MA221-FinalExam-Answers

Which of the following statements about ordinary differential equations is true?

Choose one answer.

	a. They may contain one independent variable.
	b. They are composed of one or more derivatives with respect to the independent variable.
	c. They do not contain partial derivatives.
	d. All of the above

Question 2

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 3

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 4

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 5

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 6

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 7

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 8

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 9

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 10

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 11

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 12

Which of the following relationships is an ordinary differential equation?

Choose one answer.

	a.
	b.
	c.
	d.

Question 13

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 14

Determine the order of the following differential equation.

Choose one answer.

	a. 1
	b. 2
	c. 3
	d. 4

Question 15

Which of the following classifications pertain to ordinary differential equations?

Choose one answer.

	a. Linear
	b. Nonlinear
	c. Homogeneous
	d. All of the above

Question 16

Consider the differential equation

. Suppose that a student determined the solution to be

. Based upon this information, is the student correct?

Choose one answer.

	a. No, the student should have determined the solution to be .
	b. No, the student should have determined the solution to be .
	c. No, the student should have determined that there is no solution.
	d. Yes, the student did not make any errors.

Question 17

Apply the superposition principle to find the solution for

Choose one answer.

	a.
	b.
	c.
	d.

Question 18

Find a particular solution for the following differential equation for which

Choose one answer.

	a.
	b.
	c.
	d. There is no solution.

Question 19

Find a particular solution for

which passes through the origin and through the point

Choose one answer.

	a.
	b.
	c.
	d.

Question 20

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 21

Find a 1-parameter family of solutions for the following differential equation.

Choose one answer.

	a.
	b.
	c.
	d. There is no solution.

Question 22

Find a particular solution for

where

Choose one answer.

	a.
	b.
	c.
	d. There is no solution.

Question 23

The rate at which a radioactive substance decays is proportional to the remaining number of atoms. The differential equation which can be used to describe this process follows:

represents the number of atoms remaining after

seconds. The proportionality constant

is considered the decay constant for this process. If

represents the number of remaining atoms at

seconds, find the general solution.

Choose one answer.

	a.
	b.
	c.
	d.

Question 24

For a circuit containing a capacitor and resistor, the first-order differential equation that describes a discharging capacitor follows:

represents the charge within the capacitor at

seconds, find the general solution.

is the capacitance of the capacitor.

is the resistance of the resistor.

Choose one answer.

	a.
	b.
	c.
	d.

Question 25

Earth’s atmospheric pressure

can be modeled by the following first-order differential equation.

This differential equation describes the rate at which atmospheric pressure changes with altitude

above sea level.

is a constant. If

represents the atmospheric pressure (millibars) at sea level (

kilometers), find the general solution.

Choose one answer.

	a.
	b.
	c.
	d.

Question 26

Consider

where

is defined on the interval

. It is possible to rewrite this second-order differential equation as a system of two first-order differential equations using an appropriate change of variable. Using

as the new variable, how can

be rewritten as a system of two first-order differential equations?

Choose one answer.

	a.
	b.
	c.
	d.

Question 27

Using Picard’s Method of Successive Approximations, attempt to find the first four Picard approximations to the solution for

to determine whether or not a solution exists for this first-order differential equation. It is given that

Choose one answer.

	a. ; the solution exists.
	b. ; the solution exists.
	c. ; the solution does not exist.
	d. ; the solution does not exist.

Question 28

Construct the direction field for

using integer coordinates from -5 to 5. Analyze the general shape and trajectories. Which of the following represents the equation for a possible integral curve?

Choose one answer.

	a.
	b.
	c.
	d. None of the above

Question 29

Which of the following statements about linear ordinary differential equations is true?

Choose one answer.

	a. Linear ordinary differential equations are of the form where and are either constants or functions of .
	b. Linear ordinary differential equations are of the form where and are either constants or functions of .
	c. Linear ordinary differential equations do not contain partial derivatives.
	d. All of the above

Question 30

Which of the following is a linear ordinary differential equation?

