Deck 9: Differential Equations

Question 1

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.

Answer 1

Question 2

Solve the initial-value problem. $x y ^ { \prime } - y = x \ln x , \quad y ( 1 ) = 4$

A) $y ( x ) = \frac { 1 } { 2 } ( \ln x ) ^ { 4 } + x$
B) $y ( x ) = 4 x \ln x + \frac { 1 } { 2 } x$
C) $y ( x ) = 4 \ln x + \frac { 1 } { 2 } x$
D) $y ( x ) = \frac { 1 } { 2 } x ( \ln x ) ^ { 2 } + 4 x$
E) $y ( x ) = 4 x ( \ln x ) ^ { 2 }$

Answer 2

$y ( x ) = \frac { 1 } { 2 } x ( \ln x ) ^ { 2 } + 4 x$

Question 3

Let $P ( t )$ 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 $\frac { d P } { d t } = r ( G - P ( t ) )$ as a reasonable model for learning, where r is a positive constant. Solve it as a linear differential equation.

A) $P ( t ) = G - C e ^ { - r t }$
B) $P ( t ) = G + C e ^ { - r t }$
C) $P ( t ) = G + C e ^ { rt }$
D) $P ( t ) = G C e ^ { n t }$
E) $P ( t ) = G - C e ^ { rt }$

Answer 3

$P ( t ) = G + C e ^ { - r t }$

Question 4

Solve the differential equation. $( 6 + t ) \frac { d u } { d t } + u = 6 + t , t > 0$

A) $u = \frac { t ^ { 2 } + C } { t + 6 }$
B) $u = 6 t + \frac { t ^ { 2 } } { 6 } + C$
C) $u = \frac { 6 t + t ^ { 2 } + C } { t + 6 }$
D) $u = \frac { 6 t + \frac { t ^ { 2 } } { 2 } + C } { t + 6 }$
E) $u = \frac { 6 t + \frac { 1 } { 2 } t ^ { 2 } } { \frac { 3 } { 6 } t + 2 } + C$

Question 5

We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: $$\frac { d A } { d t } = 2 A ( 1 - 0.0005 A ) - 0.01 A L$$ , $\frac { d L } { d t } = - 0.6 L + 0.0005 A L$

A) $A = 3200 , L = 110$
B) $A = 4700 , L = 90$
C) $A = 3700 , L = 120$
D) $A = 1200 , L = 80$
E) $A = 1500 , L = 100$

Question 6

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

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

Question 7

Determine whether the differential equation is linear.

Question 8

Which of the following functions is a solution of the differential equation? $y ^ { \prime \prime } + 16 y ^ { \prime } + 64 y = 0$

A) $y = e ^ { t }$
B) $y = t e ^ { - 8 t }$
C) $y = 6 e ^ { - 8 t }$
D) $y = e ^ { - 8 t }$
E) $y = t ^ { 2 } e ^ { - 8 t }$

Question 9

Solve the differential equation.

Question 10

Solve the initial-value problem.

Question 11

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

Question 12

Solve the differential equation. $y ^ { \prime } = x e ^ { - \sin x } - y \cos x$

A) $y = \left( \frac { 1 } { 2 } x ^ { 2 } + C \right) e ^ { - \sin x }$
B) $y = \frac { 1 } { 2 } x + C e ^ { - \cos x }$
C) $y = C e ^ { - \sin x }$
D) $y = e ^ { - \sin x } + C x$
E) $y = \left( 2 x ^ { 2 } + C \right) e ^ { - x } \sin x$

Question 13

Solve the initial-value problem.

Question 14

Solve the initial-value problem. $r ^ { \prime } + 2 t r = r , \quad r ( 0 ) = 2$

A) $r ( t ) = 2 e ^ { t - t ^ { 2 } }$
B) $r ( t ) = 2 e ^ { t - 2 t }$
C) $r ( t ) = e ^ { t - 2 t ^ { 2 } }$
D) $r ( t ) = \frac { e ^ { t ^ { 2 } } } { 2 }$
E) $r ( t ) = 2 e ^ { 2 t }$

Question 15

Determine whether the differential equation is linear. $y ^ { \prime } + 3 x ^ { 2 } y = 6 x ^ { 2 }$

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

Question 16

Solve the initial-value problem.

