ME003/MA221: Differential Equations « Saylor.org – Free Online Courses Built by Professors

Differential Equations

Purpose of Course show close

Differential equations are, in addition to a topic of study in mathematics, the main language in which the laws and phenomena of science are expressed. In its most basic sense, a differential equation is an expression that describes how a system changes from one moment of time to another, or from one point in space to another. When working with differential equations, the ultimate goal is to move from a microscopic view of relevant physics to a macroscopic view of the behavior of a system as a whole. Let’s look at a simple differential equation. From previous math and physics courses, we know that a car that is constantly accelerating in the x-direction, for example, obeys the equation d²x/dt² = a, where a is the applied acceleration. This equation has two derivations with respect to time, so it is a second-order differential equation; because it has derivations with respect to only one variable (in this example, time), it is known as an ordinary differential equation, or an ODE. Let’s say that we want to solve the above ODE for the position of the car as a function of time. We can do so by using direct integration: the integration of both sides with respect to time gives us dx/dt = at + c, where c is a constant of integration. If the velocity of the car is known to be a particular value at some point in time T, we can solve for c as c = [dx/dt]_t=T / aT. More simply, if the velocity is zero at time 0, then c = 0. Integrating again gives us the desired solution: x(t) = at²/2 + ct + e, where e is another constant of integration. Again, if the position of the car at t=0 is taken to be zero, then the solution for the position of the car becomes x(t) = at²/2. It is useful to note that checking the validity of a solution to an ODE is easily accomplished by substituting it back into the ODE. Unfortunately, not all differential equations are this easy to solve. Generally, an ODE is a functional relation (it would be a function, except that the “variables” are themselves functions!) between an independent variable t, a dependent function U(t), and some of its derivatives dⁱU(t)/dtⁱ. An ODE is linear if it can be written as a functional relation in which no powers of U or its derivatives appear—otherwise, the ODE is nonlinear. For the most part, nonlinear ODEs can only be solved numerically; this course will focus on linear ODEs. This course will also introduce several other subclasses and their respective properties. However, despite centuries of study, the only practical approach to the solution of complicated ODEs that has emerged is numerical approximation. Although these numerical techniques are the subject of numerical analysis courses (see MA213: Numerical Analysis), this course will introduce you to the fundamentals behind numerical solutions. The prerequisites for this course are MA101, MA102, MA103, and MA211. Considerable motivation will be gained if PHYS101 and PHYS102 are also taken as pre- or co-requisites. This course will make use of a PDF text by Paul Dawkins of Lamar University as its principal reading material. You may wish to download this PDF at the outset of this course so that you have it on hand throughout.” You can find this file by clicking here and then looking for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).”

Learning Outcomes show close

Upon successful completion of this course, students will be able to:

Identify ordinary differential equations and their respective orders.
Explain and demonstrate how differential equations are used to model certain situations.
Solve first order differential equations as well as initial value problems.
Solve linear differential equations with constant coefficients.
Use power series to find solutions of linear differential equations.
Solve linear systems of differential equations with constant coefficients.
Yse the Laplace transform to solve initial value problems.
Use select methods of numerical approximation to find solutions to differential equations.

Course Requirements show close

In order to take this course you must:

√    Have access to a computer.

√    Have continuous broadband Internet access.

√    Have the ability/permission to install plug-ins or software (e.g., Adobe Reader or Flash).

√    Have the ability to download and save files and documents to a computer.

√    Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√    Be competent in the English language.

√    Have read the Saylor Student Handbook.

Unit Outline show close

1 Introduction to Ordinary Differential Equations
1.1 Introduction to the Study of ODEs
1.2 Classification of ODEs
1.3 Examples of ODEs and their Solutions
1.4 Linear ODEs and the Superposition Principle
1.5 Initial Value Problems
1.6 Boundary Value Problems

2 First Order ODEs
2.1 Examples of First-order ODEs
2.1.1 Radioactive Decay
2.1.2 Discharge of a Capacitor
2.1.3 Atmospheric Pressure
2.2 All Nth-order ODEs Are Systems of N First-order ODEs
2.3 Picard’s Existence Theorem for First Order ODEs
2.4 Direction Fields and Trajectories as Solutions
2.5 Linear ODEs
2.6 Exact ODEs and Integrating Factors
2.7 Separable ODEs
2.8 Bernoulli Equations
2.9 Homogeneous ODEs

3 Higher-Order Linear ODEs
3.1 Examples of Second-Order Linear Differential Equations
3.1.1 Newton’s Law of Motion
3.1.2 Motion of a Mass on a Spring
3.1.3 Motion with Air Resistance
3.1.4 One-Dimensional Time-Independent Schrödinger Equation
3.2 Higher Order Differential Equations
3.2.1 Homogeneous Linear ODEs with Constant Coefficients
3.2.2 Homogeneous Linear ODEs with Variable Coefficients
3.2.3 Euler-Cauchy ODEs
3.2.4 Non-homogeneous Linear ODEs – Finding Particular Solutions
3.3 Power Series Solutions of Linear Differential Equations
3.3.1 Power Series Solutions about an Ordinary Point
3.3.2 Singular Points: The Frobenius Method
3.3.3 Bessel’s Equation

4 Systems of Linear Differential Equations
4.1 Examples of Systems of ODEs
4.1.1 Mass and Spring Systems
4.1.2 The Predator-Prey Model
4.1.3 The Double Pendulum
4.2 Existence and Uniqueness Theorems
4.3 Conversion to Systems of First-Order ODEs
4.4 Linear Algebra-Based Solutions of Systems of Linear ODEs

5 Integral Transforms
5.1 Introduction to Integral Transforms
5.2 Discrete Spectra - The Laplace Transform
5.3 The Inverse Laplace Transform
5.4 Using Laplace Transforms to Solve Initial Value Problems

6 Numerical Methods
6.1 Verlet Integration
6.2 Predictor-Corrector Methods
6.3 Runge-Kutta Methods
6.4 Adams-Bashforth and Adams-Moulton Methods

7 Nonlinear ODEs
7.1 Riccati Equations
7.2 Nurgaliev’s Law
7.3 Lotka-Volterra Equation
7.4 Segway Dynamics – Inverted Pendulum
7.5 Chaotic Dynamics – The Double Pendulum

8 Final Exam

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Unit 1: Introduction to Ordinary Differential Equations

In this unit, we will examine ODEs, their use, and their types. We will also take a look at some classic ODE-based problems. While this course will focus on the solution of linear initial-value problems expressed as ODEs, you should know the limits and characteristics of the various classes of ODE as well.

