Deck 7: Differential Equations

Question 1

Suppose that we model populations of predators and preys (in millions) with the system of differential equations: $\begin{array} { l } \frac { d x } { d t } = 2 x - 1.2 x y \\\frac { d y } { d t } = - y + 0.9 x y\end{array}$ Find the equilibrium solution.

A) $x = \frac { 10 } { 9 } , y = \frac { 3 } { 5 }$
B) $x = \frac { 9 } { 10 } , y = \frac { 3 } { 5 }$
C) $x = \frac { 5 } { 9 } , y = \frac { 3 } { 10 }$
D) $x = \frac { 5 } { 3 } , y = \frac { 10 } { 9 }$
E) $x = \frac { 10 } { 9 } , y = \frac { 5 } { 3 }$
F) $x = \frac { 10 } { 3 } , y = \frac { 9 } { 5 }$
G) $x = \frac { 3 } { 9 } , y = \frac { 3 } { 5 }$
H) $x = \frac { 3 } { 5 } , y = \frac { 10 } { 9 }$

Answer 1

$x = \frac { 10 } { 9 } , y = \frac { 5 } { 3 }$

Question 2

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.0001 P ( 1000 - P )$ with P(0) = 10. Find the formula for the population after t years.

A) $P ( t ) = \frac { 1000 } { 1 + 99 e ^ { - 0.01 t } }$
B) $P ( t ) = \frac { 1000 } { 1 + 10 e ^ { - 0.01 t } }$
C) $P ( t ) = \frac { 1000 } { 1 + e ^ { - 0.01 t } }$
D) $P ( t ) = \frac { 100 } { 1 + 9 e ^ { - 0.01 t } }$
E) $P ( t ) = \frac { 1000 } { 1 + 9 e ^ { - 0.01 t } }$
F) $P ( t ) = \frac { 100 } { 1 - 99 e ^ { - 0.01 t } }$
G) $P ( t ) = \frac { 1000 } { 1 + 99 e ^ { - 0.1 t } }$
H) $P ( t ) = 1000 e ^ { 0.1 t }$

Answer 2

$P ( t ) = \frac { 1000 } { 1 + 99 e ^ { - 0.1 t } }$

Question 3

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.0001 P ( 1000 - P )$ with P(0) = 10. Find the population after 50 years.

A)50
B)500
C)600
D)700
E)80
F)1000
G)350
H)300

Answer 3

600

Question 4

Consider the predator-prey system , where x and y are in millions of creatures and t represents time in years.(a) Find equilibrium solutions for this system.(b) Explain why it is reasonable to approximate this predator-prey system as , if the initial conditions are x(0) = 0.001 and y(0) = 0.002.(c) Describe what this approximate system tells about the rate of change of each of the specie populations x(t) and y(t) near (0, 0).(d) Find the solution for the approximate system given in part (b).(e) Sketch x (t) and y (t) as determined in part (d) on the same coordinate plane.(f) Sketch a phase trajectory through (0.001; 0.002) for the predator-prey system. Describe in words what happens to each population of species and the interaction between them.

Question 5

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.01 P - 0.0001 P ^ { 2 }$ with P(0) = 10. Find the formula for the population after t years.

A) $P ( t ) = \frac { 100 } { 1 + 9 e ^ { - 0.01 t } }$
B) $P ( t ) = \frac { 100 } { 1 + 10 e ^ { - 0.01 t } }$
C) $P ( t ) = \frac { 100 } { 1 + e ^ { - 0.01 t } }$
D) $P ( t ) = \frac { 10 } { 1 + 9 e ^ { - 0.01 t } }$
E) $P ( t ) = \frac { 10 } { 1 + e ^ { - 0.01 t } }$
F) $P ( t ) = \frac { 100 } { 1 - 9 e ^ { - 0.01 t } }$
G) $P ( t ) = \frac { 100 } { 1 + 9 e ^ { - 0.1 t } }$
H) $P ( t ) = 100 e ^ { 0.01 t }$

Question 6

Consider the following predator-prey system where x and y are in millions of creatures and t represents time in years: (a) Show that (4, 2) is the nonzero equilibrium solution.(b) Find an expression for .(c) The direction field for the differential equation is given below: (i) Locate (4, 2) on the graph.(ii) Sketch a rough phase trajectory through P indicated in the graph.(d) With the aid of the phase trajectory, answer the following questions:
(i) For the region and 0 < y < 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.(ii) For the region x > 4 and 0 < y < 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.(iii) For the region x > 4 and y > 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.(iv) For the region 0 < x < 4 and y > 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.(e) Suggest a pair of species which might interact in the manner described by this system.

