Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 7C / c / Solution

For either eigenpair above, ${\lambda }_i\in \{0,1\}$ and hence ${\lambda }_i^n={\lambda }_i$ for every positive integer $n$. Thereforeso both terms in the solution from part (a), and every linear combination of them, solve the higher-order system.

A system of two scalar differential equations of order $n$ has a $2n$-dimensional solution space. The displayed family supplies two independent solutions, so there must befurther linearly independent solutions.