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# What is an example in which differential equations are used?

There are definitely many real-life examples using differential equations; for example, the problem of bacterial growth, which involves quantity and its derivative. Scientists know that bacterial growth depends on how many bacteria exists initially; therefore, if one takes two bacteria, the first increase is by two, then by four, then by eight, and so on. If P (t) is the population (P) of the bacteria at any given time (t), one way to express the rate of change in the number of bacteria over time using differential equations is as follows:

dP / dt

Since it depends on the population P, the equation then becomes:

dP / dt = aP (t)

in which a is a constant that can be used to fit the situation; for example, how long it takes the little critters to reproduce. This is called an equations that relates a certain quantity to it’s derivative—and a classic way of looking at how a differential equation describes exponential growth.

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