Deck 8: First-Order Differential Equations

Question 1

An object's cooling constant, k, is often taken to be proportional to the its surface area. An iron ball cools from 200^oC to 100^oC in 50 seconds in a fast flowing stream of 30^oC water. Use Newton's Law of Cooling to estimate how cool a similar volume of hot iron would become in 50 seconds if it were in the shape of a rod with surface area 4 times larger than the ball's.

A) 32.0^oC
B) 145.3^oC
C) 30.0^oC
D) 34.9^oC

Answer 1

D

Question 2

Find the solution of the differential equation, y' = -y, satisfying the initial condition, y(0) = -5.

A)
B)
C)
D)

Answer 2

B

Question 3

The number of stores in a particular chain of coffee bars was 100 in 1996 and began growing exponentially with a growth constant of 0.35 year^-1. In what year would one predict the number of stores to reach 10,000?

A) 1999
B) 2004
C) 2009
D) 2014

Answer 3

C

Question 4

You select an insurance policy for your company that agrees to pay you the depreciated value of a piece of equipment if it is destroyed in an accident, but you have to choose one of two depreciation schedules when you begin the policy. One depreciation schedule is linear and depreciates the value to zero over 10 years. The other depreciation schedule is exponential and depreciates the value at a constant rate of 14% per year. An accident destroys the equipment after 6 years of ownership. Calculate the insurance company's obligation for each plan if the piece of equipment was valued initially at $100,000. Which depreciation schedule would have paid the most? (The cost of the policy is the same, no matter which depreciation schedule you choose.)

Question 5

The rate at which water flows out of a drain in the bottom of a certain tank is proportional to the height of water in the tank. The tank is a vertical cylinder with cross-sectional area of 1.0 m², so that every 1 cm in height represents 10 L. If the flow is 10 L/min (i.e. 1 cm/min) when the water level is 700 cm, how long will it take for the level to go from 700 cm to 30 cm?

A) 4586 minutes
B) 670 minutes
C) 9 minutes
D) 2205 minutes

Question 6

The conversion of sucrose (table sugar) to glucose and fructose is first order in the concentration of sucrose, which means that the rate of reaction is proportional to the concentration of the sucrose. The rate of disappearance of sucrose can be expressed as , where c represents the concentration of the sucrose, and k is called the rate constant and is mathematically identical to the negative of the decay constant. If it takes 4 hours for the sucrose concentration to drop by a factor of 8, what is the rate constant?

A) 0.69 hour^-1
B) 0.50 hour^-1
C) 0.52 hour^-1
D) 2.00 hour^-1

Question 7

The Polymerase Chain Reaction (PCR) is used to replicate segments of DNA. It is used to make DNA samples big enough for testing, starting from very small samples collected, for instance, from a crime scene. PCR can double the number of a particular DNA segment every two minutes. Write an equation for the number of segments as a function of the number of minutes, t, if there is initially just one segment.

A)
B)
C)
D)

Question 8

A treasury note that will be worth $100,000 in 25 years currently sells for $48,432. What constant interest rate does that correspond to?

A) 2.90%
B) 4.26%
C) 0.03%
D) 5.33%

Question 9

The Polymerase Chain Reaction (PCR) is used to replicate segments of DNA. It is used to make DNA samples big enough for testing, starting from very small samples collected, for instance, from a crime scene. PCR can double the number of a particular DNA segment every two minutes. If one wants a DNA sample with copies of a particular segment, how long must the PCR process be carried out to produce them? Assume that there is just one segment in the original sample.

A) 106 minutes
B) 53 minutes
C) 18 minutes
D) minutes

Question 10

The population of New Zealand grew exponentially through the 20th century at a rate of 1.6% year^-1. If the population in 2000 was 3.9 million, when was the population 1.0 million?

A) 1915
B) 1920
C) 1925
D) 1930

Question 11

Find the solution of the differential equation, y' = -2y, satisfying the initial condition, y(3) = 0.

