Deck 9: Differential Equations

Question 1

Find the general solution of the differential equation. Use C, C1, C2, . . . to denote arbitrary constants. If used, do not simplify hyperbolic functions.

-y'(t) = 3 +

A) y = 3t - + C
B) y = 3t + + C
C) y = -4 + C
D) y = 3 - + C

Answer 1

y = 3t - + C

Question 2

Find the general solution of the differential equation. Use C, C1, C2, . . . to denote arbitrary constants. If used, do not simplify hyperbolic functions.

-y'(t) = 14 - 16 + 8 - 12

A)
B)
C)
D)

Answer 2

Question 3

Find the general solution of the differential equation. Use C, C1, C2, . . . to denote arbitrary constants. If used, do not simplify hyperbolic functions.

-p'(x) = - 5 + 12

A)

B)

C)

D)

Answer 3

Question 4

Find the general solution of the differential equation. Use C, C1, C2, . . . to denote arbitrary constants. If used, do not simplify hyperbolic functions.

-y''(t) = 45 + sin 5t

A)

B)

C)

D)

Question 5

Find the general solution of the differential equation. Use C, C1, C2, . . . to denote arbitrary constants. If used, do not simplify hyperbolic functions.

-v'(t) =

A)

B)

C)

D)

Question 6

Solve the initial value problem.

-y'(x) = 18 - 9 , y(1) = 0

A) y = 6x³ + 3x^-3
B) y = 6x³ + 3x ^-3- 9
C) y = 6x³ + 3x ^-3+ 9
D) y = 6x³ - 3x ^-3+ 9

Question 7

Solve the initial value problem.

-y'(x) = 4 2x, y(0) = 5

A) y = 2 tan 2x - 5
B) y = 2 tan 2x + 4
C) y = 2 tan 2x + 5
D) y = 4 tan 2x + 5

Question 8

Solve the initial value problem.

-u''(x) = 12 - 12 , u(0) = 12, u'(0) = 16

A)

B)

C)

D)

Question 9

Solve the initial value problem.

-u'(x) = - 6, u(0) = 5

A)

B)

C)

D)

Question 10

Solve the problem.

-An object is fired vertically upward with an initial velocity v(0) = v₀ from an initial position s(0) = s₀ . Assuming no air resistance, the position of the object is governed by the differential equation is the acceleration due to gravity (in the downward direction). For the following values of v₀ and s₀ , find the position and velocity functions for all times at which the object is above the ground. Then find the time at which the highest point of the trajectory is reached and the height of the object at that time.
v₀ = 39.2 m/s , s₀ = 40 m

A) s = -9.8 $t^{2}$ + 39.2t + 40, v = -9.8t; Highest point of 118.4 m is reached at t = 5 s
B) s = -9.8 $t^{2}$ + 39.2t + 40, v = -9.8t + 39.2; Highest point of 118.4 m is reached at t = 4 s
C) s = -4.9 $t^{2}$ + 39.2t + 40, v = -9.8t + 39.2; Highest point of 118.4 m is reached at t = 5 s
D) s = -4.9 $t^{2}$ + 39.2t + 40, v = -9.8t + 39.2; Highest point of 118.4 m is reached at t = 4 s

Question 11

Solve the problem.

-A simple model of a harvested resource follows the equation where p(t) is the amount (or population) of the resource at time is the natural growth rate of the resource, and H > 0 is the harvesting rate. If for what values of H is the amount of the resource increasing? For what value of H is the amount of the resource constant?

A) H < 25; H = 25
B) H < 25; H > 25
C) H > 25; H = 25
D) H < 50; H < 50

Question 12

Solve the problem.

-Imagine a large tank with cross-sectional area A. The bottom of the tank has a circular drain with cross-sectional area a. Assume the tank is initially filled with water to a height h(0) = H. The height of the water as it flows out of the tank is described by the equation where is the acceleration due to gravity. Find the water height function for Then determine the approximate time at which the tank is first empty. If necessary, round to two decimal places.

A)

B)

C)

D)

Question 13

Match the differential equation with the appropriate direction field.

- (x) = y + 2

A)

B)

C)

D)

Question 14

Match the differential equation with the appropriate direction field.