Question 7

A phase portrait of a predator-prey system is given below in which F represents the population of foxes (in thousands) and R the population of rabbits (in thousands). (a) Referring to the graph, what is a reasonable non-zero equilibrium solution for the system?
(b) Write down a possible system of differential equations which could have been used to produce the given graph.(c) Describe how each population changes as time passes, using the initial condition P indicated on the graph.(d) Use your description in part (c) to make a rough sketch of the graph of R and F as functions of time.

Question 8

Suppose that we model populations of aphids and ladybugs with the system of differential equations: $\frac { d A } { d t } = 3 A - 0.01 A L$ $\frac { d L } { d t } = - 0.5 L + 0.0001 A L$ Find the equilibrium solution.

A) $A = 5000 , L = 300$
B) $A = 100 , L = 6$
C) $A = 30,000 , L = 50$
D) $A = 60 , L = 100$
E) $A = 300 , L = 5000$
F) $A = 6 , L = 100$
G) $A = 50 , L = 30,000$
H) $A = 100 , L = 60$

Question 9

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.01 P - 0.0001 P ^ { 2 }$ . What is the carrying capacity?

A)90
B)10
C)50
D)1000
E)100
F)60
G)20
H)10,000

Question 10

Suppose that we model populations (in millions) of predators and preys with the system of differential equations: $\begin{array} { l } \frac { d x } { d t } = 2 x - 1.2 x y \\\frac { d y } { d t } = - y + 0.9 x y\end{array}$ Find the expression for $\frac { d y } { d x }$ .

A) $\frac { 2 x - 1.2 x y } { - y + 0.9 x y }$
B) $\frac { - 2 x + 1.2 x y } { - y + 0.9 x y }$
C) $\frac { 2 x - 1.2 x y } { y - 0.9 x y }$
D) $\frac { 2 x - 1.2 x y } { - y - 0.9 x y }$
E) $\frac { y - 0.9 x y } { 2 x - 1.2 x y }$
F) $\frac { - y + 0.9 x y } { 2 x - 1.2 x y }$
G) $\frac { - y + 0.9 x y } { - 2 x + 1.2 x y }$
H) $\frac { - y + 0.9 x y } { - 2 x - 1.2 x y }$

Question 11

A predator-prey system is modeled by the system of differential equations , , where a, b, c, and d are positive constants.(a) Which variable, x or y, represents the predator? Defend your choice.(b) Show that the given system of differential equations has the two equilibrium solutions and .(c) Explain the significance of each of the equilibrium solutions.

Question 12

Suppose a population growth is modeled by the logistic differential equation with the carrying capacity 2000 and the relative growth rate k = 0.06 per year. If the initial population is P(0) = 500, and P(10).

A)309
E)308
B)756
F)755
C)310
G)307
D)757
H)800

Question 13

A rumor tends to spread according to the logistic differential equation , where y is the number of people in the community who have heard the rumor and t is the time in days.(a) Describe the population for this sociological study.(b) Assume that there were 10 people who knew the rumor at initial time t = 0. Find the solution for the differential equation.(c) How many days will it take for half of the population to hear the rumor?

Question 14

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.0001 P ( 100 - P )$ . What is the relative growth rate?

A)0.0001
B)-0.01
C)0.001
D)0.01
E)0.0002
F)-0.02
G)0.002
H)0.02

Question 15

Suppose a population growth is modeled by the logistic equation . Solve this differential equation with the initial condition P(0) = 20.

Question 16

Suppose that we model populations of aphids and ladybugs with the system of differential equations: $\begin{array} { l } \frac { d A } { d t } = 3 A - 0.01 A L \\\frac { d L } { d t } = - 0.5 L + 0.0001 A L\end{array}$ Find the expression for $\frac { d A } { d L }$ .

