We can use differential equations to talk about things like how quickly a disease spreads, how fast a population grows, and how fast the temperature of cookies rises in an oven. Translating between English and differential equations takes a bit of practice, but a good starting place is to think "derivative" whenever you see the word "rate."

Make sure you remember what proportionality and inverse proportionality are, because these words come up a lot around differential equations.

Using differential equations to describe real-life situations in this way is called **modeling**. The differential equation is a model of the real-life situation.

We'd like to take this opportunity to discuss constants and their signs (by which we mean positive and negative, not Capricorn and Sagittarius.

Suppose a population *P* is increasing at a rate proportional to *P*. This means

where *k* is the constant of proportionality. Since the population is *increasing*, the rate must be *positive*. Since the population *P* will be a positive number, the constant *k* must also be a {positive number.

Now suppose instead that the population *P* is *decreasing* at a rate proportional to *P*. Since the rate of population change is proportional to *P*, by the definition of proportionality we have

Since we're told the population is decreasing, its rate of change must be *negative*. This means *k* must be negative, since that's the only way the product *kP* will turn out negative.

There's another way to write a differential equation for the situation where *P* is decreasing at a rate proportional to *P*. We can write

and think of *k* as positive. Then (-*k*) is the constant of proportionality. The product of (-*k*) and *P* is negative, so the rate will come out negative like it's supposed to.

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