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Deck 10: Differential Equations
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Question
Consider the differential equation y' = y - 
. Which of the following statements is/are true?
A) The function f(t) =
is a solution to this differential equation with initial condition 
B) This differential equation has infinitely many solutions.
C) The constant function f(t) = 1 is a solution to this differential equation.
D) If f(t) is a solution to the differential equation satisfying the initial condition y(0) = 0, then
E) All of these statements are true.
. Which of the following statements is/are true?A) The function f(t) =
is a solution to this differential equation with initial condition B) This differential equation has infinitely many solutions.
C) The constant function f(t) = 1 is a solution to this differential equation.
D) If f(t) is a solution to the differential equation satisfying the initial condition y(0) = 0, then
E) All of these statements are true.
Question
Consider the differential equation 
= 
(y + 3). Which of the following statements is/are true?
(I) f(t) = -3 is a constant solution to this differential equation.
(II) f(t) = 0 is a constant solution to this differential equation.
(III) If f(t) is a solution to the differential equation with initial conditions y(1) = 0, then
(1) = 3.
A) I and III
B) II only
C) III only
D) I only
E) I, II, and III
= (y + 3). Which of the following statements is/are true?(I) f(t) = -3 is a constant solution to this differential equation.
(II) f(t) = 0 is a constant solution to this differential equation.
(III) If f(t) is a solution to the differential equation with initial conditions y(1) = 0, then
(1) = 3.A) I and III
B) II only
C) III only
D) I only
E) I, II, and III
Question
Solve the differential equation: y' = 
sin t.
A) y = cos(ln t) + C
B) y =
+ C
C) y = ln(-sin t) + C
D) y = -ln(cos t + C
sin t.A) y = cos(ln t) + C
B) y =
+ CC) y = ln(-sin t) + C
D) y = -ln(cos t + C
Question
Given the differential equation: 
, is this the solution 
, is this the solution Question
Which of the following functions solves the differential equation: 
A) y = -
B) y = -
C) y = ln
D) y =
A) y = -
B) y = -
C) y = ln
D) y =
Question
Find f'(1) if f(t) is a solution to the initial value problem: 
Enter just a real number (no approximations).
Enter just a real number (no approximations). Question
Which of the following functions solves the differential equation: 
?
A) y =
B) y =
C) y = 7
D) y = 7
?A) y =
B) y =
C) y = 7
D) y = 7
Question
Find a constant solution of 
Enter just a reduced fraction of form 
.
Enter just a reduced fraction of form . Question
Given the differential equation: 
, is this the solution 
, is this the solution Question
Given the differential equation: 
, is this the solution 
, is this the solution Question
Find f'(1) if f(t) is a solution to the initial value problem: 
Enter just an integer.
Enter just an integer. Question
Which of the following functions solves the differential equation: 
A) y =
+ 3
B) y =
+ 3
C) y = -
+ 3x
D) none of these
A) y =
+ 3B) y =
+ 3C) y = -
+ 3xD) none of these
Question
Given the differential equation: y' is this the solution 
Question
Find f'(0) if f(t) is a solution to the initial value problem: 
Enter just an integer.
Enter just an integer. Question
Given the differential equation: 
is this the solution 
is this the solution Question
Given the differential equation: 
, is this the solution: 
, is this the solution: Question
Find a constant solution of 
Enter just an integer.
Enter just an integer. Question
Given the differential equation: 
, is this the solution 
, is this the solution Question
Which of the following functions solves the differential equation: 
?
A) y =
B) y = -
C) y = ln 4t
D) none of these
?A) y =
B) y = -
C) y = ln 4t
D) none of these
Question
Question
Solve the differential equation with the given initial condition.
A) y =
- 2
B) y = 2
-1
C) y =
+ 1
D) y = t + 1
A) y =
- 2B) y = 2
-1C) y =
+ 1D) y = t + 1
Question
Solve the differential equation with the given initial condition.
A) y =
- 2t + 1
B) y =
- 2t
C) y =
D) y = 0
E) none of these
A) y =
- 2t + 1B) y =
- 2tC) y =
D) y = 0
E) none of these
Question
Given the differential equation with the given initial condition: 
is this the solution 
is this the solution Question
Solve the differential equation with the given initial condition.
, y(0) = 0
A) y = -ln
B) y = 0
C) y =
D) y =
E) none of these
, y(0) = 0A) y = -ln
B) y = 0
C) y =
D) y =
E) none of these
Question
Given the differential equation with the given initial condition: 
is this the solution 
is this the solution Question
Given the differential equation with the given initial condition: 
is this the solution 
is this the solution Question
Solve the differential equation with the given initial condition.
