Problem 1.
The secretion of hormones into the blood is often a periodic activity. If a hormone is secreted on a $24$-hour cycle, then the rate of change of the level of the hormone in the blood may be represented by the initial value problem \[ x'(t)=\alpha-\beta \cos\left(\frac{\pi t}{12} \right)-kx(t), \qquad x(0)=x_0, \] where $x(t)$ is the amount of the hormone in the blood at time $t$, $\alpha$ is the average secretion rate, $\beta$ is the amount of daily variation in the secretion, and $k$ is a positive constant reflecting the rate at which the body removes the hormone from the blood.If $\alpha,\beta=1$, $k=2$, and $x_0=10$, solve for $x(t)$.