Unit 1 Learning Outcomes show close

1.1 Introduction to the Study of ODEs

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read only the section titled “Differential Equation” on page 2. This reading will be used again in Unit 3.

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Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “0.2 Introduction to Differential Equations”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “0.2 Introduction to Differential Equations” (PDF)

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Instructions: Click on the link above and then click on the link that says “Download the book as PDF.” Please only read the content of section 0.2.1 on page 7.

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1.2 Classification of ODEs

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read only the sections titled “Order” and “Ordinary and Partial Differential Equations” on pages 2 and 3.

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1.3 Examples of ODEs and their Solutions

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts “
Link: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read only the section titled “Solution” on pages 3 to 5.

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Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “0.2 Introduction to Differential Equations”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “0.2 Introduction to Differential Equations” (PDF)

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Instructions: Click on the link above and then click on the link that says “Download the book as PDF.” Please only read the content of section 0.2.2 and 0.2.3 on page 7 to 11.

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Reading: MIT: Professor Haynes Miller’s 18.03 Supplementary Notes: “Chapter 1”
Link: MIT: Professor Haynes Miller’s 18.03 Supplementary Notes: “Chapter 1” (PDF)

Instructions: Click on the PDF linked above and read the whole chapter.

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Web Media: YouTube: Dr. Chris Tisdell’s “What is a Differential Equation?”
Link: YouTube: Dr. Chris Tisdell’s “What is a Differential Equation?” (YouTube)

Instructions: Watch this 23-minute video for a summary of some of the concepts that you have studied so far.

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1.4 Linear ODEs and the Superposition Principle

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read only the section titled “Linear Differential Equations” on page 3.

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Reading: Efunda’s “Higher Order Linear Differential Equations”
Link: Efunda’s “Higher Order Linear Differential Equations” (HTML)

Instruction: Please click on the above link and read the entire section.

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Web Media: Khan Academy: “What is a Differential Equation”
Link: Khan Academy: “What is a Differential Equation” (YouTube)

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Instructions: Please watch the lecture, which is an introduction to differential equations.

Watching this lecture should take approximately 10 minutes.

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1.5 Initial Value Problems

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read only the sections titled “Initial Condition,” “Initial Value Problem,” “Interval of Validity,” “Actual Solution,” and “Implicit/Explicit Solution,” which can be found on pages 5 to 7.

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1.6 Boundary Value Problems

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Boundary Value Problems & Fourier Series”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Boundary Value Problems & Fourier Series” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read only the first three pages of the section titled “Boundary Value Problems” on pages 375 to 377.

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Unit 2: First Order ODEs

In this unit, we examine the simplest class of ODEs: the linear first-order ODEs. There are at least three reasons for starting here. First, these are the simplest ODEs that exist. Second, they obey rather general existence and uniqueness theorems—that is, you can be determine whether a solution can be obtained, and can ensure that there is only one solution. Finally, any n^th order linear ODE can be converted into a system of n first-order ODEs. This is why the theory of first-order ODEs serves as a foundation for the entire field.

Unit 2 Learning Outcomes show close

2.1 Examples of First-order ODEs

2.1.1 Radioactive Decay

Reading: University of British Columbia: Leah Keshet’s Math 102 Course Notes: “Chapter 9”
Link: University of British Columbia: Leah Keshet’s Math 102 Course Notes: “Chapter 9” (PDF)

Instructions: Click on the link above, then click on the link for “9. Exponential Growth and Decay: Differential Equations.” Go to page 11. Please read sections 9.9 and 9.10 on pages 11 and 12.

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Reading: San Diego State University: Joseph M. Mahaffy’s Linear Differential Equations
Link: San Diego State University: Joseph M. Mahaffy’s Linear Differential Equations (HTML)

Instructions: Click on the link above and read the sections titled: “Introduction,” “Malthusian Growth,” and “Radioactive Decay.”

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Reading: Richard Williamson’s Introduction to Differential Equations and Dynamical Systems: “Supplementary Applications”: “Radioactive Decay”
Link: Richard Williamson’s Introduction to Differential Equations and Dynamical Systems: “Supplementary Applications”: “Radioactive Decay” (HTML)

Instructions: Click on the link above, then click on “1. Radioactive Decay (Chapter 2).” Read the entire section.

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2.1.2 Discharge of a Capacitor

Reading: Georgia State University: Department of Physics and Astronomy’s Hyperphysics: “Capacitor Discharge”
Link: Georgia State University: Department of Physics and Astronomy’s Hyperphysics: “Capacitor Discharge” (HTML)

Instructions: Click on the link above and the scroll down to the section titled: “Capacitor Discharge.”

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2.1.3 Atmospheric Pressure

Reading: Home Climate Analysis Blog’s “Atmospheric Pressure”
Link: Home Climate Analysis Blog’s “Atmospheric Pressure” (HTML)

Instructions: Click on the link above and read the entire document.

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Reading: Clinton Community College: Elizabeth Wood’s Math 155 Supplemental Notes 5: “Growth and Decay”
Link: Clinton Community College: Elizabeth Wood’s Math 155 Supplemental Notes 5: “Growth and Decay” (PDF)

Instructions: Read from the beginning to the end of “Example 1.”

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2.2 All Nth-order ODEs Are Systems of N First-order ODEs

Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Chapter 3”: “3.1 Introduction to Systems of ODEs”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Chapter 3”: “3.1 Introduction to Systems of ODEs “ (PDF)

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Instructions: Click on the link above and then click on the link titled “Download the book as PDF.” Go to page 85 and read only pages 85-88 from section 3.1.

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2.3 Picard’s Existence Theorem for First Order ODEs

Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Chapter 1”: “First Order ODE’s”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Chapter 1”: “First Order ODE’s” (PDF)

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Instructions: Click on the link above and then click on the link that says “Download the book as PDF.” Go to page 13. Please read sections 1.1 and 1.2 from pages 13 to 21. Note that section 1.2.1 will also cover subunit 2.4 below.

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Reading: SOS Math: “Existence and Uniqueness of Solution”
Link: SOS Math: “Existence and Uniqueness of Solution” (HTML)

Instructions: Click on the link above and read the entire section.

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Reading: SOS Math: “Picard Iterative Process”
Link: SOS Math: “Picard Iterative Process” (HTML)

Instructions: Click on the link above and read the entire section.