A) y = 4 -
B) y =
C) y =
+ 
D) y = 4 +
A) y = 4 -
B) y =
C) y =
+ D) y = 4 +
Question
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)
= ky; There is not enough information given to determine initial conditions.
B)
= ky; y(0) = 0
C)
= k(1 - y); There is not enough information given to determine initial conditions.
D)
= k(1 - y); y(0) = 0
E)
= (1 + y); y(0) =0
A)
= ky; There is not enough information given to determine initial conditions.B)
= ky; y(0) = 0C)
= k(1 - y); There is not enough information given to determine initial conditions.D)
= k(1 - y); y(0) = 0E)
= (1 + y); y(0) =0 Question
Given the differential equation with the given initial condition: 
is this the solution 
is this the solution Question
Solve the differential equation with the given initial condition.
A) t + 1
B)
C) t - 1
D)
A) t + 1
B)
C) t - 1
D)
Question
Find the constant solutions to the differential equation: 
.
Enter just one integer or two separated by a comma (no label).
.Enter just one integer or two separated by a comma (no label).
Question
Solve the differential equation with the given initial condition.
A) y =
- 
B) y =
C) y = 16 +
D) y = 64 -
t
A) y =
- B) y =
C) y = 16 +
D) y = 64 -
t Question
Given the differential equation with the given initial condition: 
is this the solution 
is this the solution Question
Given the differential equation with the given initial condition: 
is this the solution 
is this the solution Question
Solve the differential equation with the given initial condition.
A) y = ln
+ 1
B) y =
+ 1
C) y = tan t + 1
D) y =
+ 
A) y = ln
+ 1B) y =
+ 1C) y = tan t + 1
D) y =
+ Question
Solve the differential equation with the given initial condition.
A) y =
B) y = ±
C) y = ±
D) y =
A) y =
B) y = ±
C) y = ±
D) y =
Question
Given the differential equation: 
is this the solution 
is this the solution Question
Given the differential equation with the given initial condition: 
is this the solution 
is this the solution Question
Solve the differential equation with the given initial condition.
A) y =
B) y = -
C) y =
D) y = -1 +
A) y =
B) y = -
C) y =
D) y = -1 +
Question
Given the differential equation: 
is this the solution 
is this the solution Question
Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. 
- 4y = -2 
; y(0) = -1
A) integrating factor:
General solution:
Particular solution: y = -
- 2 
B) integrating factor:
General solution: y =
+ C 
Particular solution: y =
- 2 
C) integrating factor:
General solution: y = -2t + C
Particular solution: y = -2t -
D) integrating factor:
General solution: y =
+ C 
Particular solution: y =
- 
Solve the equation using an integrating factor.
- 4y = -2 ; y(0) = -1A) integrating factor:
General solution:
Particular solution: y = -
- 2 B) integrating factor:
General solution: y =
+ C Particular solution: y =
- 2 C) integrating factor:
General solution: y = -2t + C
Particular solution: y = -2t -
D) integrating factor:
General solution: y =
+ C Particular solution: y =
- Solve the equation using an integrating factor.
Question
Solve the initial value problem using an integrating factor.
t
+ 3y = 5t; 
, t > 0
A) y =
t - 
B) y =
+ 1 - 
C) y = 5
- 4t
D) y = 5
- 4 
t
+ 3y = 5t; , t > 0A) y =
t - B) y =
+ 1 - C) y = 5
- 4tD) y = 5
- 4 Question
The annual sales y (in millions of dollars) of a company satisfy the differential equation 
. 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 annual sales.
B) The annual sales are increasing at $0.2 million ($200,000) per year.
C) The annual sales are decreasing at a rate proportional to the annual sales.
D) The annual sales are increasing at a rate proportional to $0.2 million ($200,000) per year.
. 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 annual sales.
B) The annual sales are increasing at $0.2 million ($200,000) per year.
C) The annual sales are decreasing at a rate proportional to the annual sales.
D) The annual sales are increasing at a rate proportional to $0.2 million ($200,000) per year.
Question
Combine the terms y and 
into the derivative of a product, then solve the equation. 
Is this the solution: 
into the derivative of a product, then solve the equation. Is this the solution: Question
Solve the problem.
