A biologist starts with 100 cells in a culture. After one day, there are 250 cells in the culture. Write a differential equation that models that population \(P(t)\) of a bacteria culture if the population grows at a rate proportional to the number of bacteria present at any time \(t\) (in days).
Section 1.2 Malthusian growth and decay
The rate of growth (or decay) of a population is proportional to the size of that population at time \(t\text{.}\) Let \(P(t)\) be the size of the population at time \(t\text{.}\) We write this with the \(\propto\) symbol: \(\dfrac{\mathrm{d}P}{\mathrm{d}t} \propto P\text{.}\) To get an equality, we write
\begin{equation}
\dfrac{\mathrm{d}P}{\mathrm{d}t} = rP,\tag{1.2.1}
\end{equation}
where \(r\) is a constant of proportionality, called the growth or decay rate in the context of this model. We can check that for any constant \(P_0\) that all functions of the form \(P(t)=P_0e^{rt}\) are solutions to (1.2.1).
