**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.*

**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.

```
Terms of Use: The linked material above has been reposted with the
kind permission of Leah Keshet. Please note that this material is
under copyright and cannot be reproduced in any capacity without
explicit permission from the copyright holder.
```

**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.”

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.”

Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.

**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.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.

**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.”Terms of Use: The linked material above had been reposted with the kind permission of Elizabeth Wood, and can be viewed in its original form here. Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

**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)

Also available in:

HTML

```
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA221-book-Notes-on-Diffy-Qs-Jiri-Lebl.epub)
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.
Terms of Use: The work above is released under a [Creative Commons
Attribution-Share-Alike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/us/) (HTML).
It is attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/diffyqs/) (PDF).
```

**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)

Also available in:

HTML

```
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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.
Terms of Use: The work above is released under a [Creative Commons
Attribution-Share-Alike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/us/) (HTML).
It is attributed to Jiri Lebl and the original version can be found
[here](http://www.jirka.org/diffyqs/) (PDF)
```

**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.Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.

**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)

Also available in:

HTMLInstructions: 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**Link: MIT: Professor Arthur Mattuck’s*18.03 Differential Equations*: Notes and Exercises: “Graphical and Numerical Methods”*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.”

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.**Assessment: MIT: Professor Arthur Mattuck’s**Link: MIT: Professor Arthur Mattuck’s*18.03 Differential Equations*: Notes and Exercises: “First Order ODE’s”: “1C. Graphical and Numerical Methods”*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.Terms of Use: Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

**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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displayed on the webpages above.

**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)

Also available in:

HTMLEPUB

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.

**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.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.

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

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

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**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”.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**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.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.

**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.**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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**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.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.

**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)

Also available in:

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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”

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**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.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.

**Web Media: Khan Academy: “First Order Homogeneous Equations”**Link: Khan Academy: “First Order Homogeneous Equations” (YouTube)Also available in:

iTunes UInstrucitons: 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)Also available in:

iTunes U

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.”

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.**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.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.