

, separation gives

sec^{2} y d y = \frac{2}{x ^{2}} d x, tan y = C - \frac{2}{x} .

The limiting condition gives

C = 1

. Taking

tan^{- 1}

with range

(- π /2, π /2)

and selecting the branch approaching

9 π /4

yields

y (x) = 2 π + tan^{- 1} (1 - \frac{2}{x})

for

x > 0

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