- (x) =

A)

B)

C)

D)

Question 15

Match the differential equation with the appropriate direction field.

-

A)

B)

C)

D)

Question 16

Use the given window to sketch a direction field for the equation. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed.

-y'(x) = y(3 - y), y(0) = ; [0, 5] × [0, 4]

Question 17

Consider the differential equation. A detailed direction field is not needed. Find the solutions that are constant, for all (the equilibrium solutions). In what regions are solutions increasing? Decreasing?

-y'(t) = (y - 4)( 4 + y)

A)
B)
C)
D)

Question 18

Consider the differential equation. A detailed direction field is not needed. Find the solutions that are constant, for all (the equilibrium solutions). In what regions are solutions increasing? Decreasing?

-y'(t) = (y - 2)(y + 7)

A) y = -2, y = 7; increasing for y < -7 or y > 2; decreasing for -7 < y < 2
B) y = 2, y = -7; increasing for y < -7 or y > 2; decreasing for -7 < y < 2
C)y = -2, y = 7; increasing for -7 < y < 2; decreasing for y < -7 or y > 2
D) y = 2, y = -7; increasing for -7 < y < 2; decreasing for y < -7 or y > 2

Question 19

Consider the differential equation. A detailed direction field is not needed. Find the solutions that are constant, for all (the equilibrium solutions). In what regions are solutions increasing? Decreasing?

-y'(t) = y(y + 4)( 6 - y)

A) y = -4, y = 6; increasing for -4 < y < 6; decreasing for y < -4, y > 6
B) y = 0, y = -4, y = 6; increasing for y < -4, 0 < y < 6; decreasing for -4 < y < 0, y > 6
C) y = 0, y = -4, y = 6; increasing for -4 < y < 0; decreasing for y < -4, y > 0
D) y = -4, y = 6; increasing for y < -4, 0 < y < 6; decreasing for -4 < y < 0, y > 6

Question 20

Consider the logistic equation. Find the equilibrium solutions. A detailed direction field is not needed. Assume and

-P'(t) = 0.1P

A) P = 0 and P = 450
B) P = 450
C) P = 0 and P = 225
D) P = 0 and P = 900

Question 21

Consider the logistic equation. Find the equilibrium solutions. A detailed direction field is not needed. Assume and

-P'(t) = 0.05P

A) P = 0 and P = 1500
B) P = 0 and P = 3000
C) P = 0 and P = 750
D) P = 1500

Question 22

Consider the logistic equation. Find the equilibrium solutions. A detailed direction field is not needed. Assume and

-P'(t) = 0.1P - 0.002

A) P = 0 and P = 60
B) P = 50
C) P = 0 and P = 50
D) P = 0 and P = 100

Question 23

For the initial value problem, compute the first two approximations and given by Euler's method using the given time step.

-y'(t) = 6y, y(0) = 4; $\Delta$ t = 0.5

A) = 16; = 112
B) = 16; = 64
C) = 28; = 112
D) = 4; = 16

Question 24

For the initial value problem, compute the first two approximations and given by Euler's method using the given time step.

-y'(t) = 4 - y, y(0) = 4; $\Delta$ t = 0.1

A) = 4; = 4
B) = 4.8; = 4
C) = 4.8; = 4.72
D) = 4; = 4.72

Question 25

For the initial value problem, compute the first two approximations and given by Euler's method using the given time step.

-y'(t) = -y, y(0) = -3; $\Delta$ t = 0.1

A) = -2; = - 2.61
B) = -2.7; = -2.7
C) = -2.7; = -2.43
D) = -3; = -2.7

Question 26

Consider the initial value problem. Find the approximations of y(0.2) and y(0.4) using Euler's method with a time step of Then using the exact solution given, compute the errors in the Euler approximations at t = 0.2 and t = 0.4 Round answers to five decimal places, do not use intermediate rounding.

-y'(t) = 5 - y, y(0) = 4; y(t) = 5 -

A)
B)
C)
D)

Question 27

Consider the initial value problem. Find the approximations of y(0.2) and y(0.4) using Euler's method with a time step of Then using the exact solution given, compute the errors in the Euler approximations at t = 0.2 and t = 0.4 Round answers to five decimal places, do not use intermediate rounding.