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2.4 Direction Fields and Trajectories as Solutions

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts”: “Direction Fields”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts”: “Direction Fields” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Please read the entire section titled “Direction Fields” on pages 8 to 18.

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Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “First Order ODE’s”: “1.2 Slopes Field”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “First Order ODE’s”: “1.2 Slopes Field” (PDF)

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Instructions: Click on the link above and then click on the link that says “Download the book as PDF.” Go to page 18. Read section 1.2 up to the end of subsection 1.2.1 on pages 18 and 19.

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Reading: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Notes”: “Graphical and Numerical Methods”
Link: MIT: Professoressor Arthur Mattuck’s 18.03 Notes and Exercises: “Notes”: “Graphical and Numerical Methods” (PDF)

Instructions: Please open the PDF above, and read the first two pages (Pages 0 and 1), from the beginning up to the end of the section titled “1. Graphical Methods.”

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1C. Graphical and Numerical Methods”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1C. Graphical and Numerical Methods” (PDF)

Instructions: Please click on the PDF above. Work through all items in exercise 1C-1 on pages 3 and 4. When you finish, please check your work for exercise 1C-1 with the answers found in “Section 1 Solutions”

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2.5 Linear ODEs

Reading: Lamar University: Paul Dawkins’ Differential Equations: “First Order Differential Equations”: “Linear Differential Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations: “First Order Differential Equations”: “Linear Differential Equations” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Linear Differential Equations” on pages 21 to 33. Try to work through the examples by yourself first and then read through the text to check your work.

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Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “First Order ODE’s”: “1.4 Linear Equations and the Integrating Factor”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “First Order ODE’s”: “1.4 Linear Equations and the Integrating Factor” (PDF)

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Instructions: Click on the link above and the click on the link that says “Download the book as PDF.” Go to page 27. Please read the section titled “1.4 Linear Equations and the Integrating Factor” on pages 27 to 30.

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Reading: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.3: First Order Differential Equations”
Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.3: First Order Differential Equations” (PDF)

Instructions: Click on the link above, scroll down to “Chapter 8: Differential Equations.” Under that title, click on the link for “First Order Linear Differential Equations.” Read the entire section.

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Assessment: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.3: First Order Differential Equations: “Problems”
Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations:“Chapter 8: Differential Equations”: “Section 8.3: First Order Differential Equations”: “Problems” (PDF)

Instructions: Click on the link above and scroll down to “Chapter 8: Differential Equations.” Then click on the link for “First Order Linear Differential Equations.” Go to page 5 and work with problems: 1-a, c, d; 2-a; 3-b, c, d; 5-b; 6-b, c. After you finish, click here for the solutions.

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Assessment: Carleton University: A. Mingarelli’s Calculus: “Functions and Their properties”: “Exercises: Differential Equations”
Link: Carleton University: A. Mingarelli’s Calculus: “Functions and Their Properties”: “Exercises: Differential Equations” (PDF)

Instructions: Click on the link above and scroll down until you see “Fall, Winter 2001, 2002. TENTATIVE OUTLINE.” Continue scrolling down until you reach the row corresponding to week 23. Go to the last cell in that row and click on the link “Exercises on Differential Equations.” Work with the exercises, looking at the answers only after you finish them yourself first.

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2.6 Exact ODEs and Integrating Factors

Reading: Lamar University: Paul Dawkins’ Differential Equations: “First Order Differential Equations”: “Exact Differential Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations: “First Order Differential Equations”: “Exact Differential Equations” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Please read the section titled: “Exact Differential Equations” on pages 45 to 55. Try to solve examples 2, 3, 4 and 5 on your own before reading the solution explained in the reading.

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Reading: Cliff Notes: “Differential Equations: “First Order Equations”: “Integrating Factors”
Link: Cliff Notes: “Differential Equations: “First Order Equations”: “Integrating Factors” (HTML)

Instructions: Click on the link above and read the entire article.

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Reading: Efunda’s “Exact First Order Differential Equations”
Link: Efunda’s “Exact First Order Differential Equations” (HTML)

Instructions: Click on the link above and read the entire page.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1B. Standard First-Order Methods”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1B. Standard First-Order Methods ” (PDF)

Instructions: Please, click on the PDF link above. Work with all items in exercise 1B-1 and 1B-2 on page 1. When you are finished, please compare your answers for these exercises with “Section 1 Solutions”.

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Assessment: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Exact Equations”
Link: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Exact Equations” (PDF)

Instructions: Click on the link above, then scroll down until you find the tutorial titled “Exact Differential Equations.” Click on the corresponding link. Go to page 4 and work on exercises 1 to 11 on pages 4 to 6. After finishing each exercise, click on the “EXERCISE” link for full worked solution.

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2.7 Separable ODEs

Reading: Lamar University: Paul Dawkins’ Differential Equations: “First Order Differential Equations”: “Separable Differential Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations: “First Order Differential Equations”: “Separable Differential Equations” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Separable Differential Equations” on pages 34 to 44. Try to solve examples 3, 4, 5, and 6 on your own. After finishing, check your work against the solution provided by the text.

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Reading: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.2: Separation of Variables”
Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations:“Chapter 8: Differential Equations”: “Section 8.2: Separation of Variables” (PDF)

Instructions: Click on the link above, scroll down to “Chapter 8: Differential Equations.” Click on the link for “Separation of Variables.” Read the entire section.

Terms of Use: The work above is released under a Creative Commons Attribution-Share-Alike License 1.0 (HTML). It is attributed to Dan Sloughter, and the original version can be found here (PDF).

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Assessment: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.2: Separation of Variables: Problems”
Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations:“Chapter 8: Differential Equations”: “Section 8.2: Separation of Variables”: “Problems” (PDF)

Instructions: Click on the link above and scroll down to “Chapter 8: Differential Equations.” Click on the link for “Separation of Variables.” Go to page 7 and work through the following problems: 1-a, 1-c, 1-e, 1-g, 2-a, 2-c, 3-a, 4-a, 4-c, 4-e, and 5-b. After you finish, check your work against the solutions provided here.