A)
B)
C)
D)

Question 12

The conversion of sucrose (table sugar) to glucose and fructose is first order in the concentration of sucrose, which means that the rate of reaction is proportional to the concentration of the sucrose. The rate of disappearance of sucrose can be expressed as , where c represents the concentration of the sucrose, and k is called the rate constant and is mathematically identical to the negative of the decay constant. If the rate constant is 0.20 hour^-1, how long will it take for the sucrose concentration to diminish to 1/20 of its initial concentration?

A) 14.98 hours
B) 0.080 hours
C) 100.00 hours
D) 32.19 hours

Question 13

Find the solution of the differential equation, y' = -y, satisfying the condition, y(2) = -4.

A)
B)
C)
D)

Question 14

Find the solution of the differential equation, y' = -y, satisfying the initial condition, y(4) = 3.

A)
B)
C)
D)

Question 15

In 1965, Fairchild Semiconductor's R&D Director, Gordon Moore, noted that the maximum number of transistors that could be included cost effectively in one integrated circuit had been doubling yearly since 1959. While growth has not increased quite as quickly since 1965, it has still been exponential, and the continued exponential growth has become known as "Moore's Law". Considering that Intel's 8086 processor, introduced in 1978, had 29,000 transistors, and their Itanium processor, introduced in 2002, had 220,000,000 transistors, on average how long has it taken for processors to double the number of transistors they use.

A) 1.9 years
B) 0.8 years
C) 16.0 years
D) 0.4 years

Question 16

The number of bacteria in a culture increases exponentially with a growth constant of 0.2 hour^-1. How long will it take for the population to increase from 4000 to 20,000 ?

A) 25.00 hours
B) 0.12 hour
C) 8.05 hours
D) 0.04 hour

Question 17

At the cafeteria, two identical glasses of juice were poured at the same time and put on the counter waiting for customers to take them. Their temperatures were 33^oF when they were poured, and the cafeteria was a stable 72^oF. One glass was 40^oF when it was taken after 40 seconds. What was the temperature of the other when it was taken after 200 seconds?

A) 57.5^oF
B) 59.2^oF
C) 64.7^oF
D) 54.3^oF

Question 18

Suppose the income tax structure is as follows: the first $28,000 is taxed at 20%, the remainder is taxed at 30%. Compute the tax on an income of $40,000. Now suppose that inflation is 5% and you receive a cost of living (5%) raise to $42,000. Compute the tax on this income. To compare the taxes you should adjust the tax for inflation (add 5%).

Question 19

$50,000 that was invested in 1990 was worth $134,100 in 2000. What annual interest rate did the investment earn in that 10 year period? Assume continuous compounding.

A) 118.06%
B) 10.91%
C) 108.20%
D) 9.87%

Question 20

Find the solution of the differential equation, y' = 5y, satisfying the initial condition, y(0) = -4.

A)
B)
C)
D)

Question 21

The differential equation is separable. Find the general solution in an explicit form.

A)
B)
C)
D)

Question 22

Calculate how much you would need to invest now in order to fund a year of college twenty years from now, assuming a year of college costs $26,000 now and is inflating at 6%, and your investment will earn 10%. Assume continuous compounding.

A) $11,683
B) $3519
C) $7831
D) $57,864

Question 23

Find the solution to the following separable differential equation. , a is a constant.

A)
B)
C)
D)

Question 24

The text describes logistic growth with an equation for the actual population in terms of a growth constant and a maximum population (carrying capacity), . The equation could also be written for a fraction of the maximum population in terms of a fractional growth constant. Re-express the differential equation in terms of the fractional population, . Compare the time it takes for the population to go from 60% of the maximum to 80% of the maximum with the time it takes to go from 80% of the maximum to 90% of the maximum,
if k = 0.00100 day^-1 and M = 4000?

Question 25

Solve the following initial value problem explicitly.

A)
B)
C)
D)

Question 26

Is the following differential equation separable or not?

A) separable
B) not separable

Question 27

Is the following differential equation separable or not?

A) separable
B) not separable

Question 28

In 1995 an investor put $2000 in an account which paid 8%. In 2005 she withdrew $1000 from the account. What will the account be worth in 2020 ?

A) $13,778.11
B) $11,458.00
C) $7389.06
D) $3451.08

Question 29

Is the following differential equation separable or not?

A) separable
B) not separable

Question 30

Find the solution to the following separable differential equation.