A) $\frac { - 0.5 L + 0.0001 A L } { 3 A - 0.01 A L }$
B) $\frac { 0.5 L - 0.0001 A L } { 3 A - 0.01 A L }$
C) $\frac { - 3 A + 0.01 A L } { - 0.5 + 0.0001 A L }$
D) $\frac { 3 A - 0.01 A L } { - 0.5 L + 0.0001 A L }$
E) $\frac { - 0.5 L + 0.0001 A L } { - 3 A + 0.01 A L }$
F) $\frac { - 0.5 L + 0.0001 A L } { - 3 A - 0.01 A L }$
G) $\frac { 3 A - 0.01 A L } { 0.5 L - 0.0001 A L }$
H) $\frac { 3 A - 0.01 A L } { - 0.5 L - 0.0001 A L }$

Question 17

Suppose that a population of bacteria grows according to the logistic equation , where P is the population measured in thousands and t is time measured in days.(a) What is the carrying capacity? What is the value of k?
(b) A direction field for this equation is given below. Where are the slopes close to 0? Where are the slope values the largest? Where are the solutions increasing? Where are the solutions decreasing? (c) Use the direction field to sketch solutions for initial populations of 10, 30, 50, and 70. What do these solutions have in common? How do they differ? Which solutions have inflection points? At what population levels do they occur?
(d) What are the equilibrium solutions? How are the other solutions related to these solutions?

Question 18

In each of the given systems, x and y are populations of two different species which are solutions to the differential equations. For each system, describe how the species interact with one another (for example, do they compete for the same resources, or cooperate for mutual benefit?) and suggest a pair of species that might interact in a manner consistent with the given system of equations.(a) (d) (b) (e) (c) (f)

Question 19

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.01 P - 0.0001 P ^ { 2 }$ with P(0) = 10. Find the population after 500 years.

A)50
B)94
C)70
D)500
E)80
F)100
G)35
H)30

Question 20

The population of two species is modeled by the system of equations .(a) Find an expression for .(b) A possible direction field for the differential equation in part (a) is given below: Use this graph to sketch a phase portrait with each of P, Q, R, and S as an initial condition. Describe the behavior of the trajectories near the nonzero equilibrium solutions.(c) Graph x and y as function of t. What happens to the population of the two species as the time t increases without bound?

Question 21

The radioactive isotope Bismuth-210 has a half-life of 5 days. How many days does it take for 87.5% of a given amount to decay?

A)15 days
E)11 days
B)8 days
F)9 days
C)10 days
G)12 days
D)13 days
H)14 days

Question 22

When a child was born, her grandparents deposited $1000 in a saving account at 5% interest compounded continuously. The amount of money after t years is:

A) $1000 \left( 2 ^ { t } \right)$
B) $1000 \left( e ^ { t } \right)$
C) $500 \left( e ^ { t } \right)$
D) $500 \left( 3 ^ { t } \right)$
E) $1000 \left( e ^ { - 0.05 t } \right)$
F) $1000 \left( e ^ { 0.05 t } \right)$
G) $500 \left( e ^ { 0.1 t } \right)$
H) $1000 \left( e ^ { 2 t } \right)$

Question 23

Suppose that a population grows according to a logistic model.(a) Write the differential equation for this situation with k = 0.01 and carrying capacity of 60 thousand.(b) Solve the differential equation in part (a) with the initial condition t = 0 (hours) and population P = 1 thousand.(c) Find the population for t = 10 hours, t = 100 hours, and t = 1000 hours.(d) After how many hours does the population reach 2 thousand? 30 thousand? 55 thousand?
(e) As the time t increases without bound, what happens to the population?
(f) Sketch the graph of the solution of the differential equation.

Question 24

A bacteria culture starts with 200 bacteria and triples in size every half hour. The population of the bacteria after $t$ hours is:

A) $200 \left( 9 ^ { - t } \right)$
B) $200 \left( 9 ^ { t } \right)$
C) $200 \left( 3 ^ { - t } \right)$
D) $200 \left( 3 ^ { t } \right)$
E) $200 \left( e ^ { - t } \right)$
F) $200 \left( e ^ { t } \right)$
G) $200 \left( e ^ { - 3 t } \right)$
H) $200 \left( e ^ { 3 t } \right)$

Question 25

In a model of epidemics, the number of infected individuals in a population at a time is a solution of the logistic differential equation , where y is the number of infected individuals in the community and t is the time in days.(a) Describe the population for this situation.(b) Assume that 10 people were infected at the initial time t = 0. Find the solution for the differential equation.(c) How many days will it take for half of the population to be infected?

