Differential Equation for practical problem I have the following problem at hand, which I want to model with a differential equation:
A person is consuming a constant amount of alcohol m per day.
The alcohol in the person's blood is being decomposed with a constant rate k.
Now I would like to model a differential equation for the amount of Alcohol in the person blood - A(t), and also a solution with the property A'(t) = 0.
Any advice/hints welcome.

Assume that the drinking is done continuously, day and night, night and day.  
It is not entirely clear what constant rate $k$ means. Probably what  is meant is that when there is an amount $A$ in the blood, the rate of change due to decay (metabolization) is $kA$. We assume the units are chosen so they are compatible with those of $m$, say ml per day.  Then the model is 
$$A'(t)=m-kA(t)$$
(rate in minus rate out). 
There are many ways to solve this DE. One simple way is to let $y=m-kA(t)$. Then $$\frac{dy}{dt}=-k\frac{dA}{dt}=-ky.$$
The differential equation $\frac{dy}{dt}=-ky$ is the familiar DE of exponential decay. Its general solution is $y=Ce^{-kt}$. Now one can write down the general formula for $A(t)$.
The solution that has $A'(t)=0$ can be written down without solving the DE. If our friend starts off pretty high, with $A(0)=\frac{m}{k}$, then her blood alcohol level remains constant, with ingestion matching decay. 
