Deck 9: Differential Equations

Question 1

Find an explicit expression for the particular solution that satisfies the given initial condition.
ln

A)
B)
C)
D) y(x) = 3 - ln(x + 2)

Answer 1

Question 2

Solve the problem.
The annual sales y of a company are increasing at a rate of $150,000 per year.

A)
B)
C)
D)

Answer 2

Question 3

Solve the problem.
The deer population y in a forest is growing at a rate proportional to the difference between the population and an upper limit of 1500 deer.

A)
B)
C)
D)

Answer 3

Question 4

Use the given slope field to graph the particular solutions obtained by letting C = -2, -1, 0, 1, and 2 for the general
solution shown.

Question 5

Use the given slope field to graph the particular solutions obtained by letting C = -2, -1, 0, 1, and 2 for the general
solution shown.

A)
B)
C)
D)

Question 6

Use the given slope field to graph the particular solutions obtained by letting C = -2, -1, 0, 1, and 2 for the general
solution shown.

Question 7

Use the given slope field to graph the particular solutions obtained by letting C = -2, -1, 0, 1, and 2 for the general
solution shown.

A)
B)
C)
D)

Question 8

Solve the problem.
The price p(t) of a produce is assumed to satisfy the differential equation (A) Show that is the general solution of this differential equation, and evaluate (B) Graph the particular solutions that satisfy the initial conditions p(0) = 3 and p(0) = 12.
(C) Discuss the long-term behavior of the price of this product.

Question 9

Determine which of the slope fields (A)-(D) is associated with the given differential equation. Briefly justify your

A)
B)
C)
D)

Question 10

Solve the problem.
The population N(t) of a certain species of animal in a controlled habitat at time t is assumed to satisfy the
logistic growth model (A) Show that is the general solution of this differential equation, and evaluate (B) Graph the particular solutions that satisfy N(0) = 750 and N(0) = 3000.
(C) Discuss the long-term behavior of this population.

Question 11

Use the general solution y of the differential equation to find a particular solution that satisfies the indicated initial
condition. Graph the particular solutions for x ≥ 0.

Question 12

Use the given slope field to graph the particular solutions obtained by letting C = -2, -1, 0, 1, and 2 for the general
solution shown.

A)
B)
C)
D)

Question 13

Show that the technique of separation of variables is applicable by writing the differential equation in the form =
g(x).

A)
B)
C)
D)

Question 14

Find the general solution for the differential equation. Then find the particular solution satisfying the initial condition.

A)
B)
C)
D)

Question 15

Show that the technique of separation of variables is applicable by writing the differential equation in the form =
g(x).

A)
B)
C)
D)

Question 16

Use the general solution y of the differential equation to find a particular solution that satisfies the indicated initial
condition. Graph the particular solutions for x ≥ 0.

Question 17

Determine which of the slope fields (A)-(D) is associated with the given differential equation. Briefly justify your

A)
B)
C)
D)

Question 18

Find an explicit expression for the particular solution that satisfies the given initial condition.

A)
B)
C)
D)

Question 19

Use the general solution y of the differential equation to find a particular solution that satisfies the indicated initial
condition.

Question 20

Use the general solution y of the differential equation to find a particular solution that satisfies the indicated initial
condition. Graph the particular solutions for x ≥ 0.

Question 21

Solve the problem.
A culture of bacteria is growing in a medium that can support a maximum of 1,500 bacteria. The rate of change of the number of bacteria is proportional to the product of the number present and the difference between 1,500 and the
Number present. The culture initially contains 250 bacteria. After 1 hour, there are 350 bacteria.
a. How many bacteria are present after 3 hours? Round to the nearest whole number as needed.
b. When will the culture contain 1,250 bacteria? Round to the nearest tenth as needed.

A) a.) 620 bacteria b.) 7.7 hours
B) a.) 1,125 bacteria b.) 6.7 hours
C) a.) 1,750 bacteria b.) 10.2 hours
D) a.) 850 bacteria b.) 5.2 hours

Question 22

Find the general solution for the differential equation. Then find the particular solution satisfying the initial condition.

A)
B)
C)
D)

Question 23

Find the integrating factor, the general solution, and the particular solution satisfying the given initial condition.

A)
B)
C)
D)

Question 24

Replace the ? with a function of x that will make the integrand equal to the derivative of a product, and then find the
antiderivative. Choose 0 for the constant of integration.

A)
B)
C)
D)

Question 25

Find the general solution for the differential equation. Do not attempt to find an explicit expression for the solution.

A)
B)
C)
D)

Question 26

Is the differential equation first-order linear? If so, find f(x) and g(x) so that the equation can be expressed in standard + f(x)y = g(x).

A)
B)
C) No
D)

Question 27

Find the integrating factor I(x) for each equation, and then find the general solution.

