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Question : Solve the initial value problem. 21) y' + ty = 2t; y(0) = 5 : 2163509

Solve the initial value problem.

21) y' + ty = 2t; y(0) = 5

A) y = 2e^{-t^2/2} + 3

B) y = 3e^{t^2/2} + 2

C) y = 2e^{t^2/2} + 3

D) y = 3e^{-t^2/2} + 2

22) 2y'- 4ty = 8t; y(0) = 11

A) y = -2 + 13e^{-t^2}

B) y = -2 + 13e^{t^2}

C) y = -1 + 12e^{t^2}

D) y = 2 + 11e^{t^2}

23) ty' + (1 + t)y = 2; y(4) = 2

A) y = (2 + 6e^{4 - t}/t)

B) y = (2 + 6e^{-4 - t}/t)

C) y = (2 + 8e^{6 - t}/t)

D) y = (2 + 8 e^{4 - t}/t)

24) y' + 10ty - e^{-5t^2} = 0; y(0) = 2

A) y = te^{-5t} + 2

B) y = (t + 2)e^{-5t^2}

C) y = (t + 2)e^{-5t}

D) y = te^{-5t^2} + 2

Solve the problem.

25) An initial deposit of $24,000 is made into an account that earns 5% compounded continuously. Money is then withdrawn at a constant rate of $4000 a year until the amount in the account is 0. Find the equation for the amount in the account at any time t. When is the amount 0?

A) A = 60,000 - 36,000e^{0.05t}

10.017 years

B) A = 80,000 - 56,000e^{0.05t}

8.352 years

C) A = 60,000 - 36,000e^{0.05t}

8.352 years

D) A = 80,000 - 56,000e^{0.05t}

7.134 years

26) An initial deposit of $8,000 is made into an account earning 6.5% compounded continuously. Thereafter, money is deposited into the account at a constant rate of $2600 per year. Find the amount in this account at any time t. How much is in this account after 5 years?

A) A = 52,000e^{0.065t} - 44,000

$27,969.59

B) A = 44,000e^{0.065t} - 36,000

$24,897.34

C) A = 48,000e^{0.065t} - 40,000

$26,433.47

D) A = 60,000e^{0.065t} - 52,000

$31,041.84

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

27) Suppose that $1000 is deposited in a savings account that pays 6% annual interest compounded continuously. At what rate (in dollars per year) is it earning interest after 5 years? Enter just an integer representing the amount to the nearest dollar (no units).

28) How much would you need to invest per month - in effect, continuously - in an investment account that pays an annual interest rate of 9%, compounded continuously, in order for the account to be worth $100,000 after 20 years? Enter just an integer representing dollars to the nearest dollar (no units)

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

29) Richard deposits $2000 in an IRA at 10% interest compounded continuously for his retirement in 25 years. He intends to make continuous deposits at the rate of $2500 a year. How much will he have accumulated in 20 years? Round your answer to the nearest dollar.

A) $180,505

B) $174,505

C) $177,505

D) $170,505

30) A tank contains 2000 L of a solution consisting of 50 kg of salt dissolved in water. Pure water is pumped into the tank at the rate of 10L/s, and the mixture (kept uniform by stirring) is pumped out at the same rate. How long will it be until only 5 kg of salt remain in the tank?

A) approximately 460 seconds

B) approximately 689 seconds

C) approximately 276 seconds

D) approximately 703 seconds

31) A nutritionist proposes the following model for weight loss on a program she is developing:

(dw/dt) + 0.006w = (1/3600)C

where w(t) is a person's weight (in pounds) after t days of consuming exactly C calories per day. A person weighing 180 pounds goes on this diet program consuming 2400 calories per day. Use the above model to predict how long will it take this person to lose 15 pounds.

A) 35 days

B) 39 days

C) 37 days

D) 41 days

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

32) There is a differential equation that is a mathematical model of the situation in which the time rate of change in the population of a certain organism is proportional to the product of the current population and the difference between the current population and the limiting factor of 100,000. Is this the equation (dP/dt) = kP(100,000 - P)? Enter "yes" or "no".

33) A fly population increases at a rate proportional to the amount present. After two years the population has doubled. After three years it is 20,000. Find the number of flies initially present. Enter just an integer.

34) A jug of milk at 50° is placed outdoors at a temperature of 100°. If after 5 minutes the temperature of the milk is 60°, write the equation giving the temperature of the milk as a function of time. Enter your answer exactly as: T = ae^{b} + c

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

35) Suppose an isolated island has a native population of 8000 and a person from a visiting ship introduces a disease which has an infection rate of 0.00005. Assume that the rate of spread of the disease satisfies the following logistic equation:

(dy/dt) = k(1 - (y/N))y,

where N is the size of the population and y is the number infected at time t.

Write an equation for the number of infected natives after t days.

A) y = (8000/1 + 7999e^{-0.00005t})

B) y = (7999/1 + 8000e^{-0.00005t})

C) y = (7999/1 + 8000e^{-0.4t})

D) y = (8000/1 + 7999e^{-0.4t})

36) Suppose an isolated island has a native population of 10,000 and a person from a visiting ship introduces a disease which has an infection rate of 0.00005. Assume that the rate of spread of the disease satisfies the following logistic equation:

(dy/dt) = k(1 - (y/N))y,

where N is the size of the population and y is the number infected at time t.

How many individuals are infected after 15 days?

A) 1481

B) 1531

C) 1409

D) 1331

37) Suppose the graph below gives a solution to the differential equation (dP/ds) = g(P) where P is the price of a product and s is the weekly sales.

Which of the following statements is/are true?

(I) g(M) = 0

(II) g'(M) =0

(III) g((M/2)) = 0

(IV) g(P_{0}) > 0

A) I, III, and IV

B) IV only

C) I and IV

D) I only

E) I, II, and IV

38) Suppose the following is a graph of z = g(y).

Which of the following can then be said about the solution y = f(t) to the initial value problem y' = g(y); y(0) = -1?

(I) f(t) is an increasing function

(II) f(t) is always positive

(III) f(t) has an inflection point when y = 2.

A) II only

B) I and III

C) III only

D) I only

E) I, II, and III

39) Suppose y' = ky + b and the graphs of several solutions of the differential equation are as below:

Then

(I) k is negative.

(II) k is positive.

(III) b is positive.

(IV) b is negative.

A) II and III

B) I and IV

C) I and III

D) II and IV

E) not enough information given