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 - 36,000
= 10)017 years
B) A = 80,000 - 56,000
= 8)352 years
C) A = 80,000 - 56,000
= 7)134 years
D) A = 60,000 - 36,000
= 8)352 years
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 - 36,000
= 10)017 yearsB) A = 80,000 - 56,000
= 8)352 yearsC) A = 80,000 - 56,000
= 7)134 yearsD) A = 60,000 - 36,000
= 8)352 years Question
Combine the terms y and 
into the derivative of a product: 
.
Is this derivative correct:
?
into the derivative of a product: .Is this derivative correct:
? Question
Solve the equation using an integrating factor.
+ 
y = 4 
, t > 0
A) y = 4x + C
B) y = 4x + C
C) y = 4 + C
D) y = 4 + C
+ y = 4 , t > 0A) y = 4x + C
B) y = 4x + C
C) y = 4 + C
D) y = 4 + C
Question
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: 
Question
Solve the initial value problem using an integrating factor.
+ y = 3; y(0) = 0.
A) y =
(3t - 
)
B) y =
C) y = 3 - 3
D) y = 3t
+ y = 3; y(0) = 0.A) y =
(3t - )B) y =
C) y = 3 - 3
D) y = 3t
Question
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)
= 
, k < 0
B)
= 
, k > 0
C)
= 
, k < 0; y(0) =0
D)
= 
, k > 0; y(0) = 0
E) none of these
A)
= , k < 0B)
= , k > 0C)
= , k < 0; y(0) =0D)
= , k > 0; y(0) = 0E) none of these
Question
Solve the equation using an integrating factor.
, t > 0
A) y = t + 1/2 + C
B) y = t - C
C) y = Ct +
D) y = 1/2 + C
, t > 0A) y = t + 1/2 + C
B) y = t - C
C) y = Ct +
D) y = 1/2 + C
Question
Combine the terms y and 
into the derivative of a product, then solve the equation. 
Is this the solution: 
into the derivative of a product, then solve the equation. Is this the solution: Question
The annual sales y (in millions of dollars) of a company satisfy the differential equation 
. 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 $0.2 million ($200,000) per year to an upper limit of $5 million.
C) The annual sales are increasing at a rate proportional to the difference between the annual sales and an upper limit of $10 million.
D) The annual sales are increasing at a rate proportional to $0.2 million ($200,000) per year.
. 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 $0.2 million ($200,000) per year to an upper limit of $5 million.
C) The annual sales are increasing at a rate proportional to the difference between the annual sales and an upper limit of $10 million.
D) The annual sales are increasing at a rate proportional to $0.2 million ($200,000) per year.
Question
Solve the equation using an integrating factor.
, t > 0
A) y = C
+ C
B) y = C
C) y = C
- 2
D) y = C
, t > 0A) y = C
+ CB) y = C
C) y = C
- 2D) y = C
Question
Suppose the relationship between the price p, of a product and the weekly sales, s, of the product is given by the differential equation 
Then
A) as sales increase, the price increases.
B) as the price increases the rate of change of the price also increases.
C) the rate of the decrease of the price is proportional to the sales.
D) s = 0 is a constant solution to this differential equation.
E) all of these
ThenA) as sales increase, the price increases.
B) as the price increases the rate of change of the price also increases.
C) the rate of the decrease of the price is proportional to the sales.
D) s = 0 is a constant solution to this differential equation.
E) all of these
Question
v
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 = 60,000
- 52,000= $31,041.84
B) A = 52,000
- 44,000= $27,969.59
C) A = 48,000
- 40,000= $26,433.47
D) A = 44,000
- 36,000= $24,897.34
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 = 60,000
- 52,000= $31,041.84B) A = 52,000
- 44,000= $27,969.59C) A = 48,000
- 40,000= $26,433.47D) A = 44,000
- 36,000= $24,897.34 Question
Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. 2t 
- y = 
; y(1) = -1, t > 0
A) integrating factor:\
General solution: y = -
+ C 
Particular solution: y = -
+ 
B) integrating factor: 2t
General solution: y = -
+ 
Particular solution: y = -
+ 
C) integrating factor:
General solution: y = -4 + C
Particular solution: y = -4 + 3
D) integrating factor: 2t
General solution: y = -
+ 
Particular solution: y = -
+ 
- y = ; y(1) = -1, t > 0A) integrating factor:\
General solution: y = -
+ C Particular solution: y = -
+ B) integrating factor: 2t
General solution: y = -
+ Particular solution: y = -
+ C) integrating factor:
General solution: y = -4 + C
Particular solution: y = -4 + 3
D) integrating factor: 2t
General solution: y = -
+ Particular solution: y = -
+ Question
Solve the equation using an integrating factor.