A)
B)
C)
D)

Question 31

A stirred tank with volume, V, has a feed stream of concentrated floor cleaner flowing into it at a rate of f . The flow stream has a concentration of c_i. The outlet stream also flows at f but with a concentration of c, the concentration of the floor cleaner solution in the tank. The rate of change of the concentration is proportional to the difference between c_i and c with proportionality constant of f/V: . If the tank volume is 2000 L, the flow rate is 6 L/min, the inlet concentration is 0.9, and the initial concentration of the floor cleaner in the tank is 0.0, how long until the concentration in the tank is 0.7?

A) 9.0 minutes
B) 501.4 minutes
C) 333.3 minutes
D) 83.8 minutes

Question 32

The differential equation is separable. Find the general solution in an explicit form.

A)
B)
C)
D)

Question 33

Find the solution to the following separable differential equation. , a is a constant

A)
B)
C)
D)

Question 34

Solve the following initial value problem explicitly.

A)
B)
C)
D)

Question 35

A bank offers to sell a bank note that will reach a maturity value of $12,000 in 12 years. How much should you pay for it now if you wish to receive an 8% return on your investment?

A) $925.93
B) $4594.71
C) $11,040.00
D) $920.00

Question 36

Solve the following initial value problem explicitly. y(1) = -3

A)
B)
C)
D)

Question 37

Is the following differential equation separable or not?

A) separable
B) not separable

Question 38

A particular investment program involves continuously making small investments adding up to $3600 each year. If the investment pays 12%, and the account started off with an initial balance of $500, how much will the account be worth in 40 years?

A) $3,676,068
B) $3,615,313
C) $60,755
D) $3,706,068

Question 39

Find the solution to the following separable differential equation.

A)
B)
C)
D)

Question 40

Solve the following initial value problem explicitly. .

A)
B)
C)
D)

Question 41

Match the appropriate slope field with the differential equation .

A)
B)
C)
D)

Question 42

Use Euler's method with h = 0.1 to approximate y(1.0) and y(2.0) for the differential equation , .

A) -0.03685, 0.53979
B) -0.06263, 0.51232
C) -0.38500, -1.87000
D) 0.07900, 0.55182

Question 43

The University of XYZ has a goal to increase its endowment from the initial value of $100,000,000, to $150,000,000 over 4 years. If the interest rate earned by the endowment (after expenses) is 5% each year (compounded continuously), and the contributions become available continuously and at a constant rate, how much will they actually have to collect from contributors over those 4 years to meet their goal?

A) $6,291,639
B) $40,535,078
C) $1,061,963,001
D) $25,166,556

Question 44

Match the appropriate slope field with the differential equation .

A)
B)
C)
D)

Question 45

Construct a direction field for the differential equation.

Question 46

Consider a chemical system containing species A, B, and C; and that A and B can react to make C in a bimolecular reaction with rate constant of k₁, and C can decompose to make A and B in a first order reaction with rate constant of k_-1. If the instantaneous amounts of A, B, and C are represented as a, b, and c, and the initial amounts are given as A₀, B₀, and C₀, the change in C can be represented with the differential equation . If A₀ = 2 , B₀ = 2, C₀ = 0, k₁ = 0.04, and k_-1 = 0.04, solve the differential equation for c and graph the solution. [Note: c can never be larger than C₀ plus the smaller of A₀ or B₀. Nor can it be smaller than 0.]

Question 47

Identify the equilibrium solutions for , and determine if they are stable or unstable.

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

Question 48

Use your calculator to construct the direction field for the following differential equation. [Use the standard zoom window.]

Question 49

Consider a chemical system containing species A, B, and C; and that A and B can react to make C in a bimolecular reaction with rate constant of k₁, and C can decompose to make A and B in a first order reaction with rate constant of k_-1. If the instantaneous amounts of A, B, and C are represented as a, b, and c, and the initial amounts are given as A₀, B₀, and C₀, the change in C can be represented with the differential equation . If A₀ = 0 , B₀ = 0, C₀ = 4, k₁ = 0.02 s ^-1, and k_-1 = 0.04 s ^-1, how much C is present after 15 seconds? [Note: c cannot be larger than C₀ plus the smaller of A₀ or B₀. Nor can it be smaller than 0.]