Question 26

When a child was born, her grandparents placed $1000 in a savings account at 10% interest compounded continuously, to be withdrawn at age 20 to help pay for college. How much money is in the account at the time of withdrawal?

A) $1000 e$
B) $500 e$
C) $500 e ^ { 2 }$
D) $2000 e ^ { 2 }$
E) $4000 e$
F) $2000 e$
G) $1000 e ^ { 2 }$
H) $4000 e ^ { 2 }$

Question 27

An object cools at a rate (measured in ${ } ^ { \circ } \mathrm { C } / \mathrm { min }$ ) equal to $k$ times the difference between its temperature and that of the surrounding air. Suppose the object takes 10 minutes to cool from 60 $^\circ$ C to 40 $^\circ$ C in a room kept at 20 $^\circ$ C. Find the value of $k$ .

A) $e ^ { - 20 }$
B) $\ln 2$
C) $10 e ^ { - 20 }$
D) $40 \ln 10$
E) $\frac { 1 } { 2 }$
F) $e ^ { - 1 / 20 }$
G) $\frac { 1 } { 10 } \ln \frac { 1 } { 2 }$
H) $60 \ln \frac { 1 } { 2 }$

Question 28

Radium has a half-life of 1600 years. How many years does it take for 90% of a given amount of radium to decay?

A) $\frac { 1600 } { \ln 5 }$
B) $1600 \ln 2$
C) $\frac { 1600 \ln 10 } { \ln 2 }$
D) $1600 \ln 5$
E) $1600 \ln 10$
F) $\frac { 1600 } { \ln 2 }$
G) $1500 \ln 6$
H) $\frac { 1600 \ln 2 } { \ln 10 }$

Question 29

The radioactive isotope Bismuth-210 has a half-life of 5 days. Suppose we have an initial amount of 100 mg. The amount of Bismuth-210 remaining after $t$ days is

A) $100 \left( 2 ^ { 0.2 t } \right)$
B) $5 \left( 2 ^ { 0.2 t } \right)$
C) $50 \left( 2 ^ { - 0.2 t } \right)$
D) $100 \left( 2 ^ { - 0.2 t } \right)$
E) $100 \left( e ^ { 0.2 t } \right)$
F) $5 \left( e ^ { 0.2 t } \right)$
G) $50 \left( e ^ { - 0.2 t } \right)$
H) $100 \left( e ^ { - 0.2 t } \right)$

Question 30

A bacteria culture starts with 200 bacteria and in 1 hour contains 400 bacteria. How many hours does it take to reach 2000 bacteria?

A) $\ln 400$
B) $\ln 10$
C) $10$
D) $\ln 1600$
E) $\ln 2000$
F) $\ln 200$
G) $5$
H) $\frac { \ln 10 } { \ln 2 }$

Question 31

Assume that a population grows at a rate summarized by the equation , where b and k are positive constants (b > 1), and P is the population at time t. Show that is the general solution for the differential equation (where is the initial population). [Note: This is known as the monomolecular growth curve.]

Question 32

Carbon 14, with a half-life of 5700 years, is used to estimate the age of organic materials. What fraction of the original amount of carbon 14 would an object have if it were 2000 years old?

A) $e ^ { - ( 57 / 20 ) \ln 2 }$
B) $\frac { 57 } { 20 } \ln 2$
C) $e ^ { - ( 20 / 57 ) \ln 2 }$
D) $\frac { 20 } { 57 } \ln 2$
E) $e ^ { ( 57 / 20 ) \ln 2 }$
F) $\frac { 1 } { 57 } \ln 20$
G) $e ^ { ( 20 / 57 ) \ln 2 }$
H) $\frac { 1 } { 20 } \ln 57$

Question 33

Suppose that a population, P, grows at a rate given by the equation , where P is the population (in thousands) at time t (in hours), and b and k are positive constants.(a) Find the solution to the differential equation when b = 0.04, k = 0.01 and P (0) = 1.(b) Find P (10), P (100), and P (1000).(c) After how many hours does the population reach 2 thousand? 30 thousand? 54 thousand?
(d) As time t increases without bound, what happens to the population?
(e) Sketch the graph of the solution of the differential equation.

