# Differential Equations

### Topics

## Introduction to Differential Equations - At A Glance:

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 |

#### 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

Ben receives 20 pieces of junk mail every day.

#### Exercise 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.

#### 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 (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?

#### 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 (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?

#### 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

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

#### Exercise 15

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