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At a Glance - Word Problems

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.

Example 1

 Translate the following English statement into a differential equation: Jenna's savings account grows at a rate of \$300 per year.

Example 2

 Translate the following English statement into a differential equation: "The population is increasing at a rate proportional to the size of the population."

Example 3

 Translate the following English statement into a differential equation:The population is increasing at a rate of 2 percent per year.

Example 4

 A population P has a birth rate of 1 percent and 200 people die every year. Find the overall rate of change of the population.

Example 5

 Julie makes \$200 dollars per week and puts it in her bank account. After making the deposit, she spends 20 percent of the money in her bank account. Model this situation with a differential equation.

Exercise 1

Translate the English statement into a differential equation. Be sure to specify what your variables are.

The population is increasing at a rate of 1,000 people per year.

Exercise 2

Translate the English statement into a differential equation. Be sure to specify what your variables are.

The number of bunnies in the forest is increasing at a rate proportional to the number of bunnies there already.

Exercise 3

Translate the English statement into a differential equation. Be sure to specify what your variables are.

Tamara spends \$40 per week.

Exercise 4

Translate the English statement into a differential equation. Be sure to specify what your variables are.

Ben receives 20 pieces of junk mail every day.

Exercise 5

Translate the English statement into a differential equation. Be sure to specify what your variables are.

A batch of cookies is placed in a 375°F oven. The temperature of the cookies increases at a rate proportional to the difference between the temperature of the cookies and the temperature of the oven.

Exercise 6

The population of bunnies B is increasing at a rate proportional to the size of B. This situation can be modeled with the differential equation

Is the constant k positive or negative?

Exercise 7

The population of geese G is decreasing at a rate proportional to G. This situation can be modeled with the differential equation

Is the constant k positive or negative?

Exercise 8

If Q is a positive quantity and

where k < 0, is Q increasing or decreasing?

Exercise 9

Cookies are placed in a 375°F degree oven to bake. Newton's Law of Heating  says that the temperature t of the cookies will increase at a rate proportional to the difference between the temperature of the surrounding oven and the temperature of the cookies. If we model this situation by the differential equation

is the constant k positive or negative?

Exercise 10

A hot cup of coffee is placed on the kitchen table in a room that is 68°F. Newton's Law of Cooling says that the temperature t of the coffee will decrease at a rate proportional to the difference between the temperature of the surrounding room and the temperature of the coffee. This situation can be modeled by the differential equation

Is the constant k positive or negative?

Exercise 11

Model the situation using a differential equation. State the units of each variable and the units of the derivative.

Water is rushing into a tank at a rate of 5 gallons per minute and rushing out again at a rate of 3 gallons per minute.

Exercise 12

Model the situation using a differential equation. State the units of each variable and the units of the derivative.

The birth rate of the wolf population is 5 percent per year and hunters kill 200 wolves per year.

Exercise 13

Model the situation using a differential equation. State the units of each variable and the units of the derivative.

Every month Donna's savings account earns 2 percent interest and she deposits an additional \$100 dollars.

Exercise 14

Model the situation using a differential equation. State the units of each variable and the units of the derivative.

A beach is eroding by 10 percent per year. Every month the waves deposit an extra 150 cubic feet of sand.

Exercise 15

Model the situation using a differential equation. State the units of each variable and the units of the derivative.

A wasp population has a 10 percent birth rate and a 9 percent death rate. Every year people swat an additional 300 wasps.