-y'(t) = , y(0) = 7; y(t) = 7

A)
B)
C)
D)

Question 28

Use a calculator or computer program to approximate the value of y(T) using Euler's method with the given time step on the interval [0, T]. Then, using the exact solution given, find the error in the approximation to y(T) (only at the right endpoint of the time interval). Round answers to five decimal places, do not use intermediate rounding.

-y'(t) = -4y, y(0) = 1; $\Delta$ t = 0.2, T = 1; y(t) =

A) y(1) $\approx$ 0.00032; 0.01800
B) y(1) $\approx$ 0.00032; 0.20000
C)y(1) $\approx$ 0.00160; 0.01672
D) y(1) $\approx$ 0.00032; 0.01832

Question 29

Use a calculator or computer program to approximate the value of y(T) using Euler's method with the given time step on the interval [0, T]. Then, using the exact solution given, find the error in the approximation to y(T) (only at the right endpoint of the time interval). Round answers to five decimal places, do not use intermediate rounding.

-y'(t) = , y(0) = 4; $\Delta$ t = 0.2, T = 1; y(t) =

A) y(1) $\approx$ 4.12311; 0.024824
B) y(1) $\approx$ 4.14793; 0.024824
C) y(1) $\approx$ 4.14793; 4.12311
D) y(1) $\approx$ 4.14793; 0.20000

Question 30

Solve the problem.

-The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation where m(t) is the mass of the drug in the blood at time k is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate. Let For what initial values m(0) = A are solutions increasing? decreasing? What is the equilibrium solution?

A) increasing for A < 50 and decreasing for A > 50; m(t) = 0
B) increasing for A > 50 and decreasing for A < 50; m(t) = 0
C) increasing for A < 50 and decreasing for A > 50; m(t) = 50
D) increasing for A > 50 and decreasing for A < 50; m(t) = 50

Question 31

Solve the problem.

-A model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation where v(t) is the velocity of the object, for is the acceleration due to gravity, and b > 0 is a constant that involve the mass of the object and the air resistance. Let b = 0.2s^-1 For what initial values v(0) = A are solutions increasing? decreasing? What is the equilibrium solution?

A) increasing for A < 49 and decreasing for A > 49; v(t) = 49
B) increasing for A < 49 and decreasing for A > 49; v(t) = 0
C) increasing for A > 49 and decreasing for A < 49; v(t) = 49
D) increasing for A > 49 and decreasing for A < 49; v(t) = 0

Question 32

Find the general solution of the equation. Express the solution explicitly as a function of the independent variable.

-

A) y = -
B) y = -
C) y =
D) y =

Question 33

Find the general solution of the equation. Express the solution explicitly as a function of the independent variable.

-

A)
B) y = 42 + C
C) y = 49 + C
D)

Question 34

Find the general solution of the equation. Express the solution explicitly as a function of the independent variable.

-

A)

B)

C)

D) y = 56 + C

Question 35

Find the general solution of the equation. Express the solution explicitly as a function of the independent variable.

-

A)

B)

C)

D)

Question 36

Find the general solution of the equation. Express the solution explicitly as a function of the independent variable.

-

A)

B)

C)

D)

Question 37

Find the general solution of the equation. Express the solution explicitly as a function of the independent variable.

-

A)

B)

C)

D)

Question 38

Solve the initial value problem, if possible.

-csc x (t) = 1, y = 1

A) y = -cos t + 1
B) y = -cot t + 1
C) y = -sin t + 1
D) y = cos t +

Question 39

Solve the initial value problem, if possible.

-

A) y =
B) y = 5
C) y =
D) y =

Question 40

Solve the initial value problem, if possible.

-

A) y = + 2
B) y =
C) y = 6
D) Not separable

Question 41

Solve the initial value problem, if possible.

-

A) y = 2
B) y = 2 ln( + )
C)y = 2
D) y =

Question 42

Solve the initial value problem, if possible.

-

A) y =
B) y =
C) Not separable
D) y =

Question 43

Solve the initial value problem and leave the solution in implicit form.

- (t) = , y( 0) = 2

A)

B)

C)

D)

Question 44

Solve the initial value problem and leave the solution in implicit form.