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Assessment: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Separation of Variables”
Link: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Separation of Variables” (PDF)

Instructions: Click on the link above, then scroll down until you find the tutorial titled “Separation of Variables.” Click on the corresponding link. Go to page 4 and work on exercises 1 to 16 on pages 4 to 8. After finishing each exercise, click on the “EXERCISE” link for full worked solution.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1A. Introduction; Separation of Variables”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1A. Introduction; Separation of Variables” (PDF)

Instructions: Please click on the PDF above. Work through all items in exercises 1A-3, 1A-4, and 1A-5 on page 1. When you finish please check your answers for these exercises against “Section 1 Solutions”
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2.8 Bernoulli Equations

Reading: Lamar University: Paul Dawkins’ Differential Equations: “First Order Differential Equations”: “Bernoulli Differential Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations: “First Order Differential Equations”: “Bernoulli Differential Equations” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Bernoulli Differential Equations” on pages 56 to 62. Try to work through examples 2, 3, and 4 on your own before reading through the text.

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Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “First Order ODE’s”: “1.5 Substitution”: “1.5.2 Bernoulli Equations”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “First Order ODE’s”: “1.5 Substitution”: “1.5.2 Bernoulli Equations” (PDF)

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Instructions: Click on the link above and then click on the link titled “Download the book as PDF.” Go to page 33. Please read the section titled: “1.5.2 Bernoulli Equations” on pages 33 and 34.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1B. Standard First-Order Methods”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1B. Standard First-Order Methods” (PDF)

Instructions: Click on the link above and then scroll down to the section titled “Exercises.” Click on the PDF link for “1. First-order ODE’s.” Work with the two items in exercise 1B-9 on page 2. When you finish, please compare your answers to these exercises with “Section 1 Solutions”

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Assessment: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Bernoulli Equations”
Link: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Bernoulli Equations” (PDF)

Instructions: Click on the link above, then scroll down until you find the tutorial titled “Bernoulli Equations.” Click on the corresponding link. Go to page 4 and work on exercises 1 to 9 on pages 4 to 6. After finishing each exercise, click on the “EXERCISE” link for full worked solution.

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2.9 Homogeneous ODEs

Reading: Lamar University: Paul Dawkins’ Differential Equations: “First Order Differential Equations”: “Substitutions”
Link: Lamar University: Paul Dawkins’ Differential Equations: “First Order Differential Equations”: “Substitutions” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Substitutions” on pages 63 to 67, up to the end of the example 2.

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Web Media: Khan Academy: “First Order Homogeneous Equations”
Link: Khan Academy: “First Order Homogeneous Equations” (YouTube)

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Instrucitons: Please watch the lecture, which is about first order homogenous equations.

Watching this lecture should take approximately 10 minutes.

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Web Media: Khan Academy: “First Order Homogeneous Equations 2”
Link: Khan Academy: “First Order Homogeneous Equations 2” (YouTube)

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Instrucitons: Please watch the lecture, which is about first order homogenous equations.

Watching this lecture should take approximately 10 minutes.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1B. Standard First-Order Methods”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1B. Standard First-Order Methods” (PDF)

Instructions: Please click on above PDF link. Work with all the items in exercise 1B-3 on page 2. When you finish, please check your answers for these exercises with “Section 1 Solutions.”

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Assessment: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Homogeneous Functions”
Link: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Homogeneous Functions” (PDF)

Instructions: Click on the link above, then scroll down until you find the tutorial titled “Homogeneous Functions.” Click on the corresponding link. Go to page 5 and work on exercises 1 to 11 on pages 5 to 8. After finishing each exercise, click on the “EXERCISE” link for full worked solution.

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Unit 3: Higher-Order Linear ODEs

While one can always convert a higher-order ODE into a system of first-order ODEs, this is not always the best approach toward solving the ODE. Homogeneous linear ODEs with constant coefficients are a particularly easy-to-solve class of ODEs. In order to find a solution of an N^th-order linear ODE with constant coefficients, you need to find the roots of an N^th-order polynomial.

Finding solutions is not quite as simple for the case of linear ODEs with variable coefficients—that is, with coefficients that are functions of the independent variable. These ODEs are not generally solvable in closed form, but methods for solving some special cases do exist.

The general solution to non-homogeneous linear ODEs is the addition of the general solution of the related homogeneous ODE plus a particular solution of the non-homogeneous ODE. There are a number of approaches to finding a particular solution, and we will sample several of the most straightforward. However, finding particular solutions is less a process of following an algorithm than it is an art form.

You are already familiar with the idea that nearly any “well-behaved” function can be approximated by a power series. You can use this concept to find solutions to a wide range of ODEs. In most cases, the solution itself is expressed as a power series, a fact that has led to the development of an enormous field of applied mathematics known as “special functions theory,” to which we will return later. In this unit, we will examine the solution of a specific example—Bessel’s equation (which was actually introduced by Bernoulli!).

Unit 3 Learning Outcomes show close

3.1 Examples of Second-Order Linear Differential Equations

3.1.1 Newton’s Law of Motion

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Basic Concepts” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read only the section titled “Differential Equation” on page 2.

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Reading: NYU: Mark Tuckerman’s: “G25.2600: Molecular Dynamics”: “Notes for Lecture 1”: “Newton’s Laws of Motion”
Link: NYU: Mark Tuckerman’s: “G25.2600: Molecular Dynamics”: “Notes for Lecture 1”: “Newton’s Laws of Motion” (HTML)

Instructions: Click on link above and read the notes included.

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3.1.2 Motion of a Mass on a Spring

Web Media: YouTube: Jason Gregersen’s “Modeling Spring Motion Using Differential Equations Part One”
Link: YouTube: Jason Gregersen’s “Modeling Spring Motion Using Differential Equations Part One” (YouTube)

Instructions: Click on the link above to watch this video (5:11 minutes), which models spring motion (assuming that there is no air resistance).

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3.1.3 Motion with Air Resistance

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Mechanical Vibrations”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Mechanical Vibrations” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to page 162 and read the section titled “Mechanical Vibrations,” (pages 162-164).

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3.1.4 One-Dimensional Time-Independent Schrödinger Equation

Reading: Georgia Institute of Technology: C. David Sherrill’s A Brief Review of Elementary Quantum Chemistry: “The Schrödinger Equation”: “The Time-Independent Schrödinger Equation”
Link: Georgia Institute of Technology: C. David Sherrill’s A Brief Review of Elementary Quantum Chemistry: “The Schrödinger Equation”: “The Time-Independent Schrödinger Equation” (HTML)

Instructions: Click on the link above and read the entire section.

