Answer
$y(t) = 3 + Ce^{-t}$.
Work Step by Step
We are to construct a first order differential equation all of whose solutions $\rightarrow 3$ as $t \rightarrow \infty$.
We first look for a function which satisfies this condition. After a few trial and error tries, we find that
$$y(t) = 3 + Ce^{-t}$$,
(where $C$ is a constant) will satisfy the given condition.
In order to find a differential equation for which $y(t) = 3 + Ce^{-t}$ is a solution, we first multiply this solution equation through by $e^t$ and get, equivalently,
$$e^t y(t) = 3e^t + C$$.
We then differentiate this last equation and get
$$ e^t y'(t) + e^t y(t) = 3e^t$$.
Then, to simplify, we divide this last equation by $e^t$ (which now becomes an integrating factor):
Thus,
$$y'(t) + y(t) = 3$$,
which can be our desired differential equation.