Choose one answer.

	a.
	b.
	c.
	d. None of the above

Question 31

Find the general solution for the following exact differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 32

Find the particular solution for the following separable differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 33

Find the general solution for the following Bernoulli differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 34

Find the particular solution for the following homogeneous differential equation.

where

and

Choose one answer.

	a.
	b.
	c.
	d.

Question 35

Find the implicit solution to the following differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 36

Solve the following differential equation by separating it:

Choose one answer.

	a.
	b.
	c.
	d.

Question 37

Solve the following differential equation by separating it:

Choose one answer.

	a.
	b.
	c.
	d.

Question 38

Find the solution for the following differential equation for

Choose one answer.

	a.
	b.
	c.
	d.

Question 39

Find the solution for the following second-order ordinary differential equation.

where

and

Choose one answer.

	a.
	b.
	c.
	d.

Question 40

Consider a mass m that oscillates at the end of a spring having a spring constant

. The following second-order differential equation describes the vertical displacement y of this spring-mass system.

This differential equation neglects the influence of air resistance or frictional forces. Find the general solution which describes the vertical displacement of this spring as a function of time

Choose one answer.

	a.
	b.
	c.
	d.

Question 41

Consider a mass m that oscillates at the end of a spring having a spring constant k. The following second-order differential equation describes the vertical displacement y of this spring-mass system.

Such a differential equation implies that mass m, once started, will simply oscillate up and down forever! This differential equation neglects the influence of frictional forces. Let us assume that there is a retarding force which is proportional to the velocity of motion where s is treated as a proportionality constant (

). The aforementioned differential equation now becomes the following.

Let

and

where

.
There are three possible types of solutions which depend upon the relative size of

and

, including the following:
Overdamped if

,
Critically damped if

,
Underdamped or oscillatory if

.
Find the general solution for this differential equation, which describes the vertical displacement of this spring as a function of time t for “overdamped” motion.

Choose one answer.

	a.
	b.
	c.
	d.

Question 42

Consider a mass m that oscillates at the end of a spring having a spring constant k. The following second-order differential equation describes the vertical displacement y of this spring-mass system.

Let

and

where

.
There are three possible types of solutions which depend upon the relative size of

and

, including the following:
Overdamped if

,
Critically damped if

,
Underdamped or oscillatory if

.
Find the general solution for this differential equation, which describes the vertical displacement of this spring as a function of time t for the “critically damped” motion.

Choose one answer.

	a.
	b.
	c.
	d.

Question 43

The one-dimensional time-independent Schrodinger equation is given by the following.

is a wave function which describes the displacement

of a single particle of mass

where

is the total energy,

is the potential energy, and

is a well-known constant. Since

for the particle-in-a-box model where

, this second-order differential equation becomes the following.

Find the general solution for this differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 44

Find the general solution for this higher-order differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 45

Find the general solution for this homogeneous ordinary differential equation with constant coefficients (zero right hand side).

Choose one answer.

	a.
	b.
	c.
	d.

Question 46

Find the general solution for this ordinary differential equation with constant coefficients (nonzero right hand side).

Choose one answer.

	a.
	b.
	c.
	d.

Question 47

Find the general solution for this homogeneous ordinary differential equation with variable coefficients. This is an example of an Euler-Cauchy differential equation (zero right hand side).

Choose one answer.

	a.
	b.
	c.
	d.

Question 48

Find the general solution for this ordinary differential equation with variable coefficients. This is an example of an Euler-Cauchy differential equation (nonzero right hand side).

Choose one answer.

	a.
	b.
	c.
	d.

Question 49

Find the particular solution for the following non-homogeneous ordinary differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 50

Apply the linear differential operator

to evaluate the following expression.

Choose one answer.

	a.
	b.
	c.
	d.

Question 51

Find the general solution for the following differential equation using the method of undetermined coefficients.

Choose one answer.

	a.
	b.
	c.
	d.

Question 52

Which of the following expressions is of the correct form to serve as a trial solution

for the following differential equation?

Choose one answer.

	a.
	b.
	c.
	d.

Question 53

Find the general solution for the following differential equation using the method of variation of parameters.

Choose one answer.

	a.
	b.
	c.
	d.

