 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/14/b/solution

Differentiate the Fourier series term by term and collect cosine and sine terms. Substitution into the equation gives

\frac{a _{0}}{2} + n = 1 \sum \infty [(a_{n} + 3 n b_{n}) cos (n x) + (b_{n} - 3 n a_{n}) sin (n x)] = 4 sin x .

For a smooth periodic solution, differentiation of the series is justified. Alternatively, multiply the differential equation by each trigonometric mode and integrate by parts; periodicity cancels the boundary terms and gives the same coefficient constraints without first differentiating an infinite sum.

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