Deck 9: Differential Equations

Question 1

Find the orthogonal trajectories of the family of curves.

Answer 1

Question 2

Solve the initial-value problem.

Answer 2

Question 3

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

Answer 3

Question 4

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

Question 5

An object with mass 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 is a positive constant, and Newton's Second Law gives Find the limiting velocity.

Question 6

Solve the initial-value problem.

Question 7

Let c be a positive number.A differential equation of the form where is a positive constant, is called 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 8

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 9

Let What are the equilibrium solutions?

Question 10

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 11

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 12

Solve the initial-value problem.

Question 13

Solve the differential equation.

Question 14

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

Question 15

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 16

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 17

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

Question 18

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

Question 19

Solve the differential equation.

Question 20

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

Question 21

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

Question 22

Determine whether the differential equation is linear.

Question 23

Solve the differential equation.

Question 24

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

Question 25

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

Question 26

Find the solution of the differential equation hat satisfies the initial condition

Question 27

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 28

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 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 29

Solve the differential equation.

Question 30

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

Question 31

Solve the initial-value problem.

Question 32

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

Question 33

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

Question 34

Determine whether the differential equation is linear.

Question 35

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

Question 36

Which equation does the function satisfy?

Question 37

Solve the differential equation.

Question 38

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

Question 39

Let What are the equilibrium solutions?

Question 40

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

Question 41

Select the correct Answer: for each question.
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 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 42

Select the correct Answer: for each question.
Solve the differential equation.

A)
B)
C)
D)
E)

Question 43

Select the correct Answer: for each question.
Let be the performance level of someone learning a skill as a function of the training time The graph of P is called a We propose the differential equation as a reasonable model for learning, where is a positive constant.Solve it as a linear differential equation.

A)
B)
C)
D)
E)

Question 44

Select the correct Answer: for each question.
Solve the differential equation.

A)
B)
C)
D)
E)

Question 45

Select the correct Answer: for each question.
For what values of does the function atisfy the differential equation
a.
b.
c.
d.
e.

Question 46

Select the correct Answer: for each question.
A population is modeled by the differential equation. For what values of P is the population increasing?

A)
B)
C)
D)
E)

Question 47

Select the correct Answer: for each question.
A common inhabitant of human intestines is the bacterium 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 48

Select the correct Answer: for each question.
Suppose that a population grows according to a logistic model with carrying capacity and per year.Choose the logistic differential equation for these data.

A)
B)
C)
D)
E)

Question 49

Select the correct Answer: for each question.
Solve the differential equation.

A)
B)
C)
D)
E)

Question 50

Select the correct Answer: for each question.
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 number of rabbits rebounds to 500.
B)At the number of foxes reaches a maximum of about 2400.
C)At the population of foxes reaches a minimum of about 30.

Question 51

Select the correct Answer: for each question.
Solve the differential equation.

A)
B)
C)
D)
E)

Question 52

Select the correct Answer: for each question.
We modeled populations of aphids and ladybugs with a Lotka-Volterra system.Suppose we modify those equations as follows: Find the equilibrium solution.

A)
B)
C)
D)
E)

Question 53

Select the correct Answer: for each question.
A curve passes through the point and has the property that the slope of the curve at every point is times the y-coordinate What is the equation of the curve?

A)
B)
C)
D)
E)

Question 54

Select the correct Answer: for each question.
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 yourAnswer in billions to one decimal place.(Because the initial population is small compared to the carrying capacity, you can take to be an estimate of the initial relative growth rate.)

A)78.3 billion
B)27.0 billion
C)17.1 billion
D)59.2 billion
E)32.9 billion

Question 55

Select the correct Answer: for each question.
For what nonzero values of does the function satisfy the differential equation for all values of A and B?
a.
b.
c.
d.
e.

Question 56

Select the correct Answer: for each question.
be a positive number.A differential equation of the form where 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 57

Select the correct Answer: for each question.
Solve the differential equation.

A)
B)
C)
D)
E)

Question 58

Select the correct Answer: for each question.
Which of the following functions is a solution of the differential equation?
a.
b.
c.
d.
e.

Question 59

Select the correct Answer: for each question.
Solve the differential equation.

A)
B)
C)
D)
E)

Question 60

Select the correct Answer: for each question.
Which of the following functions are the constant solutions of the equation
a.
b.
c.
d.
e.

Question 61

Determine whether the differential equation is linear.

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

Question 62

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 63

Solve the differential equation.

A)
B)
C)
D)
E)

Question 64

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 65

Choose the differential equation corresponding to this direction field.

A)
B)
C)
D)
E)

Question 66

Which equation does the function satisfy?

A)
B)
C)
D)
E)

Question 67

Solve the initial-value problem.

A)
B)
C)
D)
E)

Question 68

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

A)
B)
C)
D)
E)

Question 69

Solve the initial-value problem.

A)
B)
C)
D)
E)

Question 70

Find the orthogonal trajectories of the family of curves.

A)
B)
C)
D)
E)

Question 71

A common inhabitant of human intestines is the bacterium 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 72

Solve the differential equation.

A)
B)
C)
D)
E)

Question 73

Solve the initial-value problem.

A)
B)
C)
D)
E)

Question 74

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 yourAnswer in billions to one decimal place.(Because the initial population is small compared to the carrying capacity, you can take to be an estimate of the initial relative growth rate.)

A)78.3 billion
B)27.0 billion
C)17.1 billion
D)59.2 billion
E)32.9 billion

Question 75

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

A)
B)
C)
D)
E)

Question 76

Use Euler's method with step size 0.25 to estim ate where is the solution of the initial-value problem.Round yourAnswer to four decimal places.

A)
B)
C)
D)
E)

Question 77

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

A)
B)
C)
D)
E)

Question 78

Let be a positive number.A differential equation of the form where is a positive constant is called 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 79

Solve the initial-value problem.

A)
B)
C)
D)
E)

Question 80

Solve the differential equation.

A)
B)
C)
D)
E)