Question 34

(a) Solve the differential equation , with b = 2 and k = 0.1, and = 1.(b) Sketch a graph of the solution you produced for part (a) and discuss the major characteristics of this monomolecular growth curve.

Question 35

Suppose that a certain population grows according to an exponential model.(a) Write the differential equation for this situation with a relative growth rate of k = 0.01. Produce a solution for the initial condition t = 0 (in hours) and population P = 1 (in thousands).(b) Find the population when t = 10 hours, t = 100 hours, and t = 1000 hours.(c) After how many hours does the population reach 2 thousand? 30 thousand? 55 thousand?
(d) As the time t increases without bound, what happens to the population?
(e) Sketch the graph of the solution of the differential equation.

Question 36

An outbreak of a previously unknown influenza occurred on the campus of the University of Northern South Dakota at Roscoe during the first semester. Due to the contagious nature of the disease, the campus was quarantined and the disease was allowed to run its course. The table below shows the total number P of infected students for the first four weeks of the outbreak on this campus of 2,500 students. (a) Find a logistic model for the data. Complete the table with predicted values using this model.(b) Find an exponential model for these data. Complete the table with predicted values using this model.(c) Compare your findings in parts (a) and (b) above. For what values would you consider both models to be a good fit for the data? Which model provides the best fit for the data? Justify your choice.

Question 37

The half-life of Carbon 14 is 5700 years. A wooden table is measured with 80% of Carbon 14 compared with newly cut tree. Find the age of the table.

A)2, 933 years
E)13,235 years
B)1,000 years
F)4,200 years
C)500 years
G)1,835 years
D)2,000 years
H)3,000 years

Question 38

The following table contains population data for a Minnesota county for the decades from 1900 to 1980: (a) Produce a scatter plot for the data.(b) Find an exponential model using the data from 1900 through 1950.(c) Find a logistic model using the data from 1900 through 1950. (Assume the carrying capacity is 440,000.)
(d) Use your models to estimate the population for 1960, 1970, and 1980. Enter your data in the table provided above.

Question 39

A bacteria population grows at a rate proportional to its size. The initial count was 400 and 1600 after 1 hour. In how many minutes does the population double?

A)20
E)40
B)25
F)45
C)30
G)50
D)35
H)55

Question 40

A bacteria culture starts with 200 bacteria and triples in size every half hour. After 2 hours, how many bacteria are there?

A)17,800
E)19,300
B)16,200
F)14,800
C)23,500
G)15,700
D)24,000
H)21,000

Question 41

$2000 is invested at 5% annual interest. Find the value of A(t) at the end of t years if:
(a) the interest compounds monthly.(b) the interest compounds continuously.

Question 42

An object cools at a rate (in ${ } ^ { \circ } \mathrm { C } / \mathrm { min }$ ) equal to $\frac { 1 } { 10 }$ of the difference between its temperature and that of the surrounding air. If a room is kept at 20 $^\circ$ C and the temperature of the object is 28 $^\circ$ C, what is the temperature of the object 5 minutes later?

A)22
B)24
C) $20 + 5 e ^ { - 1 / 10 }$
D) $20 + 8 e ^ { - 1 / 2 }$
E) $20 + 5 e ^ { - 4 / 5 }$
F) $20 + 8 e ^ { - 1 / 10 }$
G) $28 - 8 e ^ { - 1 / 10 }$
H) $28 - 10 e ^ { - 1 / 2 }$

Question 43

Assume the half-life of carbon 14 is 5700 years. A wooden statue is measured with 70% of the carbon-14. How old is the statue?

Question 44

It takes money 20 years to triple at a certain rate of interest. How long does it take for money to double at this rate?

Question 45

The following data approximate the results obtained by subjecting Hela-S cells to 250 kvp x-rays: Assume that these data fit an exponential model.(a) Find the appropriate exponential model.(b) Add another line to the table using your population model for the given doses of radiation.(c) Compare the model entries to the given data and explain any discrepancy.

