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Solve the differential equation with the given initial condition. 23) y' = y + 1, y(0) = 1

Question : Solve the differential equation with the given initial condition. 23) y' = y + 1, y(0) = 1 : 2163505

Solve the differential equation with the given initial condition.

23) y' = y + 1, y(0) = 1

A) y = 2et -1

B) y = t2 + 1

C) y = t + 1

D) y = et - 2

24) y' = ty, y(0) = -1

A) y = -et^2/2

B) y = -1 + et^2/2

C) y = (t2/2)

D) y = et + 1

25) y' = sintcos3t, y((π/3)) = 0

A) y = 16 + (cos4t/4)

B) y = (1/64) - (cos4t/4)

C) y = (cos4t/4)

D) y = 64 - cos4t

26) y' = 3t2(4 - y)2, y(0) = 2

A) y = 4 - (1/t3 + (1/2))

B) y = 4 + t3

C) y = t3 + (1/2)

D) y = (1/t3 + 2)

27) y' = -e-y, y(0) = 0

A) y = e-1

B) y = -ln|t + 1|

C) y = 0

D) y = et

E) none of these

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

28) Given the differential equation with the given initial condition: y' = (3t2 + 1/2y) ; y(1) = -5

is this the solution y = - √(t3 + t +2) ?

Enter "yes" or "no".

29) Given the differential equation with the given initial condition: (dy/dt) = y2lnt; y(1) = (1/3)

is this the solution y = (1/2 - tlnt + t) ?

Enter "yes" or "no".

30) Given the differential equation with the given initial condition: y' = √((t + 1/y)) ; y(0) = 4

is this the solution y = ((t + 1)3/2 + 4)2/3 ?

Enter "yes" or "no".

31) Given the differential equation with the given initial condition: y' = tcost; y(0) = 0

is this the solution y = tsint + cost - 1 ?

Enter "yes" or "no".

32) Given the differential equation with the given initial condition: yy' = tet^2; y(0) = 1

is this the solution y = et^2/2 ?

Enter "yes" or "no".

33) Given the differential equation with the given initial condition: y' = e-y ; y(0)= 0

is this the solution y = ln|t + 1| ?

Enter "yes" or "no".

34) Given the differential equation with the given initial condition: (dy/dt) = 3t2 + sint; y(0) = 2

is this the solution y = t3 - cost + 1 ?

Enter "yes" or "no".

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

Solve the problem.

35) Let t represent the number of hours that a packing machine is operated and y(t) represent the probability that the machine breaks down at least once during the t hours of operation. It has been observed that the rate of increase of the probability of a breakdown is proportional to the probability of not having a breakdown. Find a differential equation describing this situation.

A) y' = k(1 - y); y(0) = 0

B) y' = k(1 - y); There is not enough information given to determine initial conditions.

C) y' = (1 + y); y(0) =0

D) y' = ky; y(0) = 0

E) y' = ky; There is not enough information given to determine initial conditions.

36) Suppose water is seeping from an underground storage facility at a rate that is proportional to the square amount of water present. If f(t) = y is the amount of water present at time t, find a differential equation describing the situation.

A) y' = ky2, k > 0; y(0) = 0

B) y' = ky2, k < 0

C) y' = ky2, k > 0

D) y' = ky2, k < 0; y(0) =0

E) none of these

37) Suppose the relationship between the price p, of a product and the weekly sales, s, of the product is given by the differential equation (dp/ds) = - (1/10)(s + 3). Then

A) as the price increases the rate of change of the price also increases.

B) as sales increase, the price increases.

C) s = 0 is a constant solution to this differential equation.

D) the rate of the decrease of the price is proportional to the sales.

E) all of these

38) The annual sales y (in millions of dollars) of a company satisfy the differential equation (dy/dt) = 0.2y; y(0) = 2. Which of the following is a verbal description of the rate of change of annual sales ?

