Let $t$ be a real independent variable, and let $a ( t ) , x ( t )$ and $y ( t )$ be real-valued functions. Answer the following questions.
(1) Let $a ( t )$ be a continuous and periodic function of $t$ whose period is $T$. Find the initial-value-problem solution $x ( t )$ of the ordinary differential equation
$$\frac { \mathrm { d } } { \mathrm {~d} t } x ( t ) = a ( t ) x ( t ) ,$$
where $x ( 0 ) = x _ { 0 } \neq 0$.
(2) Find the necessary and sufficient condition on $a ( t )$ such that the solution $x ( t )$ in Question (1) is a periodic solution with a period $T$.
(3) Find the initial-value-problem solutions $x ( t )$ and $y ( t )$ of the simultaneous ordinary differential equations
$$\begin{aligned} \frac { \mathrm { d } } { \mathrm {~d} t } x ( t ) & = - k x ( t ) + \sin ( t ) \cos ( t ) y ( t ) \\ \frac { \mathrm { d } } { \mathrm {~d} t } y ( t ) & = ( - k + \sin ( t ) ) y ( t ) \end{aligned}$$
where $k$ is a real constant, $x ( 0 ) = x _ { 0 } \neq 0$ and $y ( 0 ) = y _ { 0 } \neq 0$.
(4) Explain briefly how $x ( t )$ and $y ( t )$ converge as $t \rightarrow \infty$ when $k > 0$ in Question (3).
(5) Draw a schematic graph of the solution trajectory in Question (3) on the $y x$-plane where $k = 0 , x _ { 0 } = 2$ and $y _ { 0 } = 1$.