Deck 10: Differential Equations

Question 1

Consider the differential equation y' = y - $y ^ { 2 }$ . Which of the following statements is/are true?

A) The function f(t) = $\frac { 1 } { \left( 1 + e ^ { - t } \right) }$ is a solution to this differential equation with initial condition $y ( 0 ) = \frac { 1 } { 2 }$
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 $f ^ { \prime } ( 0 ) = 0$
E) All of these statements are true.

Answer 1

All of these statements are true.

Question 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

Answer 2

I and III

Question 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

Answer 3

y = -ln(cos t + C

Question 4

Given the differential equation: , is this the solution

Question 5

Which of the following functions solves the differential equation:

A) y = -
B) y = -
C) y = ln
D) y =

Question 6

Find f'(1) if f(t) is a solution to the initial value problem: Enter just a real number (no approximations).

Question 7

Which of the following functions solves the differential equation: ?

A) y =
B) y =
C) y = 7
D) y = 7

Question 8

Find a constant solution of Enter just a reduced fraction of form .

Question 9

Given the differential equation: , is this the solution

Question 10

Given the differential equation: , is this the solution

Question 11

Find f'(1) if f(t) is a solution to the initial value problem: Enter just an integer.

Question 12

Which of the following functions solves the differential equation:

A) y = + 3
B) y = + 3
C) y = - + 3x
D) none of these

Question 13

Given the differential equation: y' is this the solution

Question 14

Find f'(0) if f(t) is a solution to the initial value problem: Enter just an integer.

Question 15

Given the differential equation: is this the solution

Question 16

Given the differential equation: , is this the solution:

Question 17

Find a constant solution of Enter just an integer.

Question 18

Given the differential equation: , is this the solution

Question 19

Which of the following functions solves the differential equation: ?

A) y =
B) y = -
C) y = ln 4t
D) none of these

Question 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

Question 21

Solve the differential equation with the given initial condition.

A) y = - 2
B) y = 2 -1
C) y = + 1
D) y = t + 1

Question 22

Solve the differential equation with the given initial condition.

- $y ^ { \prime } = y ( t - 2 ) ; y ( 0 ) = 1$

A) y = $e \left[ \left( t ^ { 2 } / 2 \right) - 2 t \right]$ - 2t + 1
B) y = $\frac { t ^ { 2 } } { 2 }$ - 2t
C) y = $e \left[ \left( t ^ { 2 } / 2 \right) - 2 t \right]$
D) y = 0
E) none of these

Question 23

Given the differential equation with the given initial condition: is this the solution

Question 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

Question 25

Given the differential equation with the given initial condition: is this the solution

Question 26

Given the differential equation with the given initial condition: is this the solution

Question 27

Solve the differential equation with the given initial condition.

A) y = 4 -
B) y =
C) y = +
D) y = 4 +

Question 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

Question 29

Given the differential equation with the given initial condition: is this the solution

Question 30

Solve the differential equation with the given initial condition.

- $y ^ { \prime } = y ^ { 2 } , y ( 0 ) = 1$

A) t + 1
B) $\frac { 1 } { t + 1 }$
C) t - 1
D) $\frac { 1 } { 1 - t }$

Question 31

Find the constant solutions to the differential equation: .
Enter just one integer or two separated by a comma (no label).

Question 32

Solve the differential equation with the given initial condition.

A) y = -
B) y =
C) y = 16 +
D) y = 64 - t

Question 33

Given the differential equation with the given initial condition: is this the solution

Question 34

Given the differential equation with the given initial condition: is this the solution

Question 35

Solve the differential equation with the given initial condition.

A) y = ln + 1
B) y = + 1
C) y = tan t + 1
D) y = +

Question 36

Solve the differential equation with the given initial condition.

A) y =
B) y = ±
C) y = ±
D) y =

Question 37

Given the differential equation: is this the solution

Question 38

Given the differential equation with the given initial condition: is this the solution

Question 39

Solve the differential equation with the given initial condition.

A) y =
B) y = -
C) y =
D) y = -1 +

Question 40

Given the differential equation: is this the solution

Question 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.

Question 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

Question 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.

Question 44

Combine the terms y and into the derivative of a product, then solve the equation. Is this the solution:

Question 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

Question 46

Combine the terms y and into the derivative of a product: .
Is this derivative correct: ?

Question 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

Question 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:

Question 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

Question 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

Question 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

Question 52

Combine the terms y and into the derivative of a product, then solve the equation. Is this the solution:

Question 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.

Question 54

Solve the equation using an integrating factor.
, t > 0

A) y = C + C
B) y = C
C) y = C - 2
D) y = C

Question 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

Question 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

Question 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 = - +

Question 58

Solve the equation using an integrating factor.
, t > 0

A) y = + C
B) y = C
C) y = +
D) y = + C

Question 59

Solve the equation using an integrating factor.
, t > 0

A) y = C +
B) y = - + + C
C) y = +
D) y = C +

Question 60

Combine the terms y and into the derivative of a product: .
Is this derivative correct: ?

Question 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

Question 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

Question 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.

Question 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)

Question 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:

Question 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

Question 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).

Question 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

Question 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

Question 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

Question 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).

Question 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

Question 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?

Question 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

Question 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) .

Question 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

Question 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.

Question 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.

Question 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

Question 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.