, t > 0
A) y =
+ C 
B) y = C
C) y =
+ 
D) y =
+ C
, t > 0A) y =
+ C B) y = C
C) y =
+ D) y =
+ C Question
Solve the equation using an integrating factor.
, t > 0
A) y = C
+ 
B) y = -
+ 
+ C
C) y =
+ 
D) y = C
+ 
, t > 0A) y = C
+ B) y = -
+ + CC) y =
+ D) y = C
+ Question
Combine the terms y and 
into the derivative of a product: 
.
Is this derivative correct:
?
into the derivative of a product: .Is this derivative correct:
? Question
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 = a
+ c
T = a
+ c Question
Question
Question
Question
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: 
Question
Consider the differential equation 
= g(y) where g(y) is the function whose graph is shown below: 
Indicate whether the following statements are true or false.
Let
= 
. 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 only
B) I and II
C) II only
D) III only
E) none of these
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
Let
= . 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 only
B) I and II
C) II only
D) III only
E) none of these
Question
Question
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
For what y value(s) does a solution of
= 
- 3y + 2 have inflection points?
A) y = 2
B) y = 0
C) y = 2 and y = 1
D) y =
E) none of these
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
For what y value(s) does a solution of
= - 3y + 2 have inflection points?A) y = 2
B) y = 0
C) y = 2 and y = 1
D) y =
E) none of these
Question
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 
?
(I) f(t) is an increasing function
(II) f(t) is always positive
(III) f(t) has an inflection point when y = 2.
A) III only
B) I, II, and III
C) I only
D) II only
E) I and III
Which of the following can then be said about the solution y = f(t) to the initial value problem ? (I) f(t) is an increasing function
(II) f(t) is always positive
(III) f(t) has an inflection point when y = 2.
A) III only
B) I, II, and III
C) I only
D) II only
E) I and III
Question
A nutritionist proposes the following model for weight loss on a program she is developing: 
+ 0.006w = 
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) 41 days
B) 37 days
C) 39 days
D) 35 days
+ 0.006w = 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) 41 days
B) 37 days
C) 39 days
D) 35 days
Question
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
y = -3, y = 1, and y = 5 are the constant solutions to
= g(y).
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
y = -3, y = 1, and y = 5 are the constant solutions to
= g(y). Question
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
Let
= 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) I only
B) III only
C) I and III
D) II only
E) none of these
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
Let
= 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) I only
B) III only
C) I and III
D) II only
E) none of these
Question
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 the situation? 
Do these graphs represent the situation? Question
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 
Question
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
y = 2 is the only constant solution of
= g(y) .
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
y = 2 is the only constant solution of
= g(y) . Question
Suppose 
= 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 IV
B) I and IV
C) II and III
D) I and III
E) not enough information given
= 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 IV
B) I and IV
C) II and III
D) I and III
E) not enough information given
Question
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
If the initial value of y(0) is greater than 6, then the corresponding solution will be an increasing function.
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
If the initial value of y(0) is greater than 6, then the corresponding solution will be an increasing function.
Question
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
If the initial value of y(0) is 3, then the corresponding solution has an inflection point.
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
If the initial value of y(0) is 3, then the corresponding solution has an inflection point.
Question
Suppose the graph below gives a solution to the differential equation 
= 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)
(III)
(IV) g(
) > 0
A) I only
B) IV only
C) I, III, and IV
D) I and IV
E) I, II, and IV
= 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)
(III)
(IV) g(
) > 0A) I only
B) IV only
C) I, III, and IV
D) I and IV
E) I, II, and IV
Question
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
If the initial value of y(0) is 2, then the corresponding solution has an inflection point.
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
If the initial value of y(0) is 2, then the corresponding solution has an inflection point.
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Deck 10: Differential Equations
1
Consider the differential equation y' = y - . Which of the following statements is/are true?
A) The function f(t) = is a solution to this differential equation with initial condition
B) This differential equation has infinitely many solutions.
C) The constant function f(t) = 1 is a solution to this differential equation.
D) If f(t) is a solution to the differential equation satisfying the initial condition y(0) = 0, then
E) All of these statements are true.
. Which of the following statements is/are true?A) The function f(t) =
is a solution to this differential equation with initial condition B) This differential equation has infinitely many solutions.
