Chapter 7.9, Problem 31E

A linear system is said to be stable if its impulse response function $h\left(t\right)$ remains bounded as $t\to +\infty$. If the linear system is governed by

$a{y}^{″}+b{y}^{\prime }+cy=g\left(t\right)$,

where $b$ and $c$ are not both zero, show that the system is stable if and only if the real parts of the roots to

$a{r}^2+br+c=0$

are less than or equal to zero.