Deck 9: Differential Equations

Question 1

Solve the initial-value problem.

A)
B)
C)
D)
E)

Answer 1

Question 2

Solve the differential equation.

A)
B)
C)
D)
E)

Answer 2

Question 3

Solve the initial-value problem.

A)
B)
C)
D)
E)

Answer 3

Question 4

Determine whether the differential equation is linear.

A)the equation is not linear
B)the equation is linear

Question 5

Solve the differential equation.

A)
B)
C)
D)
E)

Question 6

In the circuit shown in Figure, a generator supplies a voltage of volts, the inductance is 2 H, the resistance is 40 , and . Find the current 0.2 s after the switch is closed. Round your answer to two decimal places.

A) 0.75 A
B)
C)
D)
E)

Question 7

An object with mass m is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If is the distance dropped after t seconds, then the speed is and the acceleration is . If g is the acceleration due to gravity, then the downward force on the object is , where c is a positive constant, and Newton's Second Law gives .
Find the limiting velocity.

Question 8

Solve the initial-value problem.

A)
B)
C)
D)
E)

Question 9

We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: ,

A)
B)
C)
D)
E)

Question 10

A phase trajectory is shown for populations of rabbits and foxes . Describe how each population changes as time goes by. Select the correct statement.

A)At the population of foxes reaches a minimum of about 30.
B)At the number of rabbits rebounds to 500.
C)At the number of foxes reaches a maximum of about 2400.

Question 11

Which of the following functions is a solution of the differential equation?

A)
B)
C)
D)
E)

Question 12

Solve the differential equation.

A)
B)
C)
D)
E)

Question 13

Solve the differential equation.

Question 14

Determine whether the differential equation is linear.

Question 15

Solve the initial-value problem.

A)
B)
C)
D)
E)

Question 16

Solve the initial-value problem.

Question 17

Solve the initial-value problem.

Question 18

Find the solution of the initial-value problem and use it to find the population when .

Question 19

Let be the performance level of someone learning a skill as a function of the training time t. The graph of P is called a learning curve. We propose the differential equation as a reasonable model for learning, where r is a positive constant. Solve it as a linear differential equation.

A)
B)
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D)
E)

Question 20

Solve the initial-value problem.

Question 21

Suppose that a population grows according to a logistic model with carrying capacity and per year. Choose the logistic differential equation for these data.

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B)
C)
D)
E)

Question 22

Let .
What are the equilibrium solutions?

Question 23

The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s range from 35 to 40 million per year and death rates range from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. Use the logistic model to predict the world population in the 2,450 year. Calculate your answer in billions to one decimal place. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate.)

A)59.2 billion
B)32.9 billion
C)78.3 billion
D)17.1 billion
E)24.1 billion

Question 24

The rate of change of atmospheric pressure P with respect to altitude h is proportional to P provided that the temperature is constant. At the pressure is at sea level and at . What is the pressure at an altitude of ?

Question 25

Let c be a positive number. A differential equation of the form where k is a positive constant, is called a doomsday equation because the exponent in the expression is larger than the exponent 1for natural growth. An especially prolific breed of rabbits has the growth term . If such rabbits breed initially and the warren has rabbits after months, then when is doomsday?

Question 26

Consider the differential equation as a model for a fish population, where t is measured in weeks and c is a constant. For what values of c does the fish population always die out?

Question 27

One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction of the population who have heard the rumor and the fraction who have not heard the rumor. Let's assume that the constant of proportionality is . Write a differential equation that is satisfied by y.

Question 28

The Pacific halibut fishery has been modeled by the differential equation where is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be and per year. If , find the biomass a year later.

Question 29

Suppose that a population grows according to a logistic model with carrying capacity and per year. Write the logistic differential equation for these data.

Question 30

Let c be a positive number. A differential equation of the form where k is a positive constant is called a doomsday equation because the exponent in the expression is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term . If such rabbits breed initially and the warren has rabbits after months, then when is doomsday?

A)
B)
C)
D)
E)

Question 31

A curve passes through the point and has the property that the slope of the curve at every point P is times the y-coordinate P. What is the equation of the curve?