C) The constant function f(t) = 1 is a solution to this differential equation.
D) If f(t) is a solution to the differential equation satisfying the initial condition y(0) = 0, then
E) All of these statements are true.
All of these statements are true.
2
Consider the differential equation = (y + 3). Which of the following statements is/are true?
(I) f(t) = -3 is a constant solution to this differential equation.
(II) f(t) = 0 is a constant solution to this differential equation.
(III) If f(t) is a solution to the differential equation with initial conditions y(1) = 0, then (1) = 3.
A) I and III
B) II only
C) III only
D) I only
E) I, II, and III
= (y + 3). Which of the following statements is/are true?(I) f(t) = -3 is a constant solution to this differential equation.
(II) f(t) = 0 is a constant solution to this differential equation.
(III) If f(t) is a solution to the differential equation with initial conditions y(1) = 0, then
(1) = 3.A) I and III
B) II only
C) III only
D) I only
E) I, II, and III
I and III
3
Solve the differential equation: y' = sin t.
A) y = cos(ln t) + C
B) y = + C
C) y = ln(-sin t) + C
D) y = -ln(cos t + C
sin t.A) y = cos(ln t) + C
B) y =
+ CC) y = ln(-sin t) + C
D) y = -ln(cos t + C
y = -ln(cos t + C
4
Given the differential equation: , is this the solution
, is this the solution Unlock Deck
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5
Which of the following functions solves the differential equation:
A) y = -
B) y = -
C) y = ln
D) y =
A) y = -
B) y = -
C) y = ln
D) y =
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6
Find f'(1) if f(t) is a solution to the initial value problem: Enter just a real number (no approximations).
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7
Which of the following functions solves the differential equation: ?
A) y =
B) y =
C) y = 7
D) y = 7
?A) y =
B) y =
C) y = 7
D) y = 7
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8
Find a constant solution of Enter just a reduced fraction of form .
Enter just a reduced fraction of form . Unlock Deck
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9
Given the differential equation: , is this the solution
, is this the solution Unlock Deck
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10
Given the differential equation: , is this the solution
, is this the solution Unlock Deck
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11
Find f'(1) if f(t) is a solution to the initial value problem: Enter just an integer.
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12
Which of the following functions solves the differential equation:
A) y = + 3
B) y = + 3
C) y = - + 3x
D) none of these
A) y =
+ 3B) y =
+ 3C) y = -
+ 3xD) none of these
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13
Given the differential equation: y' is this the solution
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14
Find f'(0) if f(t) is a solution to the initial value problem: Enter just an integer.
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15
Given the differential equation: is this the solution
is this the solution Unlock Deck
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16
Given the differential equation: , is this the solution:
, is this the solution: Unlock Deck
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17
Find a constant solution of Enter just an integer.
Enter just an integer. Unlock Deck
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18
Given the differential equation: , is this the solution
, is this the solution Unlock Deck
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19
Which of the following functions solves the differential equation: ?
A) y =
B) y = -
C) y = ln 4t
D) none of these
?A) y =
B) y = -
C) y = ln 4t
D) none of these
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20
Write a differential equation that expresses the following description of a rate: When ice cream is removed from the freezer, it warms up at a rate proportional to the difference between the temperature of the ice cream and the room temperature of 76°. (Use y for the temperature of the ice cream, t for the time, and k for an unknown constant.)
A) y' = 76 - ky
B) y' = k(76 - y)
C) y' = k(76 - t)
D) y' = 76t - ky
A) y' = 76 - ky
B) y' = k(76 - y)
C) y' = k(76 - t)
D) y' = 76t - ky
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21
Solve the differential equation with the given initial condition.
A) y = - 2
B) y = 2 -1
C) y = + 1
D) y = t + 1
A) y =
- 2B) y = 2
-1C) y =
+ 1D) y = t + 1
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22
Solve the differential equation with the given initial condition.
A) y = - 2t + 1
B) y = - 2t
C) y =
D) y = 0
E) none of these
A) y =
- 2t + 1B) y =
- 2tC) y =
D) y = 0
E) none of these
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23
Given the differential equation with the given initial condition: is this the solution
is this the solution Unlock Deck
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24
Solve the differential equation with the given initial condition.
