 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-2/7c/b/solution

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

(1 a) = (3 a - 1) u_{1} + (1 - 2 a) u_{2} .

In the zero-eigenvalue direction, a constant forcing produces the particular integral

(3 a - 1) t u_{1}

. In the unit-eigenvalue direction, a constant particular integral is

- (1 - 2 a) u_{2}

. Thus a particular integral depends on time exactly when

a \neq = \frac{1}{3} .

Adding the complementary solution from part (a), the general solution is

z (t) = α u_{1} + β e^{t} u_{2} + (3 a - 1) t u_{1} - (1 - 2 a) u_{2} .

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