Chapter 2.2, Problem 33P

Show that if $a$ and $\lambda$ are positive constants, and b is any real number, then every solution if the equation

$y\text{'}+ay=b{e}^{-\lambda t}$

Has the property that $y\to 0$ as $t\to \infty$.

Hint: Consider the cases $a=\lambda$ and $a\ne \lambda$ separately.