Question 54

Which of the following statements regarding the method of undetermined coefficients and variation of parameters is true?

Choose one answer.

	a. The method of variation of parameters is a much more general method than the method of undetermined coefficients which can be used in many cases.
	b. Although the complementary solution is absolutely required when using the method of variation of parameters, it is not required when using the method of undetermined coefficients.
	c. When using the method of variation of parameters, it is possible that the acquired integrals cannot be determined.
	d. All of the above

Question 55

Find the general solution for the following ordinary differential equation using a method called the reduction of order.

Choose one answer.

	a.
	b.
	c.
	d.

Question 56

Given that

is a solution of

where

, it is possible to use a method called reduction of order to find the general solution of the following ordinary differential equation.

Which of the following expressions would represent

within such a determination?

Choose one answer.

	a.
	b.
	c.
	d.

Question 57

Find the particular solution

and complementary function

for the following ordinary differential equation using the method of inverse operators.

Choose one answer.

	a.
	b.
	c.
	d.

Question 58

Suppose that you were asked to find the particular solution

and complementary function

, using the method of inverse operators, for the following ordinary differential equation.

Identify the inverse operator

, which might arise within your solution to this type of problem.

Choose one answer.

	a.
	b.
	c.
	d.

Question 59

Find the power series general solution for the following ordinary differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 60

Which of the following statements are true with regard to a power series solution to the ordinary differential equation

Choose one answer.

	a. x = 0 and 1 are ordinary points.
	b. x = 0 and 1 are regular singularities.
	c. x = 0 and 1 are irregular singularities.
	d. Insufficient information has been provided to distinguish whether x = 0 and 1 are ordinary points, regular singularities, or irregular singularities.

Question 61

Find the series solution for an ordinary differential equation using the Frobenius method.

Choose one answer.

	a.
	b.
	c.
	d.

Question 62

Which of the following statements most accurately describes how the general solution depends upon the order

of the Bessel equation shown below?

Choose one answer.

	a. In order to satisfy the Bessel equation, the general solution must be , when is a positive non-integer.
	b. In order to satisfy the Bessel equation, the general solution must be , when is a positive integer.
	c. In order to satisfy the Bessel equation, the general solution must be , when is a positive non-integer.
	d. In order to satisfy the Bessel equation, the general solution must be, when is a positive integer.
	e. A and B

Question 63

Which of the following statements most accurately describes how the general solution for the Bessel equation shown below can depend upon a variety of well-known functions?

Choose one answer.

	a. When is a positive non-integer, where and can be expressed in terms of a Gamma function.
	b. When is a positive integer, where can be expressed in terms of a Neumann function.
	c. When is a positive integer, where can be expressed in terms of and .
	d. All of the above

Question 64

Find the solution for the following differential equation for

and

Choose one answer.

	a.
	b.
	c.
	d.

Question 65

Find the solution for the following differential equation for

and

Choose one answer.

	a.
	b.
	c.
	d.

Question 66

Find the solution for the following differential equation for which

and

Choose one answer.

	a.
	b.
	c.
	d.

Question 67

Find the particular solution for the following non-homogeneous ordinary differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 68

Find the particular solution for the following non-homogeneous ordinary differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 69

Find the solution for the following higher-order differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 70

Find the solution for the following higher-order differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 71

Find the general solution for the following ordinary differential equation using the variation of parameters method.

Choose one answer.

	a.
	b.
	c.
	d.

Question 72

Find the general solution for the following ordinary differential equation using the method of variation of parameters.

Choose one answer.

	a.
	b.
	c.
	d.

Question 73

Find the general solution for the following differential equation using the method of variation of parameters.

Choose one answer.

	a.
	b.
	c.
	d.

Question 74

Find the general solution for the following non-homogeneous ordinary differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 75

Find the general solution for the following non-homogeneous ordinary differential equation.

Choose one answer.

	a.
	b.
	c.
	d.

Question 76

Find the general solution for the following system of ordinary differential equations.

Choose one answer.

	a.
	b.
	c.
	d.