Question 17

Solve the initial-value problem. $x y ^ { \prime } = y + x ^ { 2 } \sin x , y ( 3 \pi ) = 0$

A) $y = x \sin x - 3 x$
B) $y = - x \cos x - x$
C) $y = - x \sin x + 3 x$
D) $y = 3 x \sin x$
E) $y = 3 x \cos x - \sin x$

Question 18

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$

A) $y = C e ^ { - 2 / x }$
B) $y = x ^ { 2 } e ^ { - 1 / x } + C e ^ { - 1 / x }$
C) $y = C x e ^ { 2 x }$
D) $y = e ^ { 1 / x } x + C e ^ { 1 / x }$
E) $\text { none of these }$

Question 19

Solve the initial-value problem. $( 1 + \cos x ) \frac { d y } { d x } = \left( 2 + e ^ { - y } \right) \sin x , y ( 0 ) = 0$

A) $y = \ln ( 2 - \cos x ) ( 2 + \cos x )$
B) $y = \frac { ( 7 - \cos x ) } { ( 2 + \cos x ) }$
C) $y = \ln \frac { \cos x } { ( 2 + \cos x ) }$
D) $y = \frac { \ln ( 2 + \cos x ) } { \ln ( 2 - \cos x ) }$
E) $y = \ln ( 7 - \cos x ) - \ln ( 1 + \cos x )$

Question 20

In the circuit shown in Figure, a generator supplies a voltage of $E ( t ) = 20 \sin 60 t$ volts, the inductance is 2 H, the resistance is 40 Ω , and $I ( 0 ) = 2 A$ . Find the current 0.2 s after the switch is closed. Round your answer to two decimal places.

A) $- 0.06 \mathrm {~A}$
B) $- 0.75 \mathrm {~A}$
C) $- 0.13 A$
D) $- 0.50 \mathrm {~A}$
E) $- 0.11 \mathrm {~A}$

Question 21

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 $20 \text { minutes }$ . The initial population of a culture is $25$ cells. Find the number of cells after $6$ hours.

A) $P = 6,673,600$
B) $P = 6,553,600$
C) $P = 6,653,600$
D) $P = 6,565,600$
E) $P = 6,693,600$

Question 22

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

Question 24

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 $2000$ inhabitants, $110$ people have a disease at the beginning of the week and $1100$ have it at the end of the week. How long does it take for $70 \%$ of the population to be infected?

A) $30 \text { days }$
B) $20 \text { days }$
C) $0 \text { days }$
D) $10 \text { days }$
E) $15 \text { days }$

Question 25

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 26

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 27

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 28

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 29

A sum of $\$ 5,000$ is invested at $35 \%$ interest. If $A ( t )$ 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 $A ( t )$ .

A) $\frac { d A } { d t } = 4,500 A , \quad A ( 0 ) = 5,000$
B) $\frac { d A } { d t } = 35 A , \quad A ( 0 ) = 3,000$
C) $\frac { d A } { d t } = 0.35 A , \quad A ( 0 ) = 500$
D) $\frac { d A } { d t } = 0.35 A , \quad A ( 0 ) = 5,000$
E) $\frac { d A } { d t } = 0.35 A ( 0 ) , \quad A ( 0 ) = 5,000$

Question 30

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 31

Let c be a positive number. A differential equation of the form $\frac { d y } { d t } = k y ^ { 1 + c }$ where k is a positive constant is called a doomsday equation because the exponent in the expression $k y ^ { 1 + c }$ is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term $k y ^ { 1.04 }$ . If $4$ such rabbits breed initially and the warren has $21$ rabbits after $8$ months, then when is doomsday?

