Chapter 2.4, Problem 29P

Consider the initial value problem

$y\text{'}+p\left(t\right)y=g\left(t\right),y\left(t_0\right)=y_0$ (i)

Show that the solution of the initial value problem (i) can be written in the form

$y=y_0\exp \left(-{\displaystyle \underset{t_0}{\overset{t}{\int }}p\left(s\right)ds}\right)+{\displaystyle \underset{t_0}{\overset{t}{\int }}\exp \left(-{\displaystyle \underset{s}{\overset{t}{\int }}p\left(r\right)dr}\right)g\left(s\right)}ds$

Assume that $p\left(t\right)\ge p_0>0$ for all $t\ge t_0$ and that $g\left(t\right)$ is bounded for $t\ge t_0$ (i.e. there is a constant $M$ such that $\left|g\left(t\right)\right|\le M$ for all $t\ge t_0$). Show that the solution of the initial value problem (i) is bounded for $t\ge t_0$.

Construct an example with nonconstant $p\left(t\right)$ and $g\left(t\right)$ that illustrates this result.