 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/13/a/iii/solution

y = C_{1} cos x + C_{2} sin x + \frac{1}{2} + \frac{1}{6} cos 2 x .

The boundary condition at zero gives

C_{1} + 2/3 = 0

, while that at

π /2

gives

C_{2} + 1/3 = 0

. Hence

y (x) = - \frac{2}{3} cos x - \frac{1}{3} sin x + \frac{1}{2} + \frac{1}{6} cos 2 x .

The two boundary conditions determine both constants uniquely; substitution at each endpoint gives zero.

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