Question 46

In an experiment, a tissue culture has been subjected to ionizing radiation. It was found that the number A of undamaged cells depends on the exposure time, in hours, according to the formula If 5000 cells were present initially and 3000 survived a 2-hour exposure, and the elapsed time of exposure after which only half the original cells survive.

Question 47

The growth of a population is modeled by the differential equation $\frac { d P } { d t } = 0.2 P ^ { 101 }$ , and the initial population is $P ( 0 ) = 2$ Find $P ( 50 )$

A)37
B)90
C)44,053
D)81,350
E)30
F)80
G)90,000
H)37,648

Question 48

Solve the differential equation $\frac { d y } { d t } = y ^ { 2 }$ , $y ( 0 ) = 1$ . From your solution, and the value of $y ( 2 )$ .

A) $\frac { 1 } { 3 }$
B)1
C) $- \frac { 1 } { 3 }$
D) $- 3$
E) $- 1$
F)3

G) $- \frac { 1 } { 5 }$
H) $\frac { 1 } { 5 }$

Question 49

$2000 is invested at 5% annual interest. Find the value at the end of 18 years if:
(a) the interest compounds monthly.(b) the interest compounds continuously.

Question 50

In 1970, the Brown County groundhog population was 100. By 1980, there were 900 groundhogs in Brown County. If the rate of population growth of these animals is proportional to the population size, how many groundhogs might one expect to see in 1995?

Question 51

In a certain medical treatment, a tracer dye is injected into a human organ to measure its function rate and the rate of change of the amount of dye is proportional to the amount present at any time. If a physician injects 0.5 g of dye and 30 minutes later 0.1 g remains, how much dye will be present in hours?

Question 52

$2000 is invested at 3% annual interest. Find the value at the end of 10 years if:
(a) the interest compounds annually.(b) the interest compounds continuously.

Question 53

A thermometer is taken outside from a room where the temperature is 72 $^\circ$ F. Outdoors, the temperature is 48 $^\circ$ F. After one minute, the thermometer reads 55 $^\circ$ F. After how many minutes does the thermometer read 50 $^\circ$ F?

A)2.107
B)1.107
C)3.100
D)1.503
E)2.017
F)1.017
G)3.010
H)1.013

Question 54

Find the solution of the initial-value problem $\frac { d y } { d x } = x \sin \left( x ^ { 2 } \right)$ , $y ( 0 ) = 0$ .

A) $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) - \frac { 1 } { 2 }$
B) $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right)$
C) $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$
D) $\frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$
E) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) - \frac { 1 } { 2 }$
F) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right)$
G) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$
H) $\frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$

Question 55

Assume that the rate of growth of a population of fruit flies is proportional to the size of the population at each instant of time. If 100 fruit flies are present initially and 200 are present after 5 days, how many will be present after 10 days?

Question 56

A lettuce leaf collected from the salad bar at the college cafeteria contains as much carbon-14 as a freshly cut lettuce leaf. How old is it? (Use 5700 years for the half-life of C.)

Question 57

Solve the differential equation $y ^ { \prime } = 5 y ( 1000 - y )$ subject to the initial condition $y ( 0 ) = 500$ . From your solution, and the value of the limit $\lim _ { t \rightarrow \infty } y ( t )$ .

A)5000
B)2500
C)1000
D)2000
E)200
F)20000
G)100
H)500

Question 58

$2000 is invested at 3% annual interest. Find the value of A(t) at the end of t years if:
(a) the interest compounds annually.(b) the interest compounds continuously.

Question 59

In an idealized experiment, the following results were obtained for a population of bacteria during a 7 hour period. The initial population is 1000 bacteria. (a) Identify the period where there is no change in the number of bacteria. (This is called the period of adaptation.)
(b) Identify the period of growth.(c) Assume that the growth rate of bacteria is proportional to the population. Find an exponential model for the data during the period of growth.(d) Add an additional line to the table using your population model to generate the entries for the given time values. Compare these entries with the given data and explain any discrepancy.

Question 60

Solve the differential equation $\frac { d y } { d t } = t ( y - 3 )$ , $y ( 2 ) = 3$ . From your solution, and the value of $y ( 5 )$ .

A) $- 2$
B)2
C)5
D)0
E) $- 3$
F)3
G) $- 5$
H) $\frac { 1 } { 5 }$

Question 61

Find the solution of the initial-value problem $\frac { d y } { d t } = \frac { y + t ^ { 2 } y } { t ^ { 2 } }$ , $y ( 1 ) = 2$ .