, y(0) = 0
A) y = -ln
B) y = 0
C) y =
D) y =
E) none of these
, y(0) = 0A) y = -ln
B) y = 0
C) y =
D) y =
E) none of these
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25
Given the differential equation with the given initial condition: is this the solution
is this the solution Unlock Deck
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26
Given the differential equation with the given initial condition: is this the solution
is this the solution Unlock Deck
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27
Solve the differential equation with the given initial condition.
A) y = 4 -
B) y =
C) y = +
D) y = 4 +
A) y = 4 -
B) y =
C) y =
+ D) y = 4 +
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28
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) = ky; There is not enough information given to determine initial conditions.
B) = ky; y(0) = 0
C) = k(1 - y); There is not enough information given to determine initial conditions.
D) = k(1 - y); y(0) = 0
E) = (1 + y); y(0) =0
A)
= ky; There is not enough information given to determine initial conditions.B)
= ky; y(0) = 0C)
= k(1 - y); There is not enough information given to determine initial conditions.D)
= k(1 - y); y(0) = 0E)
= (1 + y); y(0) =0 Unlock Deck
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29
Given the differential equation with the given initial condition: is this the solution
is this the solution Unlock Deck
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30
Solve the differential equation with the given initial condition.
A) t + 1
B)
C) t - 1
D)
A) t + 1
B)
C) t - 1
D)
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31
Find the constant solutions to the differential equation: .
Enter just one integer or two separated by a comma (no label).
.Enter just one integer or two separated by a comma (no label).
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32
Solve the differential equation with the given initial condition.
A) y = -
B) y =
C) y = 16 +
D) y = 64 - t
A) y =
- B) y =
C) y = 16 +
D) y = 64 -
t Unlock Deck
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33
Given the differential equation with the given initial condition: is this the solution
is this the solution Unlock Deck
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34
Given the differential equation with the given initial condition: is this the solution
is this the solution Unlock Deck
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35
Solve the differential equation with the given initial condition.
A) y = ln + 1
B) y = + 1
C) y = tan t + 1
D) y = +
A) y = ln
+ 1B) y =
+ 1C) y = tan t + 1
D) y =
+ Unlock Deck
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36
Solve the differential equation with the given initial condition.
A) y =
B) y = ±
C) y = ±
D) y =
A) y =
B) y = ±
C) y = ±
D) y =
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37
Given the differential equation: is this the solution
is this the solution Unlock Deck
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38
Given the differential equation with the given initial condition: is this the solution
is this the solution Unlock Deck
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39
Solve the differential equation with the given initial condition.
A) y =
B) y = -
C) y =
D) y = -1 +
A) y =
B) y = -
C) y =
D) y = -1 +
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40
Given the differential equation: is this the solution
is this the solution Unlock Deck
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41
Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. - 4y = -2 ; y(0) = -1
A) integrating factor:
General solution:
Particular solution: y = - - 2
B) integrating factor:
General solution: y = + C
Particular solution: y = - 2
C) integrating factor:
General solution: y = -2t + C
Particular solution: y = -2t -
D) integrating factor:
General solution: y = + C
Particular solution: y = -
Solve the equation using an integrating factor.
- 4y = -2 ; y(0) = -1A) integrating factor:
General solution:
Particular solution: y = -
- 2 B) integrating factor:
General solution: y =
+ C Particular solution: y =
- 2 C) integrating factor:
General solution: y = -2t + C
Particular solution: y = -2t -
D) integrating factor:
General solution: y =
+ C Particular solution: y =
- Solve the equation using an integrating factor.
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42
Solve the initial value problem using an integrating factor.
t + 3y = 5t; , t > 0
A) y = t -
B) y = + 1 -
C) y = 5 - 4t
D) y = 5 - 4
t
+ 3y = 5t; , t > 0A) y =
t - B) y =
+ 1 - C) y = 5
- 4tD) y = 5
- 4 Unlock Deck
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43
The annual sales y (in millions of dollars) of a company satisfy the differential equation . 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 annual sales.
B) The annual sales are increasing at $0.2 million ($200,000) per year.
C) The annual sales are decreasing at a rate proportional to the annual sales.
D) The annual sales are increasing at a rate proportional to $0.2 million ($200,000) per year.
. 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 annual sales.
B) The annual sales are increasing at $0.2 million ($200,000) per year.
C) The annual sales are decreasing at a rate proportional to the annual sales.
D) The annual sales are increasing at a rate proportional to $0.2 million ($200,000) per year.
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44
Combine the terms y and into the derivative of a product, then solve the equation. Is this the solution:
into the derivative of a product, then solve the equation. Is this the solution: Unlock Deck
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45
Solve the problem.
