This isn't for homework or anything - I came across a problem in which you start with one amoeba in a jar and you know that a single amoeba will spawn another one every three minutes. The original questions states that the jar is full after one hour and asks when it was half full. A common sense approach yeilds 57 minutes, but I was wondering how I'd turn this into a differential equation problem if I wanted to solve it that way.
If $A$ is the number of amoebae, and $t$ is the number of minutes that have passed since you start, I'm looking for an expression for $\frac{dA}{dt}$. If you were told that they doubled every minute then I know it'd be $\frac{dA}{dt} = 2A$, but I'm stuck on how to deal with the every three minutes thing.
Any help would be very much appreciated!
The differential equation is $$ \frac{dA(t)}{dt}=kA(t) $$ with the initial condition $A(0)=1$. So the solution is $$A=e^{kt}$$ and, since $A(3)=2$ we have $e^{3k}=2 \Rightarrow k=\dfrac{\log 2}{3}$ and the equation is $$ A(t)=e^{\frac{\log 2}{3}t} $$ with the time $t$ measured in minutes.