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Differential Equations

Differential Equations

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

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.

Example 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.

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

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.

Example 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.

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