In Problems 13 - 16, write a differential equation that fits the physical description.
$$\begin{array}{l}{\text { The rate of change of the population } p \text { of bacteria at }} \\ {\text { time } t \text { is proportional to the population at time } t .}\end{array}$$

Answer

$\frac{d p}{d t}=k p,$ where $\mathrm{k}$ is the proportionality constant.

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Calculus 2 / BC

Fundamentals of Differential Equations

Chapter 1

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Video Transcript

well in this problem was supposed to come up with the differential equation. It models it, but Syria population. So basically, I'd say it time T. The population of bacteria's peer of tea and theme model that we're trying to come up with basically says that the rate of change off the bacteria population Time T which is, by definition of narrative, dp dt right or a different addition that is P prime 50 simply the deal until our population Time T days um, rate of change. So in works, this is rate of change. In other words, it's it's it tells it tells us that time Tito, how fast the population is increasing or decreasing rate of change that trying t. And the model tells us that this must be proportional to the population at the time. So this must be proportional means that there's some constant, some constant C times depopulation of time. And this is a very simple moral. So basically, this is defy for just rewrite everything just in terms of the equation, it'll look like D B e t t. A sequel. To some constant times, Petey and C is constant

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