Question 77

Suppose that you were given two tanks filled with brine that were connected by pipes. At a rate

of 10 gallons per minute, the brine is flowing through the first pipe from the first tank to the second tank. At the same rate

, the brine is flowing through the second pipe from the second tank to the first tank. The initial volume of brine in the first tank is 50 gallons, and the amount of salt is 15 pounds. The second tank initially contains 25 gallons of pure water. If

represents the amount of salt in the first tank and

represents the amount of salt in the second tank as a function of time

, which of the following statements is true regarding the system of ordinary differential equations which model this system?

Choose one answer.

	a.
	b.
	c. pounds, pounds
	d. All of the above

Question 78

Suppose that you were tasked with creating a system of ordinary differential equations to model predator-prey dynamics. Let x and y denote the number of prey (e.g., rabbits) and predators (e.g., foxes), respectively, as a function of time

, where

, and

represent parameters that describe how some predator and prey interact with each other. Which of the following systems of first-order nonlinear ordinary differential equations describe such a system?

Choose one answer.

	a.
	b.
	c.
	d.

Question 79

Consider a mass m that oscillates at the end of a spring having a spring constant

. The following second-order differential equation describes the vertical displacement

of this spring-mass system.

Such a differential equation implies that mass

, once started, will simply oscillate up and down forever! This differential equation neglects the influence of frictional forces. Let us assume that there is a retarding force which is proportional to the velocity of motion where

is treated as a proportionality constant (

). The aforementioned differential equation now becomes the following.

Let

and

where

.
There are three possible types of solutions which depend upon the relative size of

and

, including the following:
Overdamped if

,
Critically damped if

,
Underdamped or oscillatory if

.
Which of the following differential equations can be used to describe one of these cases?

Choose one answer.

	a.
	b.
	c.
	d. Each general solution corresponds to one of the given cases.

Question 80

Which of the following statements is true regarding the Lotka-Volterra Predator-Prey Model shown below?

Choose one answer.

	a. , , , and are positive constants.
	b. is the size of the prey population at time .
	c. is the size of the predator population at time .
	d. All of the above

Question 81

Which of the following statements is true regarding the differential equations used to describe the double pendulum shown below?

Choose one answer.

	a. are all vector quantities.
	b. represent mass.
	c. represent the length of rigid weightless rods.
	d. represent tension.
	e. All of the above

Question 82

Using existence and uniqueness theorems, identify which of the following statements is true regarding the ordinary differential equation

, where

Choose one answer.

	a. There exists a solution.
	b. The solution exists in some open interval centered at 0.
	c. The solution exists and is unique in some (possibly smaller) interval centered at 0.
	d. All of the above

Question 83

It is possible to convert an nth order differential equation into an n-dimensional system of first-order differential equations. For the following 4th-order differential equation, identify the corresponding 4-dimensional system of first-order ordinary differential equations.

Use the following four new variables

to make this determination.

Choose one answer.

	a.
	b.
	c.
	d.

Question 84

There are a variety of methods borrowed from linear algebra that are very useful for finding the solution for systems of linear ordinary differential equations. One such method includes the usage of matrices. For the following system of linear ordinary differential equations, use these methods to arrive at the solution in matrix form.

Choose one answer.

	a.
	b.
	c.
	d.

Question 85

Find the general solution for the following system of ordinary differential equations.

Choose one answer.

	a.
	b.
	c.
	d.

Question 86

Find the general solution for the following system of ordinary differential equations.

Choose one answer.

	a.
	b.
	c.
	d.

Question 87

Which of the following statements describe how integral transforms can be used to reduce the complexity of selected classes of mathematics problems?

Choose one answer.

	a. An integral transform is an operator (e.g., map one function to another function).
	b. The use of selected integral transforms can be used to turn differential equations subject to particular boundary conditions into much simpler algebra problems.
	c. The solution that arises as a result of the use of selected integral transforms must be inverted to represent the solution of the original differential equation.
	d. All of the above

Question 88

Using Laplace transforms, find the solution for the following ordinary differential equation.

, where

and

Make sure to also identify the specific Laplace transform

that was used to arrive at your final solution.

Choose one answer.

	a.
	b.
	c.
	d.