A) $y = 2 e ^ { t + ( 1 / t ) }$
B) $y = 3 e ^ { t - ( 1 / t ) }$
C) $y = 2 e ^ { t }$
D) $y = 2 e ^ { 1 + \left( 1 / t ^ { 2 } \right) }$
E) $y = 2 e ^ { 1 / t }$
F) $y = c e ^ { t - ( 1 / t ) }$
G) $y = c e ^ { t + ( 1 / t ) }$
H) $y = 2 e ^ { t - ( 1 / t ) }$

Question 62

Find the solution of the initial-value problem $\frac { d y } { d t } = 2 t \sqrt { 1 - y }$ , $y ( 1 ) = 0$ .

A) $2 \sqrt { y - 1 } = 3 - t ^ { 2 }$
B) $\frac { 2 } { \sqrt { y - 1 } } = 3 - t ^ { 2 }$
C) $2 \sqrt { 1 - y } = 3 + t ^ { 2 }$
D) $\frac { 2 } { \sqrt { 1 - y } } = 3 + t ^ { 2 }$
E) $2 \sqrt { 1 - y } = - 3 + t ^ { 2 }$
F) $\frac { 2 } { \sqrt { 1 - y } } = - 3 + t ^ { 2 }$
G) $2 \sqrt { 1 - y } = 3 - t ^ { 2 }$
H) $\frac { 2 } { \sqrt { 1 - y } } = 3 - t ^ { 2 }$

Question 63

Find the solution to the differential equation that satisfies the initial condition .

Question 64

The graph of a direction field for the differential equation is given below: (a) Sketch a solution curve that satisfies the given condition, but without solving the differential equation:
(i) (ii) (iii) (b) Solve the differential equation for each of the conditions in part (a). Compare your answers to the curves you produced in part (a).(c) What is the relationship between the curves (i) and (ii) in part (a)? Explain why this occurs.

Question 65

Find the solution of the initial-value problem $y ^ { \prime } = \frac { \ln x } { x y }$ , $y ( 1 ) = 2$ .

A) $y = \frac { 1 + x } { 1 + \ln x }$
B) $y = \frac { 8 x } { ( 1 + x ) ^ { 2 } }$
C) $y = 2 + 2 \ln x$
D) $y = \sqrt { 4 + ( \ln x ) ^ { 2 } }$
E) $y = x \ln x + 2 x$
F) $y = x \left( 1 + x ^ { 2 } \right)$
G) $y = x + \sqrt { 1 + \ln x }$
H) $y = \sqrt { x } ( 1 + x )$

Question 66

Solve the initial-value problem $t \frac { d y } { d t } = y ( y - 1 )$ , $y ( 2 ) = 4$ .

A) $y = - \frac { 8 } { 8 + 3 t }$
B) $y ^ { 2 } - y = 6 t$
C) $y = - \frac { 8 } { 8 - 3 t }$
D) $y ^ { 2 } - y = 6 t$ ..
E) $y = \frac { 8 } { 8 - 3 t }$ .
F) $y ^ { 2 } = 8 t$
G) $y = \frac { 1 } { 1 - t }$
H) $y ^ { 2 } + y = 6 t$

Question 67

Find the solution to the differential equation that satisfies the initial condition .

Question 68

Find the orthogonal trajectories of the family of curves . Then draw several members of each family on the same coordinate plane.

Question 69

Find the orthogonal trajectories of the family of curves . Then draw several members
of each family on the same coordinate plane.

Question 70

Solve the initial-value problem $\frac { d y } { d t } = 2 t y ^ { 2 } + 3 y ^ { 2 }$ , $y ( 0 ) = 1$ . Then use your solution to evaluate $y ( 1 )$ .

A) $- 3$
B)1
C) $- \frac { 1 } { 3 }$
D) $\frac { 1 } { 3 }$
E)0
F)3

G) $- \frac { 1 } { 5 }$
H) $\frac { 1 } { 5 }$

Question 71

Solve the initial-value problem $\frac { d y } { d t } = \frac { 2 t } { y ^ { 2 } + t ^ { 2 } y ^ { 2 } }$ , $y ( 0 ) = 3$ . Then use your solution to evaluate $y ( \sqrt { e - 1 } )$ .

