Consider the differential equation $$ay′′ + by′ + cy = g(t)$$ where $a > 0$, $c > 0$, and $g(t)$ is a continuous function on $\mathbb R$.
(a) If $y(t)$ is a solution of the above equation with $b > 0$ and $g(t) = 0$, show that $\lim_{t\rightarrow \infty}=0$.
(b) Is the result in (a) true, if $b = 0$?
(c) If $g (t) = d$ and $b > 0$, show that $\lim_{t\rightarrow \infty}=\frac{d}{c}$ for every solution $y(t)$ to the above equation.
I am under the impression that I need to use the characteristic equation in parts a and b to show that its' roots are negative, thus $y=e^{-xt}$ is a solution to the equation in one form or the other. This would lead to the limit approaching zero.
Part C is the main difficulty, I believe it involves utilizing the form of "variation of parameters", but I am not sure what to do with that.