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Question : Consider the differential equation y' = g(y) where g(y) is the function whose graph is shown : 2163516

Consider the differential equation y' = g(y) where g(y) is the function whose graph is shown below:

Indicate whether the following statements are true or false.

40) y = -3, y = 1, and y = 5 are the constant solutions to y' = g(y).

A) True

B) False

41) y = 2 is the only constant solution of y' = g(y) .

A) True

B) False

42) If the initial value of y(0) is greater than 6, then the corresponding solution will be an increasing function.

A) True

B) False

43) If the initial value of y(0) is 3, then the corresponding solution has an inflection point.

A) True

B) False

44) If the initial value of y(0) is 2, then the corresponding solution has an inflection point.

A) True

B) False

45) For what y value(s) does a solution of y' = y^{2} - 3y + 2 have inflection points?

A) y = 2

B) y = 0

C) y = (3/2)

D) y = 2 and y = 1

E) none of these

46) Let y' = 2 - y. Which of the following properties hold for the solution y = f(t) determined by the initial condition y(0) = 1?

(I) It is always concave down.

(II) It is a constant solution.

(III) It is always decreasing.

A) II only

B) I only

C) I and III

D) III only

E) none of these

47) Let y' = y^{3}. Which of the following properties hold for the solution y = f(t) determined by the initial condition y(0) = -2?

(I) It is always increasing.

(II) It has an inflection point.

(III) It is always concave down.

A) I and II

B) II only

C) I only

D) III only

E) none of these

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

48) One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. Do these graphs represent: y' = 6 - 3y; y(0) = -1; y(0) = 3?

Enter just "yes" or "no".

49) One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. y' = 6 + 2y; y(0) = -4; y(0) = -2

Do these graphs represent the situation?

Enter just "yes" or "no".

50) One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph.. y' = y^{2} - 9; y(0) = -5; y(0) = 2

Do these graphs represent the situation?

Enter just "yes" or "no".

51) One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. y' = y^{2} - 2y - 8; y(0) = -3; y(0) = 3

Do these graphs represent the situation?

Enter just "yes" or "no".

52) One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. y' = y^{3} - 3y^{2}; y(0) = -1.5; y(0) = 2.5

Do these graphs represent the situation?

Enter just "yes" or "no".

53) One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. y' = cosy; y(0) = - (π/4); y(0) = (5π/4)

Do these graphs represent the situation?

Enter just "yes" or "no".

54) One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. y' = y^{2} - 4; y(0) = -3; y(0) = -1; y(0) =1; y(0) = 3

Do these graphs represent the situation?

Enter just "yes" or "no".

55) Given y' = y - 3. On a ty-coordinate system sketch the solutions corresponding to the initial conditions y(0) = 0 and y(0) = 4. Does this graph represent the situation?

Enter "yes" or "no".

56) Given y' = y(y + 3). On a ty-coordinate system sketch the solutions corresponding to the initial conditions y(0) = -4, y(0) = -1, and y(0) = 1. Does this graph represent the situation?

Enter "yes" or "no".

57) Given y' = e^{y} - 1. On a ty-coordinate system sketch the solutions corresponding to the initial conditions y(0) = -1 and y(0) = 1. Does this graph represent the situation?

Enter "yes" or "no".

58) Below is a sketch of f(x) = (x - 1)e^{x} .

On a ty-coordinate system, sketch the solutions to the differential equation y' = (y - 1)e^{y} corresponding to the initial conditions y(0) = 2, y(0) = (1/2), and y(0) = - (1/2). Does the following graph represent this situation?

Enter "yes" or "no".

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

59) Which of the following is a sketch of the solution of y' = y^{2} - 9; y(0) = 2 ?

A)

B)

C)

D)

60) Consider the differential equation y' = y^{2} - 4y + 3. Which of the following could be a graph of solutions to this differential equation?

A)

B)

C)

D)

61) The following could be graphs of solutions to which of the following differential equations?

A) y' = y(y + 2)

B) y' = 3y(y - 2)

C) y' = y^{2} + 2

D) y' = (y - 2)e^{y}

E) none of these

Solve the problem.

62) A skydiver's terminal velocity is 46 meters per second. That is, no matter how long the skydiver falls, his or her speed will not exceed 46 meters per second but will get arbitrarily close to that value. The velocity in meters per second, v(t), after t seconds satisfies the differential equation v'(t) = (49/5) - kv(t). What is the value of k?

A) (230/49)

B) (49/230)

C) (49/115)

D) (98/5)

63) A large boulder is placed in a river to help divert the water. Suppose the rate at which the boulder erodes is proportional to the product of its current size and the difference between its original size, B, and 10 times its current size. Give a differential equation that is satisfied by f(t), the height at time t.

A) y' = ky(10y - B)

B) y' = kBy(-10y)

C) y' = ky(B - 10y)

D) y' = 10ky(B - y)