A)
B)
C)
D)
E)

Question 32

Biologists stocked a lake with fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be . The number of fish tripled in the first year. Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years.

Question 33

Solve the differential equation.

A)
B)
C)
D)
E)

Question 34

The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s range from 35 to 40 million per year and death rates range from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. Use the logistic model to predict the world population in the 2,450 year. Calculate your answer in billions to one decimal place. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate.)

A)24.1 billion
B)32.9 billion
C)59.2 billion
D)78.3 billion
E)17.1 billion

Question 35

One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of inhabitants, people have a disease at the beginning of the week and have it at the end of the week. How long does it take for of the population to be infected?

A)
B)
C)
D)
E)

Question 36

A sum of is invested at interest. If is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by .

A)
B)
C)
D)
E)

Question 37

Suppose that a population develops according to the logistic equation , where t is measured in weeks. What is the carrying capacity?

A)
B)
C)
D)
E)

Question 38

One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of inhabitants, people have a disease at the beginning of the week and have it at the end of the week. How long does it take for of the population to be infected?

Question 39

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every . The initial population of a culture is cells. Find the number of cells after hours.

A)
B)
C)
D)
E)

Question 40

Solve the differential equation.

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D)
E)

Question 41

Experiments show that if the chemical reaction takes place at , the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows : How long will the reaction take to reduce the concentration of to 50% of its original value?

Question 42

Solve the differential equation.

Question 43

A certain small country has $20 billion in paper currency in circulation, and each day $70 million comes into the country's banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let denote the amount of new currency in circulation at time t with . Formulate and solve a mathematical model in the form of an initial-value problem that represents the "flow" of the new currency into circulation (in billions per day).

Question 44

Solve the differential equation.

Question 45

Find the solution of the differential equation that satisfies the initial condition .

Question 46

Find the orthogonal trajectories of the family of curves.

A)
B)
C)
D)
E)

Question 47

Select a direction field for the differential equation from a set of direction fields labeled I-IV.

Question 48

A population is modeled by the differential equation. For what values of P is the population increasing?

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B)
C)
D)
E)

Question 49

Choose the differential equation corresponding to this direction field.

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E)

Question 50

Find the orthogonal trajectories of the family of curves.

Question 51

Solve the differential equation.

Question 52

A tank contains L of brine with kg of dissolved salt. Pure water enters the tank at a rate of L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after minutes?

Question 53

Which equation does the function satisfy?

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B)
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E)

Question 54

Solve the initial-value problem.

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B)
C)
D)
E)

Question 55

Find the solution of the differential equation that satisfies the initial condition .

Question 56

Use Euler's method with step size 0.1 to estimate , where is the solution of the initial-value problem. Round your answer to four decimal places.

Question 57

Use Euler's method with step size 0.25 to estimate , where is the solution of the initial-value problem. Round your answer to four decimal places.

A)
B)
C)
D)
E)

Question 58

Solve the differential equation.

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B)
C)
D)
E)

Question 59

The solution of the differential equation satisfies the initial condition .
Find the limit.

Question 60

Kirchhoff's Law gives us the derivative equation .
If , use Euler's method with step size 0.1 to estimate after 0.3 second.

Question 61

For what nonzero values of k does the function satisfy the differential equation for all values of A and B?

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B)
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D)
E)

Question 62

For what values of k does the function satisfy the differential equation ?

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B)
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E)

Question 63

Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Suppose that a roast turkey is taken from an oven when its temperature has reached and is placed on a table in a room where the temperature is . If is the temperature of the turkey after t minutes, then Newton's Law of Cooling implies that . This could be solved as a separable differential equation. Another method is to make the change of variable . If the temperature of the turkey is after half an hour, what is the temperature after 35 min?

A)
B)
C)
D)
E)

Question 64

A population is modeled by the differential equation .
For what values of P is the population decreasing?

Question 65

Which of the following functions are the constant solutions of the equation
a.
b.
c.
d.
e.

Question 66

A sum of is invested at interest. If is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by .

Question 67

A function satisfies the differential equation .
What are the constant solutions of the equation?