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3.2 Higher Order Differential Equations

Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators and Higher Order ODE’s”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators and Higher Order ODE’s” (PDF)

Instructions: Please click on the link above and go to page 6. Work with exercise 2F-1 (items c, d, e, and f,); exercise 2F-2; and exercise 2F-3, item d. When you finish, please check your answers with “Section II Solutions”

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3.2.1 Homogeneous Linear ODEs with Constant Coefficients

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Second Order Differential Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Second Order Differential Equations” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to Page 104 and read the sections titled: “Basic Concepts,” “Real, Distinct Roots,” “Complex Roots,” and “Repeated Roots,” pages 104-121. Then go to page 126 and read the sections titled “Fundamental Sets of Solutions” and “More on the Wronskian” (pages 126-136).

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Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Higher Order Linear ODE’s”: “2.3 Higher Order Linear ODE’s”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Higher Order Linear ODE’s”: “2.3 Higher Order Linear ODE’s” (PDF)

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Instructions: Click on the link above and then click on the link that says “Download the book as PDF.” Go to page 57. Please read the section titled “2.3 Higher Order Linear ODE’s” (pages 57-60).

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Assessment: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Second Order (Homogeneous)”
Link: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: “Second Order (Homogenous)” (PDF)

Instructions: Click on the link above, then scroll down until you find the tutorial titled “Second Order (Homogeneous).” Click on the corresponding link. Go to page 5 and work on exercises 1 to 16 on pages 5 to 7. After finishing each exercise, click on the “EXERCISE” link for full worked solution.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2C. Second Order Linear ODE’s with Constant Coefficients”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2C. Second Order Linear ODE’s with Constant Coefficients” (PDF)

Instructions: Please click on the link above and go to page 3. Work with exercises 2C-1 and 2C-2. When you finish, please check your answers with “Section II Solutions”

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3.2.2 Homogeneous Linear ODEs with Variable Coefficients

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Reduction of Order”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Reduction of Order” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to page 122 and read the section titled “Reduction of Order” (pages 122-125).

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Assessment: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 17”: “17.7 Additional Exercises”
Link: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 17”: “17.7 Additional Exercises” (PDF)

Instructions: Please click on the link above and go to page 955. Work on exercises 17.18, 17.19, 17.20, and 17.21. After finishing each exercise, click on the “solution” link.

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3.2.3 Euler-Cauchy ODEs

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Series Solutions to Differential Equations”: “Euler Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Series Solutions to Differential Equations”: “Euler Equations” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to page 340 and read the section titled: “Euler Equations” (pages 340-344).

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Activity: University of Hartford: Virginia Noonburg’s M344 Advanced Engineering Mathematics: “Lecture 5”
Link: University of Hartford: Virginia Noonburg’s M344 Advanced Engineering Mathematics: “Lecture 5” (PDF)

Instructioons: Click on the link above and scroll down until you see “M344 Advanced Engineering Math.” Then click on the link titled “Lecture 5: Cauchy-Euler Equations, Method of Frobenius.” Go to page 4 and look for “Practice Problems.” Complete items a and b under problem 2 and then check for the answers.

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Assessment: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 17”: “17.7 Additional Exercises”
Link: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 17”: “17.7 Additional Exercises” (PDF)

Instructions: Click on the link above. Click on “PDF (Portable Document Format).” Go to page 953. Work on exercises 17.11, 17.12, 17.13, and 17.14. After finishing each exercise, click on the “solution” link.

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3.2.4 Non-homogeneous Linear ODEs – Finding Particular Solutions

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Nonhomogeneous Differential Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Nonhomogeneous Differential Equations” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Nonhomogeneous Differential Equations” on pages 137-138.

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3.2.4.1 Linear Differential Operators

Reading: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Notes”: “Linear Differential Operators”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Notes”: “Linear Differential Operators” (PDF)

Instructions: Please click on the link above and read pages 1 to 5 of “Linear Differential Operators.”

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators and Higher Order ODE’s”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators and Higher Order ODE’s” (PDF)

Instructions: Please click on the PDF linked above and go to page 6. Work with exercise 2F-1 items “a” and “b”, and exercise 2F-3 items “a” and “b”. When you finish, please check your answers for these exercises with “Section II Solutions.”

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3.2.4.2 Trial Solutions – Method of Undetermined Coefficients

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Undetermined Coefficients”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Undetermined Coefficients” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Undetermined Coefficients” on pages 139-155.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Undetermined Coefficients”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Undetermined Coefficients” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Undetermined Coefficients” on pages 355-356.

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Assessment: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: Graham McDonald’s “Second Order (Inhomogeneous)”
Link: University of Salford: “Mathematics Hyper-Tutorials”: “Ordinary Differential Equations Math Tutorials”: Graham McDonald’s “Second Order (Inhomogenous)” (PDF)

Instructions: Click on the link above, then scroll down until you find the tutorial titled “Second Order (Inhomogeneous).” Click on the corresponding link. Go to page 5 and work on exercises 1 to 13 on pages 5 and 6. After finishing each exercise, click on the “EXERCISE” link for full worked solutions.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2C. Second Order Linear ODE’s with Constant Coefficients”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2C. Second Order Linear ODE’s with Constant Coefficients” (PDF)

Instructions: Please click on the PDF linked above and go to page 3. Work with exercises 2C-7 and 2C-8 on page 3. When you finish, please check your answers for these exercises with “Section II Solutions.”

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators and Higher Order ODE’s”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2F. Linear Operators and Higher Order ODE’s” (PDF)

Instructions: Please click on the PDF linked above and go to page 6. Work with exercise 2F-6. When you finish, please check your answers to this exercise with “Section II Solutions.”

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3.2.4.3 Variation of Parameters

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Variation of Parameters”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Second Order Differential Equations”: “Variation of Parameters” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Variation of Parameters” on pages 156-161.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Variation of Parameters”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Variation of Parameters” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled: “Variation of Parameters” on pages 357-362.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2D. Variation of Parameters”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Higher-Order ODE’s”: “2D. Variation of Parameters” (PDF)

Instructions: Please click on the PDF linked above and go to page 4. Work with exercises 2D-1 and 2D-2 on page 4. When you finish, please check your answers to these exercises with “Section II Solutions.”

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Assessment: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 21”: “21.10 Exercises”
Link: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 21”: “21.10 Exercises” (PDF)

Instructions: Click on the link above. Click on “PDF (Portable Document Format).” Go to page 1117 and work on exercises 21.3, 21.4, and 21.5. After finishing each exercise, click on the “solution” link.

