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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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