A)
B)
C)
D)

Question 28

Find the integrating factor, the general solution, and the particular solution satisfying the given initial condition.

A)
B)
C)
D)

Question 29

Replace the ? with a function of x that will make the integrand equal to the derivative of a product, and then find the
antiderivative. Choose 0 for the constant of integration.

A)
B)
C)
D)

Question 30

Find an explicit expression for the particular solution for the differential equation.

A)
B)
C)
D)

Question 31

Find the general solution for the differential equation. Then find the particular solution satisfying the initial condition.
y

A)
B)
C)
D)

Question 32

Find the general solution for the differential equation. Then find the particular solution satisfying the initial condition.

A)
B)
C)
D)

Question 33

Solve the problem.
A bank offers a certificate of deposit that earns 4.5% interest compounded continuously. How much will a $4,000 investment be worth in 10 years? Round to the nearest cent as needed.

A) $4,200.00
B) $4,180.00
C) $6,273.25
D) $5,967.30

Question 34

Solve the problem.
A company wishes to analyze a new room deodorizer. The active ingredient evaporates at a rate proportional to the amount present. One third of the ingredient evaporates in the first 20 days after the deodorizer is installed. If
The deodorizer becomes ineffective after 85% of the active ingredient has evaporated, how long will one of these
Deodorizers remain effective? Round to the nearest whole number as needed.

A) 26 days
B) 45 days
C) 94 days
D) 100 days

Question 35

Find the integrating factor, the general solution, and the particular solution satisfying the given initial condition.

A)
B)
C)
D)

Question 36

Is the differential equation first-order linear? If so, find f(x) and g(x) so that the equation can be expressed in standard + f(x)y = g(x).

A) Yes; f(x) = kx and g(x) = 0
B) Yes; f(x) = 0 and g(x) = kx
C) No
D) Yes; f(x) = 1 and g(x) = kx

Question 37

Find the integrating factor, the general solution, and the particular solution satisfying the given initial condition.

A)
B)
C)
D)

Question 38

Replace the ? with a function of x that will make the integrand equal to the derivative of a product, and then find the
antiderivative. Choose 0 for the constant of integration.

A)
B)
C)
D)

Question 39

Find an explicit expression for the particular solution for the differential equation.

A)
B)
C)
D)

Question 40

Find the general solution for the differential equation. Do not attempt to find an explicit expression for the solution.

A)
B)
C)
D)

Question 41

Solve the problem.
An initial deposit of $45,000 is made into an account that earns 5.4% compounded continuously. Money is then withdrawn at a constant rate of $2,700 a year until the amount in the account is 0. Find the amount in the
Account at any time t. When is the amount 0? What is the total withdrawn from the account?

A) . The account will be depleted after about 35.1 years. A total of about $94,770 will have been withdrawn at that time.
B) . The account will be depleted after about 42.6 years. A total of about $115,130 will have been withdrawn at that time.
C) . The account will be depleted after about 15.4 years. A total of about $41,580 will have been withdrawn at that time.
D) . The account will be depleted after about 20.5 years. A total of about $55,350 will have been withdrawn at that time.

Question 42

Find the integrating factor I(x) for each equation, and then find the general solution.

A)
B)
C)
D)

Question 43

Solve the problem.
A 1000-gallon holding tank contains 100 gallons of water. Initially, each gallon of water in the tank contains 3 pounds of pollutants. Water containing 4 pounds of pollutants per gallon enters the tank at a rate of 50 gallons
Per hour, and the uniformly mixed water is released from the tank at a rate of 25 gallons per hour. How many
Pounds of pollutants are in the tank after 4 hours? At what rate (in pounds per gallon) are the pollutants being
Released after 4 hours?

A) There are 250 pounds of pollutants in the tank after 4 hours. The pollutants are being released at a rate of 1.25 pounds per gallon after 4 hours.
B) There are 650 pounds of pollutants in the tank after 4 hours. The pollutants are being released at a rate of 5.75 pounds per gallon after 4 hours.
C) There are 180 pounds of pollutants in the tank after 4 hours. The pollutants are being released at a rate of 4.35 pounds per gallon after 4 hours.
D) There are 750 pounds of pollutants in the tank after 4 hours. The pollutants are being released at a rate of 3.75 pounds per gallon after 4 hours.

Question 44

Find the general solution two ways. First use an integrating factor and then use separation of variables.

A)
B)
C)
D)

Question 45

Find the integrating factor I(x) for each equation, and then find the general solution.

A)
B)
C)
D)

Question 46

Find the general solution two ways. First use an integrating factor and then use separation of variables.

A)
B)
C)
D)

Question 47

Solve the problem.
The supply S and demand D for a certain commodity satisfy the equations ) and find the equilibrium price p at time t and the long-range equilibrium price.

A)
B)
C)
D)