 Solution

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

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

For either eigenpair above,

λ_{i} \in {0, 1}

and hence

λ_{i}^{n} = λ_{i}

n

. Therefore

\frac{d ^{n}}{d t ^{n}} (u_{i} e^{λ_{i} t}) = λ_{i}^{n} u_{i} e^{λ_{i} t} = A u_{i} e^{λ_{i} t},

so both terms in the solution from part (a), and every linear combination of them, solve the higher-order system.

n

2 n

-dimensional solution space. The displayed family supplies two independent solutions, so there must be

2 n - 2

further linearly independent solutions.

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