Exercise 5, § 2

Show by an example that a linear equation $d\mathbf{x}/dt = A.\mathbf{x} + e^{\alpha t}\mathbf{p}(t)$, where $A \in \mathscr{L}(E)$ is independent of $t$ and $\mathbf{p}$ is a polynomial (with coefficients in E) can have an integral which is equal to a polynomial of the same degree as $\mathbf{p}$, even when $\alpha$ is a characteristic root of $A$.