A) 6.51
B) 2.46
C) 3.84
D) 1.74

Question 50

An object falling freely through the atmosphere will accelerate due to the force of gravity at 9.86 m/s² [Acceleration is the derivative of velocity with respect to time.] The atmosphere, however, will commonly exert a retarding force proportional to the object's velocity squared. The proportionality constant will depend largely on the object's shape. Write and solve the differential equation, then identify a free-falling object's terminal velocity (the limiting velocity) in terms of its drag proportionality constant, k. [To solve explicitly, you may use the initial condition v(0) = 0.]

Question 51

Use Euler's method with h = 0.1 to approximate y(1.0) and y(2.0) for the differential equation , .

A) -2.8343, -4.5215
B) -1.8345, -0.8317
C) -2.8969, -4.4726
D) -2.7343, -4.4215

Question 52

Identify the equilibrium solutions for , for , and determine if they are stable or unstable.

A)
B)
C)
D)

Question 53

A volume discount on a certain item is expressed as a differential equation in which the derivative of the price per item with respect to the number purchased is proportional to the difference between the price and some base price, below which the price can never go. If the price for just one item is $60, and the base price is $30, and the price per item when buying 10 items is $54, what is the price per item when buying 100 items?

A) $33
B) $48
C) $57
D) $46

Question 54

Construct a direction field for the differential equation.

Question 55

Use Euler's method with h = 0.1 to approximate y(1.0) and y(2.0) for the differential equation , .

A) 1.4996, 0.5508
B) 2.7573, 3.0562
C) 2.7364, 3.0928
D) 2.8488, 3.1430

Question 56

Use the direction field below to sketch a solution curve and estimate the initial value y(0) for the differential equation , such that the solution curve passes through the point (-1,-2).

Question 57

Identify the equilibrium solutions for , and determine if they are stable or unstable.

A) y = (unstable); y = 0 (stable)
B) y = (stable); y = 0 (unstable)
C) y = 0 (stable)
D) y = 0 (unstable)

Question 58

Match the appropriate slope field with the differential equation .

A)
B)
C)
D)

Question 59

Use your calculator to construct the direction field for the following differential equation. [Use the standard zoom window.]

Question 60

Use your calculator to construct the direction field for the following differential equation. [Use the standard zoom window.]

Question 61

Write the following second-order equation as a system of first-order equations.

Question 62

Find all equilibrium points for the following coupled predator-prey-model equations.

A) (0, 0), (2.000, 0), (0.500, 0.375)
B) (0, 0), (0.500, 0), (2.000, 0.500)
C) (0, 0), (0.500, 2.000), (0, 0.375)
D) (0, 0), (2.000, 0), (0.500, 0.250)

Question 63

Write the following second-order equation as a system of first-order equations.

Question 64

Find all equilibrium points for the following system of equations.

A)
B)
C)
D)

Question 65

Use the following direction field to identify the stability of the equilibrium point (0.50, 0.84).

A) Stable
B) Unstable

Question 66

Find all equilibrium points.

A) where and are any real numbers
B)
C)
D)

Question 67

Find and interpret all equilibrium points for the competing species model.

Question 68

Find all equilibrium points for the following coupled equations. Identify each equilibrium point as stable or unstable.

A) (0, 0)unstable; (1.33, 1.50)stable; (0.50, 0)unstable; (0, 1.25)unstable
B) (0, 0)unstable; (0, 0.33)unstable; (0.50, 0)unstable; (0.50, 0.33)stable
C) (0, 0)unstable; (0, 0.33)unstable; (0.50, 0)unstable; (1.33, 1.50)stable
D) (0, 0)unstable; (0, 1.50)unstable; (1.33, 0)unstable; (0.50, 1.25)stable

Question 69

Find all equilibrium points for the following system of equations.

A)
B)
C)
D)

Question 70

Use the following direction field to identify the stability of the equilibrium point (0.57, 0.14).

A) Stable
B) Unstable

Question 71

Use the direction field below to sketch a solution curve and estimate the initial value y(0) for the differential equation , such that the solution curve passes through the point (2, 2).

Question 72

Write the following third-order equation as a system of equations.