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3.2.4.4 Reduction of Order

Reading: CliffsNotes: “Reduction or Order”
Link: CliffsNotes: “Reduction or Order” (HTML)

Instruction: Please click on the above link and read the entire article.

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Reading: Efunda’s “Higher Order Linear Differential Equations”: “Method of Reduction of Order”
Link: Efunda’s “Higher Order Linear Differential Equations”: “Method of Reduction of Order” (HTML)

Instruction: Please click on the above link and read the entire article.

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3.2.4.5 Method of Inverse Operators

Reading: Efunda’s “Higher Order Linear Differential Equations”: “Inverse Operators”
Link: Efunda’s “Higher Order Linear Differential Equations”: “Inverse Operators” (HTML)

Instruction: Please click on the above link and read the entire section.

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3.3 Power Series Solutions of Linear Differential Equations

Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Power Series Methods”: “7.1 Power Series”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Power Series Methods”: “7.1 Power Series” (PDF)

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Instructions: Click on the link above and then click on the link that says “Download the book as PDF.” Go to page 261. Please read the section titled “7.1 Power Series”
on pages 261-268.

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3.3.1 Power Series Solutions about an Ordinary Point

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Series Solutions to Differential Equations”: “Series Solutions to Differential Equations”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Series Solutions to Differential Equations”: “Series Solutions to Differential Equations” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Series Solutions to Differential Equations” on pages 330-339.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Series Solutions”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Higher Order Differential Equations”: “Series Solutions” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Series Solutions” on pages 370-373.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Power Series”: “6C. Solving Second-order ODE’s”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Power Series”: “6C. Solving Second-order ODE’s” (PDF)

Instructions: Pleas click on the PDF linked above and then go to page 2. Work with exercises 6C-2, 6C-3, 6C-4, 6C-5, 6C-6 and 6C-7. When you finish, please check your work with “Section II Solutions.”

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3.3.2 Singular Points: The Frobenius Method

Reading: California State University, East Bay: Massoud Malek’s Differential Equations: “Series Solutions of Linear Differential Equations”
Link: California State University, East Bay: Massoud Malek’s Differential Equations: “Series Solutions of Linear Differential Equations” (PDF)

Instructions: Click on the link above. Scroll down until you find “Series Solutions of LDE.” Click on that link and read the entire document.

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Reading: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 23”: “23.2 Regular Singular Points of Second Order Equations”
Link: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 23”: “23.2 Regular Singular Points of Second Order Equations” (PDF)

Instructions: Click on the link above and then click on “PDF (Portable Document Format).” Go to page 1198. Read the section titled “23.2 Regular Singular Points of Second Order Equations” on pages 1198-1215.

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Assessment: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 23”: “23.5 Exercises”
Link: Caltech: Sean Mauch’s Introduction to Methods of Applied Mathematics: “Ordinary Differential Equations”: “Chapter 23”: “23.5 Exercises” (PDF)

Instructions: Click on the link above and then click on “PDF (Portable Document Format).” Go to page 1220 and work on exercises 23.3, 23.5, 23.6, and 23.7. After finishing each exercise, click on the “solution” link.

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3.3.3 Bessel’s Equation

Reading: University of Hartford: Virginia Noonburg’s M344 Advanced Engineering Mathematics: “Lecture 6: Bessel’s Equation”
Link: University of Hartford: Virginia Noonburg’s M344 Advanced Engineering Mathematics: “Lecture 6: Bessel’s Equation” (PDF)

Instructions: Click on the link above and scroll down until you see “M344 Advanced Engineering Math.” Under that title, click on the link that reads: “Lecture 6: Bessel’s Equation.” Read the entire lecture.

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Assessment: Holon Institute of Technology: Professor Benzion Shklyar’s “Solution of Bessel Equation with v=1/3”
Link: Holon Institute of Technology: Professor Benzion Shklyar’s “Solution of Bessel Equation with v=1/3” (PDF)

Instructions: Click on the link above and scroll down until you find “Bessel Equation.” Under that title, click on the link for “Solution of Bessel Equation with v=1/3.” Read the question and work on it on your own. After you finish, look under the question for all the steps of the solution.

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Unit 4: Systems of Linear Differential Equations

In the previous units, we worked with differential equations with one unknown function. However, some problems involve several functions that depend on a single variable. In these cases, a system of ordinary differential equations can be created to model the problem.

The concept of the differential operator will help us develop techniques for solving systems of linear ODEs. Briefly, if we define the differentiation operator D_x = d/dx, so that D_xY = dY/dx, a general linear ordinary differential equation can be written as a polynomial in D_x. This allows us to form an equivalence between a system of linear ODEs and a matrix equation, which can then be solved as an eigenvalue problem using linear algebra. This is quite a boon, as most problems in linear algebra can indeed be solved!

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4.1 Examples of Systems of ODEs

4.1.1 Mass and Spring Systems

Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.1 Introduction to Systems of ODEs”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.1 Introduction to Systems of ODEs” (PDF)

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Instructions: Click on the link above and then click on the link that says “Download the book as PDF.” Go to page 85 and read the section titled “3.1 Introduction to Systems of ODEs” on pages 85-88.

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4.1.2 The Predator-Prey Model

Reading: Scholarpedia.org: Frank Hoppensteadt’s “Predator-Prey Model”
Link: Scholarpedia.org: Frank Hoppensteadt’s “Predator-Prey Model” (PDF)

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Instructions: Click on the link above and read the entire article.

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4.1.3 The Double Pendulum

Reading: The MathWorks: Cleve Moler’s Numerical Computing with MATLAB: “Ordinary Differential Equations”
Link: The MathWorks: Cleve Moler’s Numerical Computing with MATLAB: “Ordinary Differential Equations” (PDF)

Instructions: Click on the link above. Click on the link “Ordinary Differential Equations (53 pages).” Go to page 49 and read the item numbered “7.23.” Read only up to page 50.

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4.2 Existence and Uniqueness Theorems

Reading: Michigan State University: Sheldon Newhouse’s Math 235 Lecture Notes: “16. Systems of Differential Equations”
Link: Michigan State University: Sheldon Newhouse’s Math 235 Lecture Notes: “16. Systems of Differential Equations” (PDF)

Instructions: Click on the link above, scroll down, and, under the title “Lecture Notes,” click on the link for “16. Systems of Differential Equations.” Go to page 5. Read the section titled “2 Systems of Differential Equations” on pages 5-11.