A) $100.54 \text { months }$
B) $190.54 \text { months }$
C) $170.54 \text { months }$
D) $120.54 \text { months }$
E) $150.54 \text { months }$

Question 32

Solve the differential equation. $4 \frac { d w } { d t } + 5 e ^ { t + w } = 0$

A) $w = \frac { 5 } { 4 } e ^ { t } - C$
B) $w = \ln \left( \frac { 4 } { 5 } t \right) - t$
C) $w = - \ln \left( \frac { 5 e ^ { z } } { 4 } - C \right)$
D) $w = \ln \left( \frac { 4 } { 5 } + \frac { 4 } { 5 } C e ^ { t } \right) - t$
E) $\text { none of these }$

Question 33

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 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) 78.3 billion
C) 59.2 billion
D) 17.1 billion
E) 32.9 billion

Question 35

Suppose that a population grows according to a logistic model with carrying capacity $8,000$ and $k = 0.01$ per year. Choose the logistic differential equation for these data.

A) $\frac { d P ( t ) } { d t } = 0.01 P \left( 1 + \frac { p } { 1 } \right)$
B) $\frac { d P ( t ) } { d t } = 8,000 P \left( 1 + \frac { p } { 0.01 } \right)$
C) $\frac { d P ( t ) } { d t } = 0.01 P \left( 1 - \frac { p } { 8,000 } \right)$
D) $\frac { d P ( t ) } { d t } = 0.01 P \left( 1 + \frac { p } { 8,000 } \right)$
E) $\frac { d P ( t ) } { d t } = P \left( 1 + \frac { p } { 8,000 } \right)$

Question 36

Suppose that a population develops according to the logistic equation $\frac { d P } { d t } = 0.06 P - 0.0006 P ^ { 2 }$ , where t is measured in weeks. What is the carrying capacity?

A) $K = 600$
B) $K = 100$
C) $K = 0.0006$
D) $K = 0.06$
E) $K = 700$

Question 37

A curve passes through the point $( 8,2 )$ and has the property that the slope of the curve at every point P is $3$ times the y-coordinate P. What is the equation of the curve?

A) $y = 2 e ^ { 3 x + 24 }$
B) $y = \frac { e ^ { 3 x - 24 } } { 2 }$
C) $y = 2 e ^ { 3 x - 8 }$
D) $y = 2 e ^ { 3 x - 24 }$
E) $y = 2 e ^ { x - 24 }$

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

Solve the differential equation. $3 y y ^ { \prime } = 7 x$

A) $7 x ^ { 2 } + 3 y ^ { 2 } = C$
B) $3 x ^ { 2 } + 7 y ^ { 2 } = C$
C) $3 x ^ { 2 } - 7 y ^ { 2 } = C$
D) $7 x ^ { 2 } - 3 y ^ { 2 } = C$
E) $7 x ^ { 2 } + 3 y ^ { 2 } = 10$

Question 40

Let .
What are the equilibrium solutions?

Question 41

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

Question 42

Find the orthogonal trajectories of the family of curves. $y = k x ^ { 9 }$

A) $x ^ { 2 } - 8 y ^ { 2 } = C$
B) $x ^ { 2 } - 8 y ^ { 2 } = 0$
C) $x ^ { 2 } - 8 y ^ { 2 } = 0$ .
D) $x ^ { 2 } + 8 y ^ { 2 } = 0$
E) $x ^ { 2 } + 9 y ^ { 2 } = C$

Question 43

Which equation does the function $y = e ^ { - 6 t }$ satisfy?

A) $y ^ { \prime \prime } + y ^ { \prime } + 42 y = 0$
B) $y ^ { \prime \prime } - 3 y ^ { \prime } + 42 y = 0$
C) $y ^ { \prime \prime } - y ^ { \prime } - 42 y = 0$
D) $y ^ { \prime \prime } + y ^ { \prime } - 42 y = 0$
E) $y ^ { \prime \prime } - y ^ { \prime } + 42 y = 0$

Question 44

Use Euler's method with step size 0.25 to estimate $y ( 1 )$ , where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places. $y ^ { \prime } = 6 x + y ^ { 2 } , \quad y ( 0 ) = 0$

A) $y ( 1 ) = 3.6216$
B) $y ( 1 ) = 5.1216$
C) $y ( 1 ) = 4.1216$ .
D) $y ( 1 ) = 2.6216$
E) $y ( 1 ) = 4.1216$

Question 45

Find the orthogonal trajectories of the family of curves.

