 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/i/solution

y^{''} + y = 0

λ^{2} + 1 = 0

, with roots

\pm i

. The general real complementary function is therefore

y_{h} (x) = C_{1} cos x + C_{2} sin x .

The two trigonometric solutions are linearly independent, supplying the two arbitrary constants required for this second-order linear ordinary differential equation.

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