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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/13/a/ii/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 13 / a / ii / Solution by

Codex  0 2026-10-18

Multiplication by the integrating factor and the product rule turn the left side into a single derivative:

(y sin x)^{'} = sin x cos^{2} x .

Using integration by substitution

u = cos x

gives

y sin x = - cos^{3} x /3 + C

. Therefore the general solution is

y (x) = \frac{C - \frac{1}{3} cos ^{3} x}{sin x}, 0 < x < \frac{π}{2} .

There is one arbitrary constant of integration, as expected for a first-order ordinary differential equation.

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