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MA221: Differential Equations

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 nth 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.

Unit2 Learning Outcomes
Upon successful completion of this unit, the student should be able to:
- Find the solution for first-order ordinary differential equations. - Find the solution for first-order ordinary differential equations within applications involving radioactive decay. - Find the solution for first-order ordinary differential equations within applications involving the discharge of a capacitor. - Find the solution for first-order differential equations within applications involving atmospheric pressure. - Rewrite an Nth-order ordinary differential equation as a system of N first-order ordinary differential equations. - Determine the existence of a solution for a first-order ordinary differential equation using Picard’s Existence Theorem. - Analyze directional fields and trajectories to evaluate the qualitative behavior of solutions to selected ordinary differential equations. - Define linear ordinary differential equations. - Distinguish linear ordinary differential equations from nonlinear ordinary differential equations. - Find the solution for an exact ordinary differential equation. - Find the solution for separable ordinary differential equations. - Find the solution for Bernoulli differential equations. - Find the solution for homogeneous ordinary differential equations.

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.

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

     

  • Reading: MIT: Professor Arthur Mattuck’s 18.03 Differential Equations: Notes and Exercises: “Graphical and Numerical Methods” Link: MIT: Professor Arthur Mattuck’s 18.03 Differential Equations: Notes and Exercises: “Graphical and Numerical Methods” (PDF)
     
    Instructions: Read the first two pages (Pages 0 and 1) of these notes, up to the end of the section titled “1. Graphical Methods.”
     
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  • Assessment: MIT: Professor Arthur Mattuck’s 18.03 Differential Equations: Notes and Exercises: “First Order ODE’s”: “1C. Graphical and Numerical Methods” Link: MIT: Professor Arthur Mattuck’s 18.03 Differential Equations: Notes and Exercises: “First Order ODE’s”: “1C. Graphical and Numerical Methods” (PDF)
     
    Instructions: Work through all items in exercise 1C-1 on pages 3 and 4 of the document.  When you finish, check your work with the answers in “Section 1 Solutions”, starting on page 9. 

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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.

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

     

  • 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.

    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).

     

  • 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.

    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).

     

  • Assessment: Carleton University: A. Mingarelli’s Calculus: “Functions and Their properties”: “Exercises: Differential Equations” The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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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: “Integrating Factors” Link: Cliff Notes: Differential Equations: “Integrating Factors” (HTML)
     
    Instructions: Click on the link above and read this 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).

     

  • 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.

    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).

     

  • 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” 

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.

    Terms of Use: The work above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML).  It is attributed to Jiri Lebl, and the original version can be found here (PDF). 

     

  • 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.

    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.

  • 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.

    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.

  • 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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