Question 46

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

Question 47

Solve the differential equation.

Question 48

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

Question 49

Solve the differential equation.

Question 50

Solve the initial-value problem. $\frac { d r } { d t } + 2 t r - r = 0 , \quad r ( 0 ) = 10$

A) $r ( t ) = 10 e ^ { t - 10 t ^ { 2 } }$
B) $r ( t ) = e ^ { 10 t - t ^ { 2 } }$
C) $r ( t ) = 10 e ^ { t ^ { 2 } }$
D) $r ( t ) = 2 e ^ { t - t ^ { 2 } }$
E) $\text { none of these }$

Question 51

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

Question 52

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 53

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 54

Choose the differential equation corresponding to this direction field.

A) $y ^ { \prime } = x + y - 1$
B) $y ^ { \prime } = \sin x \sin y$
C) $y ^ { \prime } = y + x y$
D) $y ^ { \prime } = 2 - y$
E) $y ^ { \prime } = x ( 2 - y )$

Question 55

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 56

A population is modeled by the differential equation. $\frac { d P } { d t } = 1.4 P \left( 1 - \frac { P } { 4560 } \right)$ For what values of P is the population increasing?

A) $P > 1.4$
B) $P > 4560$
C) $P < 4580$
D) $0 < P < 4560$
E) $0 < P < 1.4$

Question 57

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 58

Solve the differential equation.

Question 59

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 60

Solve the differential equation. $\frac { d u } { d t } = 15 + 5 u + 3 t + u t$

A) $u = - 3 + C e ^ { \frac { t ^ { 2 } } { 2 } - 5 t }$
B) $u = - 3 + C e ^ { t ^ { 2 } - 5 t }$
C) $u = - 3 + C e ^ { \frac { t ^ { 2 } } { 2 } + 5 t }$
D) $u = - 3 + C e ^ { \frac { t ^ { 2 } } { 2 } + 5 t ^ { 2 } }$
E) $u = - 3 + C e ^ { t ^ { 2 } + 5 t }$

Question 61

For what values of k does the function $y = \cos k t$ satisfy the differential equation $9 y ^ { \prime \prime } = - 25 y$ ?

A) $k = - \frac { 1 } { 5 }$
B) $k = \frac { 1 } { 5 }$
C) $k = \frac { 1 } { 7 }$
D) $k = - \frac { 5 } { 7 }$
E) $k = \frac { 5 } { 7 }$

Question 62

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 63

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

Question 64

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 $160 ^ { \circ } \mathrm { F }$ and is placed on a table in a room where the temperature is $60 ^ { \circ } \mathrm { F }$ . If $u ( t )$ is the temperature of the turkey after t minutes, then Newton's Law of Cooling implies that $\frac { d u } { d t } = k ( u - 60 )$ . This could be solved as a separable differential equation. Another method is to make the change of variable $y = u - 60$ . If the temperature of the turkey is $150 ^ { \circ } \mathrm { F }$ after half an hour, what is the temperature after 35 min?

A) $t = 143 ^ { \circ } \mathrm { F }$
B) $t = 148 ^ { \circ } \mathrm { F }$
C) $t = 298 ^ { \circ } \mathrm { F }$
D) $\text { none of these }$
E) $t = 95 ^ { \circ } \mathrm { F }$

Question 65

For what nonzero values of k does the function $y = A \sin k t + B \cos k t$ satisfy the differential equation $y ^ { \prime \prime } + 100 y = 0$ for all values of A and B?

A) $k = 10$
B) $k = - 100$
C) $k = - 10$
D) $k = 100$
E) $k = 1$

Question 66

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

Question 67

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