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 - 36,000 = 10)017 years
B) A = 80,000 - 56,000= 8)352 years
C) A = 80,000 - 56,000= 7)134 years
D) A = 60,000 - 36,000= 8)352 years
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 - 36,000
= 10)017 yearsB) A = 80,000 - 56,000
= 8)352 yearsC) A = 80,000 - 56,000
= 7)134 yearsD) A = 60,000 - 36,000
= 8)352 years Unlock Deck
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46
Combine the terms y and into the derivative of a product: .
Is this derivative correct: ?
into the derivative of a product: .Is this derivative correct:
? Unlock Deck
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47
Solve the equation using an integrating factor.
+ y = 4 , t > 0
A) y = 4x + C
B) y = 4x + C
C) y = 4 + C
D) y = 4 + C
+ y = 4 , t > 0A) y = 4x + C
B) y = 4x + C
C) y = 4 + C
D) y = 4 + C
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48
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:
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49
Solve the initial value problem using an integrating factor.
+ y = 3; y(0) = 0.
A) y = (3t - )
B) y =
C) y = 3 - 3
D) y = 3t
+ y = 3; y(0) = 0.A) y =
(3t - )B) y =
C) y = 3 - 3
D) y = 3t
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50
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) = , k < 0
B) = , k > 0
C) = , k < 0; y(0) =0
D) = , k > 0; y(0) = 0
E) none of these
A)
= , k < 0B)
= , k > 0C)
= , k < 0; y(0) =0D)
= , k > 0; y(0) = 0E) none of these
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51
Solve the equation using an integrating factor.
, t > 0
A) y = t + 1/2 + C
B) y = t - C
C) y = Ct +
D) y = 1/2 + C
, t > 0A) y = t + 1/2 + C
B) y = t - C
C) y = Ct +
D) y = 1/2 + C
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52
Combine the terms y and into the derivative of a product, then solve the equation. Is this the solution:
into the derivative of a product, then solve the equation. Is this the solution: Unlock Deck
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53
The annual sales y (in millions of dollars) of a company satisfy the differential equation . 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 $0.2 million ($200,000) per year to an upper limit of $5 million.
C) The annual sales are increasing at a rate proportional to the difference between the annual sales and an upper limit of $10 million.
D) The annual sales are increasing at a rate proportional to $0.2 million ($200,000) per year.
. 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 $0.2 million ($200,000) per year to an upper limit of $5 million.
C) The annual sales are increasing at a rate proportional to the difference between the annual sales and an upper limit of $10 million.
D) The annual sales are increasing at a rate proportional to $0.2 million ($200,000) per year.
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54
Solve the equation using an integrating factor.
, t > 0
A) y = C + C
B) y = C
C) y = C - 2
D) y = C
, t > 0A) y = C
+ CB) y = C
C) y = C
- 2D) y = C
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55
Suppose the relationship between the price p, of a product and the weekly sales, s, of the product is given by the differential equation Then
A) as sales increase, the price increases.
B) as the price increases the rate of change of the price also increases.
C) the rate of the decrease of the price is proportional to the sales.
D) s = 0 is a constant solution to this differential equation.
E) all of these
ThenA) as sales increase, the price increases.
B) as the price increases the rate of change of the price also increases.
C) the rate of the decrease of the price is proportional to the sales.
D) s = 0 is a constant solution to this differential equation.
E) all of these
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56
v
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 = 60,000 - 52,000= $31,041.84
B) A = 52,000 - 44,000= $27,969.59
C) A = 48,000 - 40,000= $26,433.47
D) A = 44,000 - 36,000= $24,897.34
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 = 60,000
- 52,000= $31,041.84B) A = 52,000
- 44,000= $27,969.59C) A = 48,000
- 40,000= $26,433.47D) A = 44,000
- 36,000= $24,897.34 Unlock Deck
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57
Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. 2t - y = ; y(1) = -1, t > 0
A) integrating factor:\
General solution: y = - + C
Particular solution: y = - +
B) integrating factor: 2t
General solution: y = - +
Particular solution: y = - +
C) integrating factor:
General solution: y = -4 + C
Particular solution: y = -4 + 3
D) integrating factor: 2t
General solution: y = - +
Particular solution: y = - +
- y = ; y(1) = -1, t > 0A) integrating factor:\
General solution: y = -
+ C Particular solution: y = -
+ B) integrating factor: 2t
General solution: y = -
+ Particular solution: y = -
+ C) integrating factor:
General solution: y = -4 + C
Particular solution: y = -4 + 3
D) integrating factor: 2t
General solution: y = -
+ Particular solution: y = -
+ Unlock Deck
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58
Solve the equation using an integrating factor.