A) The annual sales are decreasing at a rate proportional to the annual sales.

B) The annual sales are increasing at a rate proportional to $0.2 million ($200,000) per year.

C) The annual sales are increasing at $0.2 million ($200,000) per year.

D) The annual sales are increasing at a rate proportional to the annual sales.

39) The annual sales y (in millions of dollars) of a company satisfy the differential equation (dy/dt) = 0.2(10 - y); y(0) = 2. Which of the following is a verbal description of the rate of change of annual sales.

A) The annual sales are increasing at a rate proportional to the difference between the annual sales and an upper limit of $10 million.

B) The annual sales are decreasing at a rate proportional to the annual sales.

C) The annual sales are increasing at $0.2 million ($200,000) per year to an upper limit of $5 million.

D) The annual sales are increasing at a rate proportional to $0.2 million ($200,000) per year.

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

40) A cool object is to be heated to a maximum temperature M = M°C. At any time t, the rate at which the temperature rises is proportional to the difference between the actual temperature and the maximal temperature. If the object is originally 0°C, find and solve a differential equation describing this situation. Is this the solution: y(t) = M - Me-kt? Enter "yes" or "no".

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

41) When a dead body is discovered, one of the first steps in the ensuing investigation is for a medical examiner to determine the time of death as closely as possible. If the temperature of the medium has been fairly constant and less than 48 hours have passed since death, Newton's law of cooling can be used. Newton's law of cooling states, (dT/dt) = -k(T - TM), where k is a constant, T is the temperature of the object after t hours, and TM is the (constant) temperature of the surrounding medium. Assuming the temperature of a body at death is 98.6°F, the temperature of the surrounding air is 70°F, and at the end of one hour the body temperature is 89°F, what is the temperature of the body after 4 hours? Round to the nearest tenth of a degree.

A) 89°F

B) 5.6°F

C) 70.5°F

D) 75.6°F

42) When a dead body is discovered, one of the first steps in the ensuing investigation is for a medical examiner to determine the time of death as closely as possible. If the temperature of the medium has been fairly constant and less than 48 hours have passed since death, Newton's law of cooling can be used. Newton's law of cooling states, (dT/dt) = -k(T - TM), where k is a constant, T is the temperature of the object after t hours, and TM is the (constant) temperature of the surrounding medium. Assuming the temperature of a body at death is 98.6°F, the temperature of the surrounding air is 69°F, and at the end of one hour the body temperature is 90°F, when will the temperature of the body be 73°F? Round to the nearest tenth of an hour.

A) 0.2 hr

B) 2.8 hr

C) 0.4 hr

D) 5.8 hr

43) Earth's atmospheric pressure p is often modeled by assuming that the rate dp/dh at which p changes with the altitude h above sea level is proportional to p. Suppose that the pressure at sea level is 1013 millibars and that the pressure at an altitude of 9 km is 341 millibars.

Solve the initial value problem

Differential equation: (dp/dh) = kp,

Initial condition: p = p0 when h = 0

to express p in terms of h. Determine the values of p0 and k from the given altitude-pressure data.

A) p = 1013e-0.046h

B) p = 1013e-0.094h

C) p = 1013e-0.128h

D) p = 1013e-0.121h

44) Earth's atmospheric pressure p is often modeled by assuming that the rate dp/dh at which p changes with the altitude h above sea level is proportional to p. Suppose that the pressure at sea level is 1013 millibars and that the pressure at an altitude of 12 km is 237 millibars.

What is the atmospheric pressure at an altitude of 18 km? Round to the nearest millibar.

(You will first need to solve the initial value problem

Differential equation: (dp/dh) = kp,

Initial condition: p = p0 when h = 0

and determine the values of p0 and k from the given altitude-pressure data).

A) ≈ 111 millibars

B) ≈ 121 millibars

C) ≈ 115 millibars

D) ≈ 118 millibars

Solution
5 (1 Ratings )

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