# Differential Equations

# Word Problems Exercises

### Example 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.

### Example 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.

### Example 3

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

Tamara spends $40 per week.

### Example 4

Ben receives 20 pieces of junk mail every day.

### Example 5

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.

### Example 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?

### Example 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?

### Example 8

If *Q* is a positive quantity and

where *k* < 0, is *Q* increasing or decreasing?

### Example 9

Cookies are placed in a 375°F degree oven to bake. Newton's Law of Heating (link forward) 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?

### Example 10

A hot cup of coffee is placed on the kitchen table in a room that is 68°F. Newton's Law of Cooling (link forward) 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?

### Example 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.

### Example 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.

### Example 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.

### Example 14

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

### Example 15

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