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Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.3 Linear Systems of ODEs”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.3 Linear Systems of ODEs” (PDF)

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Instructions: Click on the link above and then click on the link that says “Download the book as PDF.” Go to page 99. Please read the section titled “3.3 Linear Systems of ODEs” on pages 99-102.

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4.3 Conversion to Systems of First-Order ODEs

Reading: Wikipedia’s Ordinary Differential Equation: “Reduction to a First Order System”
Link: Wikipedia’s Ordinary Differential Equation: “Reduction to a First Order System” (PDF)

Instructions: Click on the link above. Read the sections titled: “Reduction to a First Order System” and “Linear Ordinary Differential Equations.”

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4B. General Systems; Elimination; Using Matrices”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4B. General Systems; Elimination; Using Matrices” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 4B-1 and 4B-2 on page 1. When you finish, please check your answers to these exercises with “Linear Systems Solutions.”

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4.4 Linear Algebra-Based Solutions of Systems of Linear ODEs

Reading: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.4 Eingenvalue Method”
Link: Jirka.org: Jiri Lebl’s Notes on Diffy Qs: “Systems of ODEs”: “3.4 Eingenvalue Method” (PDF)

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Instructions: Click on the link above and then click on the link that says “Download the book as PDF.” Go to page 99. Please read the section titled “3.4 Eingenvalue Method” on pages 103-109.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4B. General Systems; Elimination; Using Matrices”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4B. General Systems; Elimination; Using Matrices” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 4B-4, 4B-5, and 4B-6 on pages 1 and 2. When you finish, please check your answers for these exercises with “Linear Systems Solutions.”

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Assessment: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4C. Eingenvalues and Eigenvectors”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4C. Eingenvalues and Eigenvectors” (PDF)

Instructions: Click on the PDF linked above and go to page 2. Work with exercises 4C-1 and 4C-5. When you finish, please check your answers to these exercises with “Linear Systems Solutions.”

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4D. Complex and Repeated Eingenvalues”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4D. Complex and Repeated Eingenvalues” (PDF)

Instructions: Please click on the PDF linked above, and go to page 3. Work with exercises 4D-1, 4D-2, and 4D-3. When you finish, please check your answers to these exercises with “Linear Systems Solutions.”

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Systems of Differential Equations”: “Nonhomogeneous Systems”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Systems of Differential Equations”: “Nonhomogeneous Systems” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to Page 303 and read the section titled “Nonhomogeneous Systems” on pages 303-306.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4I. Inhomogeneous Systems”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Linear Systems”: “4I. Inhomogeneous Systems” (PDF)

Instructions: Please click on the PDF linked above and go to page 6. Work with exercises 4I-1, 4I-2, 4I-3, 4I-4, and 4I-5. When you finish, please check your answers to these exercises with “Linear Systems Solutions.”

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Unit 5: Integral Transforms

Integral transforms form another class of tools that can be used to convert linear differential equations into algebraic equations. An integral transform is a mathematical operator that transforms a function on which it operates into another function through integration over a kernel. Often, by applying an integral transform, a complex function can be represented as a sum of functions (usually known as special functions) that are simpler to manipulate in the context of the underlying ODE. This technique is closely related to the earlier discussion of power series solutions.

A wide variety of integral transforms find application in the study of linear differential equations. One useful feature of integral transforms is spectral factorization, or the representation of an arbitrary function as the sum of a series of orthogonal basis functions. The differential equation is thereby transformed into an algebraic problem, which can be solved using linear algebra. Once the solution is obtained, the solution to the original problem can be found by applying an inverse integral transform.

Unit 5 Learning Outcomes show close

5.1 Introduction to Integral Transforms

Reading: Dublin City University: Eugene O’Riordan’s “MS227. Linear Mathematics”: “Laplace Transform”
Link: Dublin City University: Eugene O’Riordan’s “MS227. Linear Mathematics”: “Laplace Transform” (PDF)

Instructions: Click on the link above. Under the title “MS227 Linear Mathematics,” click on the link for “Laplace Transform.” Read the entire document.

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5.2 Discrete Spectra - The Laplace Transform

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “The Definition”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “The Definition” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Then read the section titled: “The Definition” on pages 183-186.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Laplace Transforms”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Laplace Transforms” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to Page 187 and read the section titled: “Laplace Transforms” on pages 187-190. Work on examples 1 and 2 on your own before looking at the solutions. After finishing with your work, look at the solutions.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3A. Elementary Properties and Formulas”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3A Elementary Properties and Formulas” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 3A-1, 3A-2, 3A-4, 3A-5, 3A-7, 3A-8, and 3A-9. When you finish, please check your answers to these exercises with “Section 3 Solutions.”

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5.3 The Inverse Laplace Transform

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Inverse Laplace Transforms”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Inverse Laplace Transforms” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled: “Inverse Laplace Transforms” on pages 191 to 201. Work on examples 2 and 3 on your own before looking at the solutions. After finishing with your work, look at the solutions.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Step Functions”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Step Functions” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled: “Laplace Transforms” on pages 202 to 214. Work on examples 2 and 3 on your own before looking at the solutions. After finishing with your work, look at the solutions.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Dirac Delta Function”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Dirac Delta Function” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to page 233 and read the “Dirac Delta Function” section.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Convolution Integrals”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Convolution Integrals” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to Page 236. Read the section titled “Convolution Integrals,” beginning on page 236, and ending with example 1 on page 237. Then, go to page 239 and read pages 239 and 240. The table on page 239 will be used for the exercises, so make sure that you have it available when working with the assignments.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3A. Elementary Properties and Formulas”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3A Elementary Properties and Formulas” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 3A-3 and 3A-10. When you finish, please check your answers to these exercises with “Section 3 Solutions.”

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5.4 Using Laplace Transforms to Solve Initial Value Problems

Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Solving IVP’s with Laplace Transforms”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Solving IVP’s with Laplace Transforms” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Go to page 215 and read the section titled “Solving IVP’s with Laplace Transforms” on pages 215 to 221. Work on examples 2 and 3 on your own. After you finish, look at the solutions in the reading.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Non Constant Coefficients IVP’s”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Non Constant Coefficients IVP’s” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “Non Constant Coefficients IVP’s” on pages 222 to 225. Work on example 2 on your own. After you finish, look at the solution provided in the reading.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “IVP’s with Steps Functions”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “IVP’s with Steps Functions” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read the section titled “IVP’s with Steps Functions” on pages 226 to 232. Work on examples 2 and 3 on your own. After you finish, look at the solution in the reading.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Dirac Delta Function”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Dirac Delta Function” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read pages 234 and 235. Work on example 2 on your own. After you finish, look at the solution in the reading.

