Answer
$\frac{dy}{dt}=3-y$.
Work Step by Step
We must write an equation of the form$\frac{dy}{dt} = ay+b$ such that all solutions approach $y=3$ as $t\rightarrow\infty$.
Solution: We recognize that at $y=3$, the slope of the function $ay+b$ must be $0$:
$0=a(3)+b$,
so,
$3=-\frac{b}{a}$.
In other words, the ratio $\frac{b}{a}$ must be $-3$. So a differential equation of the required form is
$\frac{dy}{dt}=-y+3$, or
$\frac{dy}{dt}=3-y$.