, t > 0
A) y = + C
B) y = C
C) y = +
D) y = + C
, t > 0A) y =
+ C B) y = C
C) y =
+ D) y =
+ C Unlock Deck
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59
Solve the equation using an integrating factor.
, t > 0
A) y = C +
B) y = - + + C
C) y = +
D) y = C +
, t > 0A) y = C
+ B) y = -
+ + CC) y =
+ D) y = C
+ Unlock Deck
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60
Combine the terms y and into the derivative of a product: .
Is this derivative correct: ?
into the derivative of a product: .Is this derivative correct:
? Unlock Deck
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61
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 = a + c
T = a
+ c Unlock Deck
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62
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 689 seconds
B) approximately 703 seconds
C) approximately 460 seconds
D) approximately 276 seconds
A) approximately 689 seconds
B) approximately 703 seconds
C) approximately 460 seconds
D) approximately 276 seconds
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63
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.
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64
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)
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65
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:
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66
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
Let = . 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 only
B) I and II
C) II only
D) III only
E) none of these
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
Let
= . 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 only
B) I and II
C) II only
D) III only
E) none of these
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67
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).
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68
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
For what y value(s) does a solution of = - 3y + 2 have inflection points?
A) y = 2
B) y = 0
C) y = 2 and y = 1
D) y =
E) none of these
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
For what y value(s) does a solution of
= - 3y + 2 have inflection points?A) y = 2
B) y = 0
C) y = 2 and y = 1
D) y =
E) none of these
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69
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 ?
(I) f(t) is an increasing function
(II) f(t) is always positive
(III) f(t) has an inflection point when y = 2.
A) III only
B) I, II, and III
C) I only
D) II only
E) I and III
Which of the following can then be said about the solution y = f(t) to the initial value problem ? (I) f(t) is an increasing function
(II) f(t) is always positive
(III) f(t) has an inflection point when y = 2.
A) III only
B) I, II, and III
C) I only
D) II only
E) I and III
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70
A nutritionist proposes the following model for weight loss on a program she is developing: + 0.006w = 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) 41 days
B) 37 days
C) 39 days
D) 35 days
+ 0.006w = 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) 41 days
B) 37 days
C) 39 days
D) 35 days
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71
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
y = -3, y = 1, and y = 5 are the constant solutions to = g(y).
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
y = -3, y = 1, and y = 5 are the constant solutions to
= g(y). Unlock Deck
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72
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
Let = 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) I only
B) III only
C) I and III
D) II only
E) none of these
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
Let
= 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) I only
B) III only
C) I and III
D) II only
E) none of these
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73
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 the situation?
Do these graphs represent the situation? Unlock Deck
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74
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
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75
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
y = 2 is the only constant solution of = g(y) .
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
y = 2 is the only constant solution of
= g(y) . Unlock Deck
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76
Suppose = 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 IV
B) I and IV
C) II and III
D) I and III
E) not enough information given
= 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 IV
B) I and IV
C) II and III
D) I and III
E) not enough information given
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77
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
If the initial value of y(0) is greater than 6, then the corresponding solution will be an increasing function.
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
If the initial value of y(0) is greater than 6, then the corresponding solution will be an increasing function.
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78
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
If the initial value of y(0) is 3, then the corresponding solution has an inflection point.
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
If the initial value of y(0) is 3, then the corresponding solution has an inflection point.
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79
Suppose the graph below gives a solution to the differential equation = 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)
(III)
(IV) g( ) > 0
A) I only
B) IV only
C) I, III, and IV
D) I and IV
E) I, II, and IV
= 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)
(III)
(IV) g(
) > 0A) I only
B) IV only
C) I, III, and IV
D) I and IV
E) I, II, and IV
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80
Consider the differential equation = g(y) where g(y) is the function whose graph is shown below:
Indicate whether the following statements are true or false.
If the initial value of y(0) is 2, then the corresponding solution has an inflection point.
= g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false.
If the initial value of y(0) is 2, then the corresponding solution has an inflection point.
Unlock Deck
Unlock for access to all 126 flashcards in this deck.
Unlock Deck
k this deck
Unlock Deck
Unlock for access to all 126 flashcards in this deck.