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Reading: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Convolution Integrals”
Link: Lamar University: Paul Dawkins’ Differential Equations: “Laplace Transforms”: “Convolution Integrals” (PDF)

Instructions: Click on the link above and then look for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).” Read pages 237 (start with example 2) and 238.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3B. Derivative Formulas; Solving ODE’s”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3B. Derivative Formulas; Solving ODE’s” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 3B-1, 3B-3, 3B-4, 3B-5 and 3B-6. When you finish, please check your answers to these exercises with “Section 3 Solutions.”

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3C. Discontinuous Functions”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3C. Discontinuous Functions” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 3C-1, 3C-2, 3C-3, 3C-4 and 3C-5. When you finish, please check your answers to these exercises with “Section 3 Solutions.”

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3D. Convolution and Delta Function”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “Laplace Transform”: “3D. Convolution and Delta Function” (PDF)

Instructions: Please click on the PDF linked above. Work with exercises 3D-1, 3D-2, 3D-3, 3D-4, 3D-5, 3D-6, 3D-7, and 3D-8. When you finish, please check your answers to these exercises with “Section 3 Solutions.”

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Unit 6: Numerical Methods

Until now, we have studied particular kinds of equations for which we have methods to find analytical solutions. In most cases, however, it is impossible to find such solutions, and we must accordingly rely on methods of numerical approximation.

Reasonably enough, the technique for obtaining a solution for an ODE that can be written as a system of first-order ODEs is numerical integration. Roughly speaking, the value for the derivative of the desired solution is numerically integrated in order to obtain a solution to the ODE. As numerical integration is a rather stable process (meaning that the result can be made arbitrarily close to the value of the actual integral), automated solver programs can numerically integrate a wide range of ODEs. Note that nonlinear ODEs can be solved by numerical integration! Despite the scope of numerical methods, however, it is important to appreciate their limitations.

Unit 6 Learning Outcomes show close

6.1 Verlet Integration

Reading: Wikipedia’s “Verlet Integration”
Link: Wikipedia’s “Verlet Integration” (PDF)

Instructions: Click on the link above and read the entire article.

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6.2 Predictor-Corrector Methods

Reading: Wikipedia’s “Predictor-Corrector Method”
Link: Wikipedia’s “Predictor-Corrector Method” (PDF)

Instructions: Click on the link above and read the entire article.

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6.3 Runge-Kutta Methods

Reading: MIT: “Honors Differential Equations”: Ernest Ngaruiya’s “Numerical Approximations in Differential Equations: The Runge-Kutta Method”
Link: MIT: “Honors Differential Equations”: Ernest Ngaruiya’s “Numerical Approximations in Differential Equations: The Runge-Kutta Method” (PDF)

Instructions: Click on the link above. Scroll down and click on the link for “18034 Honors Differential Equations.” Read the entire paper.

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Reading: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.1: Numerical Solutions of Differential Equations”
Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.1: Numerical Solutions of Differential Equations” (PDF)

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Instructions: Click on the link above and scroll down to “Chapter 8: Differential Equations.” Then click on the link for “Numerical Solutions of Differential Equations” and read the entire section.

Terms of Use: The work above is released under a Creative Commons Attribution-Share-Alike License 1.0 (HTML). It is attributed to Dan Sloughter, and the original version can be found here (PDF).

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Reading: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.1: Numerical Solutions of Differential Equations”: “Problems”
Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations:“Chapter 8: Differential Equations”: “Section 8.1: Numerical Solutions of Differential Equations”: “Problems” (PDF)

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Instructions: Click on the link above and scroll down to “Chapter 8: Differential Equations.” Then click on the link for “Numerical Solutions of Differential Equations.” Go to page 10 and work with problems: 4-a, c, e, g; 5-a, c, e, g; 7-b, c, d; 9-b, c, d. After you finish, check your work against the solutions provided here.

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6.4 Adams-Bashforth and Adams-Moulton Methods

Reading: Wapedia’s “Wiki: Linear Multistep Method”
Link: Wapedia’s “Wiki: Linear Multistep Method” (PDF)

Instructions: Click on the link above and read the entire article.

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Unit 7: Nonlinear ODEs

This course has concentrated on the properties and solutions of linear ODEs, but we have also seen methods that apply to the occasional nonlinear ODE. While the detailed study of solution methods for nonlinear ODEs lies beyond the scope of this course, it is worthwhile looking at a few examples.

Unit 7 Learning Outcomes show close

7.1 Riccati Equations

Web Media: SOS Math: Mohamed Amine Khamsi’s “Ricatti Equations”
Link: SOS Math: Mohamed Amine Khamsi’s “Ricatti Equations” (HTML)

Instructions: Click on the link above. Read the entire article. Try to work through the examples on your own before looking at the solutions in the article.

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Assessment: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1B. Standard First Order Methods”
Link: MIT: Professor Arthur Mattuck’s 18.03 Notes and Exercises: “Exercises”: “First Order ODE’s”: “1B. Standard First Order Methods” (PDF)

Instructions: Please click on the PDF linked above and go to page 2. Work with all items in exercises 1B-10. When you finish, please check your answers to this exercise with “Section 1 Solutions.”

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7.2 Nurgaliev’s Law

Reading: Wikipedia’s “Nurgaliev’s Law”
Link: Wikipedia’s “Nurgaliev’s Law” (HTML)

Instructions: Click on the link above and read the entire article.

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7.3 Lotka-Volterra Equation

Reading: Wikipedia’s “Lotka-Volterra Equation”
Link: Wikipedia’s “Lotka Volterra Equation” (HTML)

Instructions: Click on the link above and read the entire article.

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7.4 Segway Dynamics – Inverted Pendulum

Reading: Wikipedia’s “Inverted Pendulum”
Link: Wikipedia’s “Inverted Pendulum” (HTML)

Instructions: Click on the link above and read the entire article.

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7.5 Chaotic Dynamics – The Double Pendulum

Reading: Wikipedia’s “Double Pendulum”
Link: Wikipedia’s “Double Pendulum” (PDF)

Instructions: Click on the link above and read the entire article.

Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to Wikipedia and the original version can be found here (HTML).
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Final Exam

Final Exam: The Saylor Foundation's "MA211 Final Exam"
Link: The Saylor Foundation's "MA211 Final Exam" (HTML)

Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
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