- (x) = , y( -3) = -3

A)

B)

C)

D)

Question 45

Solve the initial value problem and leave the solution in implicit form.

- (x) = , y( 3) = 24

A)

B)

C)

D)

Question 46

Solve the initial value problem and leave the solution in implicit form.

- (x) = sec u sin , u( 6 $\pi$ ) = $\pi$

A)

B)

C)

D)

Question 47

Solve the problem.

-The spread of a cold virus can be modeled using logistic equations. The key assumption is that at any given time, a fraction y of the population, where 0 ≤ y ≤ 1, has the virus, while the remaining fraction 1-y does not. Furthermore, the cold virus spreads by interactions between those who have it and those who do not. The number of such interactions is proportional to y(1 - y). Therefore, the equation that describes the spread of the virus is where k is a positive real number. Assume k = 0.7 and solve the initial value problem where the number of people who initially have the cold virus is y(0) = 0.3

A) y =
B) y =
C) y =
D) y =

Question 48

Solve the problem.

-Let y(t) be the concentration of a substance in a chemical reaction. The change in the concentration, under appropriate conditions, is modeled by the equation where k > 0 is a rate constant and the positive integer n is the order of the reaction. Solve the initial value problem for a third-order reaction , assuming y(0) = 4 and k = 0.6.

A) y =
B) y = -
C) y =
D) y =

Question 49

Find the general solution of the equation.

- (t) + 2y = 11

A) y = +
B) y = + C
C) y = 11 + C
D) y = + + C

Question 50

Find the general solution of the equation.

- (x) = y - 4

A) y = - 4
B) y =
C) y = - 4
D) y = + 4

Question 51

Find the general solution of the equation.

- (t) - = -8

A)

B)

C)

D)

Question 52

Find the general solution of the equation.

- (x) = 10u + 11

A)

B)

C)

D)

Question 53

Solve the initial value problem.

-y'(t) + 9y = 3, y(0) = 1

A)

B)

C)

D)

Question 54

Solve the initial value problem.

- (t) = 4y - 8, y(0) = 11

A)

B)

C)

D)

Question 55

Solve the initial value problem.

- (x) = 3u + 12, u(1) = -2

A)

B)

C)

D)

Question 56

Solve the initial value problem.

- (t)+ = 7, v(-1) = 0

A)

B)

C)

D)

Question 57

Find the equilibrium solution of the equation. Use a sketch of the direction field for to determine whether the equilibrium is stable

- (t) = 7y - 16

A) y = ; stable
B) y = 23; stable
C) y = ; unstable
D) y = ; unstable

Question 58

Find the equilibrium solution of the equation. Use a sketch of the direction field for to determine whether the equilibrium is stable

- (t) + = -1

A) u = -6; stable
B) u = 6; unstable
C) u = ; unstable
D) u =- ; stable

Question 59

Find the equilibrium solution of the equation. Use a sketch of the direction field for to determine whether the equilibrium is stable

- (t) + 2u + 8 = 0

A) y = - ; stable
B) y = -4; unstable
C) y = 4; stable
D) y = ; unstable

Question 60

Solve the problem.

-Use Newton's Law of Cooling to find the temperature in the following case. A loaf of bread is removed from an oven at and cooled in a room whose temperature is . If the bread cools to in 20 minutes, how much longer will it take the bread to cool to

A) $\approx$ 18 minutes
B) $\approx$ 6 minutes
C) $\approx$ 26 minutes
D) $\approx$ 7 minutes

Question 61

Solve the problem.

-The initial value problem models the payoff of a loan. Solve the initial value problem for t $\ge$ 0, and determine the first month in which the balance is zero. (t) = 0.01B - 2000, B(0) = 100,000

A) B = 20,000 - 10,000 , reaches a balance of zero after approximately 59 months
B) B = 20,000 + 10,000 , reaches a balance of zero after approximately 79 months
C) B = 200,000,000 + 20,000,000 , reaches a balance of zero after approximately 119 months
D) B = 200,000 - 100,000 , reaches a balance of zero after approximately 69 months

Question 62

Solve the problem.

