Chapter 2.2, Problem 38P

Consider the initial value problem

$y\text{'}+ay=g(t),y(t_0)=y_0.$

Assume that $a$ is a positive constant and that $g(t)\to g_0$ as $t\to \infty$. Show that $y(t)\to g_0/a$ as $t\to \infty$. Construct an example with a nonconstant $g(t)$ that illustrates this result.