A) $\sqrt [ 3 ] { 30 }$
B) $\sqrt [ 3 ] { 12 }$
C) $30$
D)1
E) $\sqrt [ 3 ] { 3 }$
F) $\sqrt [ 3 ] { 10 }$
G)12
H)27

Question 72

Solve the initial-value problem $\frac { d y } { d t } = 3 y + 1$ , $y ( 0 ) = 2$ . Then use your solution to evaluate $y ( \ln 2 )$ .

A) $\frac { 3 } { 13 }$
B) $\frac { 3 } { 55 }$
C) $\frac { 7 } { 3 }$
D) $\frac { 41 } { 3 }$
E) $\frac { 13 } { 3 }$
F) $\frac { 55 } { 3 }$
G) $\frac { 3 } { 7 }$
H)9

Question 73

Consider the differential equation .(a) Find the general solution to the differential equation.(b) Find the solution that satisfies the initial condition .

Question 74

Find the solution of the initial-value problem $\frac { d y } { d t } = \frac { 2 t } { y ^ { 2 } + t ^ { 2 } y ^ { 2 } }$ , $y ( 0 ) = 3$ .

A) $y = 3 \ln \left( 1 + t ^ { 2 } \right) + 3 C$
B) $y = \sqrt [ 3 ] { 3 \ln \left( 1 + t ^ { 2 } \right) + 3 C }$
C) $y = 3 \ln \left( 1 + t ^ { 2 } \right) + 9$
D) $y = \sqrt [ 3 ] { 3 \ln \left( 1 + t ^ { 2 } \right) + 9 }$
E) $y = 3 \ln \left( 1 + t ^ { 2 } \right) + 27$
F) $y = \sqrt [ 3 ] { 3 \ln \left( 1 + t ^ { 2 } \right) + 27 }$
G) $y = 3 \ln \left( 1 + t ^ { 2 } \right)$
H) $y = \sqrt [ 3 ] { 3 \ln \left( 1 + t ^ { 2 } \right) }$

Question 75

Consider the differential equation .(a) Find the general solution to the differential equation.(b) Find the solution with the initial-value .(c) Find the solution with the initial-value .

Question 76

Find the solution of the initial-value problem $\frac { d y } { d t } = 2 t y ^ { 2 } + 3 y ^ { 2 }$ , $y ( 0 ) = 1$ .

A) $y = - \frac { 1 } { 1 - 3 t - t ^ { 2 } }$
B) $y = \frac { 1 } { 1 - 3 t + t ^ { 2 } }$
C) $y = \frac { 1 } { 1 + 3 t - t ^ { 2 } }$
D) $y = \frac { 1 } { 1 - 3 t - t ^ { 2 } }$
E) $y = 1 + 3 t - t ^ { 2 }$
F) $y = 1 - 3 y - t ^ { 2 }$
G) $y = 1 - 3 t + t ^ { 2 }$
H) $y = - 1 + 3 t + t ^ { 2 }$

Question 77

Consider the differential equation .(a) Find the general solution to the differential equation.(b) Find the solution that satisfies the initial condition .

Question 78

Find the orthogonal trajectories of the family of curves . Then draw several members of each family on the same coordinate plane.

Question 79

Find the solution of the initial-value problem $\frac { d y } { d t } = 3 y + 1$ , $y ( 0 ) = 2$ .

A) $y = \frac { 1 } { 3 } \left( 7 e ^ { t } - 1 \right)$
B) $y = \left( 7 e ^ { t } - 5 \right)$
C) $y = \frac { 1 } { 3 } \left( e ^ { 3 t } - 1 \right)$
D) $y = \left( 3 e ^ { 3 t } - 1 \right)$
E) $y = \frac { 1 } { 3 } \left( 5 e ^ { 3 t } + 1 \right)$
F) $y = \left( 5 e ^ { 3 t } + 1 \right)$
G) $y = \frac { 1 } { 3 } \left( 7 e ^ { 3 t } - 1 \right)$
H) $y = \left( 7 e ^ { 3 t } - 1 \right)$

Question 80

Find the equation of a curve that passes through the point and whose slope at a point is .