Consider the linear system

$$\dot{\mathbf{x}}(t)-A \mathbf{x}(t)=\mathbf{z}(t)$$

where the $n$-vector $\mathbf{z}(t)$ and the $n \times n$ matrix $A$ are given; $A$ has constant real entries, and has $n$ distinct eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ and $n$ linearly independent eigenvectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}$. Find the complementary function. Given a particular integral $\mathbf{x}_{\mathbf{p}}(t)$, write down the general solution. In the case $n=2$ show that the complementary function is purely oscillatory, with no growth or decay, if and only if

$$\operatorname{trace} A=0 \quad \text { and } \quad \operatorname{det} A>0 .$$

Consider the same case $n=2$ with trace $A=0$ and $\operatorname{det} A>0$ and with

$$\mathbf{z}(t)=\mathbf{a}_{1} \exp \left(i \omega_{1} t\right)+\mathbf{a}_{2} \exp \left(i \omega_{2} t\right)$$

where $\omega_{1}, \omega_{2}$ are given real constants. Find a particular integral when

(i) $i \omega_{1} \neq \lambda_{1}$ and $i \omega_{2} \neq \lambda_{2}$;

(ii) $i \omega_{1} \neq \lambda_{1}$ but $i \omega_{2}=\lambda_{2}$.

In the case

$$A=\left(\begin{array}{cc} 1 & 2 \\ -5 & -1 \end{array}\right)$$

with $\mathbf{z}(t)=\left(\begin{array}{c}2 \\ 3 i-1\end{array}\right) \exp (3 i t)$, find the solution subject to the initial condition $\mathbf{x}=\left(\begin{array}{l}1 \\ 0\end{array}\right)$ at $t=0$.