Question 89

Which of the following statements best describes spectra associated with selected Laplace transforms?

Choose one answer.

	a. It is possible to use a Laplace transform power spectrum as a basic tool for the complex-exponential decomposition of finite-duration continuous functions.
	b. It is possible to use a Laplace transform power spectrum within selected applications which include pole-zero estimation.
	c. It is not possible to use a Laplace transform power spectrum as a basic tool for the complex-exponential decomposition of finite-duration continuous functions.
	d. A and B are both correct.

Question 90

Find the inverse Laplace transform for

Choose one answer.

	a.
	b.
	c.
	d.

Question 91

Find the solution for the following initial value problem using Laplace transforms.

, where

and

Choose one answer.

	a.
	b.
	c.
	d.

Question 92

Using Laplace transforms, find the solution for the following ordinary differential equation.

, where

and

Make sure to also identify the specific Laplace transform

that was used to arrive at your final solution.

Choose one answer.

	a.
	b.
	c.
	d.

Question 93

Using Laplace transforms, find the solution for the following ordinary differential equation.

, where

and

Make sure to also identify the specific Laplace transform

that was used to arrive at your final solution.

Choose one answer.

	a.
	b.
	c.
	d.

Question 94

The Verlet Method can be used to approximate the solutions for Newton’s Laws of Motion. If r denotes the position of a particle as a function of time with acceleration a where t represents some small time increment, determine which of the following expressions represents an approximate solution for the position of this particle.

Choose one answer.

	a.
	b.
	c.
	d.

Question 95

Predictor-Corrector Methods can be used to approximate the solution for a variety of ordinary differential equations. For the following differential equation, determine which of the following expressions represents an approximate solution. Let h denote the step size where the trapezoidal rule is employed.

where

Choose one answer.

	a.
	b.
	c.
	d.

Question 96

The Runge-Kutta Method can be used to approximate the solution for a variety of ordinary differential equations. For the following differential equation, determine which of the following expressions represents an approximate solution using the Runge-Kutta 4th Order Method.

where

Let h denote the step size where …

Choose one answer.

	a.
	b.
	c.
	d.

Question 97

The Adam-Bashforth and Adams-Moulton Methods can be used to approximate the solution for a variety of ordinary differential equations. For the following differential equation, determine an approximate solution for

using the 4 Point Adams-Bashforth Method.

where

and the step size is 0.1.

Choose one answer.

	a. -1.308179
	b. 0.308179
	c. 1.308179
	d. 2.308179

Question 98

Find an exact solution for the Lotka-Voltera equations shown below.

Which of the following represent this solution in parametric form?

Choose one answer.

	a. and where is a constant.
	b. and where is a constant.
	c. and where is a constant.
	d. and where is a constant.

Question 99

Which of the following statements correctly describes how nonlinear ordinary differential equations can be used to model selected inverted pendulum dynamics?

Choose one answer.

	a. They can be used to model inverted pendulum dynamics involving a stationary pivot point.
	b. They can be used to model inverted pendulum dynamics where the inverted pendulum is on a cart consisting of a horizontally moving base.
	c. They can be used to model inverted pendulum dynamics where the inverted pendulum is connected to a massless oscillating base.
	d. All of the above

Question 100

Which of the following statements correctly describes how nonlinear ordinary differential equations can be used to model selected double pendulum dynamics?

Choose one answer.

	a. They can be used to model double pendulums composed of two simple pendulums.
	b. They can be used to model double pendulums composed of two compound pendulums.
	c. They can be used to model double pendulums composed of two limbs of equal or unequal lengths and/or masses
	d. All of the above

Question 101

Find the general solution for the Riccati equation shown below.

Choose one answer.

	a.
	b.
	c.
	d.

Question 102

Which of the following statements describe how Nurgaliev’s law can be used to model population dynamics?

Choose one answer.

	a. The solution for this differential equation can be used to make predictions regarding the size of the population as a function of time which is measured in years.
	b. The solution for this differential equation takes into consideration the probability associated with the birth of males or females within a given year.
	c. The solution for this differential equation takes into consideration the probability associated with the death of a person within a given year.
	d. All of the above