-Use Newton's Law of Cooling to find the temperature in the following case. A glass of water with a temperature of 6°C is placed in a room with a temperature of 30°C. One minute later the water has warmed to 9°C. After how many minutes does the water have a temperature that is 90% of the ambient temperature?

A) $\approx$ 23 min
B) $\approx$ 16 min
C) $\approx$ 19 min
D) $\approx$ 27 min

Question 63

Solve the problem.

-Let y(t) be the population of a species of plant that is being harvested, for t ≥ 0. Consider the harvesting model where h is the annual harvesting rate, y₀ is the initial population of the species, and t is measured in years. If y₀ = 5000, what harvesting rate should be used to maintain a constant population of y = 5000, for t ≥ 0?

A) h = 5000
B) h = 25
C) h = 450
D) h = 45

Question 64

Make a sketch of the population function (as a function of time) that results from the given growth rate function. Assume the population at time t = 0 begins at some positive value.

-

A)

B)

C)

D)

Question 65

Make a sketch of the population function (as a function of time) that results from the given growth rate function. Assume the population at time t = 0 begins at some positive value.

-

A)

B)

C)

D)

Question 66

Make a sketch of the population function (as a function of time) that results from the given growth rate function. Assume the population at time t = 0 begins at some positive value.

-

A)

B)

C)

D)

Question 67

Write a logistic equation with the given parameter values. Then solve the initial value problem and graph the solution. Let r be the natural growth rate, K the carrying capacity, and the initial population.

-r = 0.2, K = 1680, = 80

A)

B)
C)
D)

Question 68

Write a logistic equation with the given parameter values. Then solve the initial value problem and graph the solution. Let r be the natural growth rate, K the carrying capacity, and the initial population.

-r = 0.9, K = 7670, = 590

A)
B)
C)
D)

Question 69

Find a logistic function that describes the population. Graph the population function.

-The population increases from 300 to 600 in the first year and eventually levels off at 2100.

A)
B)
C)
D)

Question 70

Find a logistic function that describes the population. Graph the population function.

-The population increases from 50 to 70 in the first month and eventually levels off at 150.

A)
B)
C)
D)

Question 71

-r = 0.06, K = 1000, = 50

A)
B)
C)
D)

Question 72

Solve the problem.

-A 500 gallon tank is filled with pure water. A salt solution with a concentration of 11 g/gal flows into the tank at a rate of 7 gal/min. The thoroughly mixed solution is drained from the tank at a rate of 7 gal/min.
(a) Write an initial value problem for the mass of the salt.
(b) Solve the initial value problem.

A)

B)

C)

D)

Question 73

Solve the problem.

-A 2000-L tank is initially filled with a copper sulfate solution with a concentration of 30 g/L. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 6 L/min. The thoroughly mixed solution is drained from the tank at a rate of 6 L/min.
(a) Write an initial value problem for the mass of the copper sulfate.
(b) Solve the initial value problem.

A)

B)

C)

D)

Question 74

Solve the problem.

-Consider the pair of differential equations that model a predator-prey system with populations x and y. (t) = 3x - 6xy
(t) = -y + 7xy
(a) Find the lines along which (t) = 0,
(b) find the lines along which (t) = 0,
(c) find the equilibrium points for the system.

A)

B)
C)

D)

Question 75

Solve the problem.

-Consider the pair of differential equations that model a predator-prey system with populations x and y. (t) = 5x - xy
(t) = -3y + xy
(a) Find the lines along which (t) = 0,
(b) find the lines along which (t) = 0,
(c) find the equilibrium points for the system.

A)
B)
C)
D)

Question 76

Solve the problem.

-According to a country's census, the population (to the nearest million) was 266 in Year 0 and 306 in Year 10. The projected population for Year 50 is 445. To construct a logistic model, both the growth and carrying capacity must be estimated.
(a) Estimate r by assuming that t = 0 corresponds to Year 0 and that the population between Year 0 and Year 10 is exponential; that is, the population is given by Round the value of r to four decimal places, if necessary.
(b) Write the solution to the logistic equation using the estimated value of r and use the projected value P(50) = 445 million to find an estimation for the value of the carrying capacity K. Round to the nearest million.

A)

B)

C)

D)