I want to prepare an example of a differential equation for bacetrial growth.
We have $\frac{dy}{dt}=ry(t)$
Multiplying by dt and moving y(t) over to the other side we get $\frac{1}{y(t)}dy=rdt$, integrating both sides, we get $y(t)=ce^{rt}$. If we say that the bacterial population at time zero is y(0)=10.000 CFU, and at time 1, y(1)=200.000, then we have the system
\begin{equation} \begin{array} f10000=ce^{r0}\\ 200000=ce^{r} \end{array} \end{equation}
We get from the first equation $c=10000$. We plug it into the second, and get that $r=ln20$.
We now have the growth equation:
$y(t)=10000e^{ln20t}$
and by Moos suggestion:
$y(t)=10000t^{20}$
Would this be correct, in its simplicity?
Thanks