Differential Equations
Math 244, Spring 2021

Textbook

Same book as last semester. There are a couple of different versions of the 9th edition (see below).

Elementary Differential Equations and Boundary Value Problems, by William Boyce and Richard DiPrima (9th Edition)

TEXT NOTE: This is NOT the current version, which is the 11th edition. There are a couple of different versions of the text; here are the links to Amazon.com for the information:
With Boundary Value Problems
The text without Boundary Value Problems
There also is an optional Student Solution Manual Available (typically, these have more detailed solutions to odd numbered problems)
We'll generally be covering topics from Chapters 1-7 except Chapter 4. Also, topics from Chapter 7, 9 as time permits.

Beginning Material

Daily Information and Links

Review Material - Read these at your leisure.

Week 1: Jan 25-29 Introduction and 2.1.
- Monday, Jan 25
  - Video Introduction (1.3), Part 1
  - Video Introduction (1.3, Part 2)
  - Video Introduction (1.3-1.4, Part 3)
  - Video Introduction (1.3-1.4, Part 4)
  - Homework: Section 1.3: 1-9 odd, 11* 14*, 15, 17, 19*, 21, 25. Turn in starred problems by Thursday, Jan 28.
- Wed, Jan 27 Section 1.1 and some review (integration by parts)
  - Video Review: Integration by Parts using a table (see the handout from Monday).
  - Would you rather watch Edwards James Olmos in a classic movie (pre Battlestar Galactica)?
  - Video: Section 1.1, Part 1 (Direction fields)
  - Video: Section 1.1, Part 2 (Direction fields)
  - Homework: Section 1.1: 1,3,5, 15-20, 22*, 23. Section 1.2: 1(a,b), 3, 8, 9*, 15. Due Friday evening (Jan 29).
- Fri, Jan 29: First order linear DEs (Section 2.1)
  - Video: Section 2.1, part 1 (Linear first order DEs)
  - Video: Section 2.1, part 2 (Linear first order DEs)
  - Homework: Sect 2.1: 1-7 odd*, 13, 15, 16, 30, 33-36
    Homework due on Tuesday Feb 2
Week 2: Feb 1 - Feb 5
- Monday's live session:
  - For a video of the live session, go to our page in Canvas, then go to "Pages", and look for "Live Session on Monday Feb 1.
    In the live session, we summarized 2.1, we looked at 2.2 (separable eqns), then we started the group work project below. The videos below will take the place of the live session after Friday.
  - Homework: Section 2.2: 1, 3*, 5*, 7, 9*, 11, 16, 20, 31, 35. HW due Thursday.
  - Group Work: The group work project is here. It will be due a week from today (next Monday). You may work with anyone you like, or you can give it a go on your own. If you do work with others, please indicate that on your solutions.
  - Substitute videos:
- Wednesday, Feb 3: Section 2.3 Modeling
  - Video 1: Introduction to Modeling
  - Video 2: Tank Mixing
  - Video 3: Money
  - Video 4: Finish money, work on skydiver.
  - Extra: Solving 2.3, #11 using Wolfram Alpha
  - Homework: Section 2.3: 3, 4*, 5, 9, 10*, 12, 13, 16, 23, 28(a)*. For 28, use the model mv'=-mg-kv. Due Tues, Feb 9.
- Fri Feb 5: Section 2.4 (Existence and Uniqueness)
  - Video 1: The theory
  - Video 2: Examples
  - Video 3: More examples
  - Video 4: Bernoulli Equations (From the homework section in 2.4)
  - Homework: Section 2.4: 1, 3, 4*, 7, 9, 13*, 15, 21, 22(a,b)*, 28*, 32*. Due Wed, Feb 10.
Week 3
- Mon Feb 8: Section 2.5: Autonomous DEs.
  - Live Session on Monday. We'll talk about the group work, remind everyone of upcoming things, then discuss autonomous DEs. The live video will be posted on Canvas, but the material we'll cover is also covered in depth below.
  - Video 1: Section 2.5, Autonomous DEs, Part 1
  - Video 2: Autonomous DEs, Part 2
  - Video 3: Autonomous DEs, Part 3
  - Video 4: Autonomous DEs, Part 4
  - Homework: Section 2.5, 1-7 odd, 8, 10, 11, 14, 15, 22-24, 25, 26. NOTE: I won't be collecting homework from you later this week as I would not be able to return it before the exam. Please try all the problems however!
- Wed Feb 10: Section 2.6: Exact equations
  - Video 1: Section 2.6, Exact Equations, Part 1 of 4
  - Video 2: Section 2.6, Exact Equations, Part 2 of 4
  - Video 3: Section 2.6, Exact Equations, Part 3 of 4
  - Video 4: Section 2.6, Exact Equations, Part 4 of 4
  - Homework: 1, 3, 4, 10, 13, 15, 17, 18, 19, 22.
- Fri Feb 12: No new material. Catch up and study for next week's exam!
Week 4
- Mon Feb 15: No classes.
- Wed Feb 17: Exam 1: You may take it any time on Wed or Thur. More details later, but the time window closes at midnight Thursday, so be sure to leave yourself time to finish before that.
- Fri Feb 19: Section 3.1- Second order DEs.
  - Quick NOTE: If we were in class, I would go through the material in the first three videos and perhaps the next one or two videos in one class period, then finish up the other videos in the next. I've included all three of the "linear operator" videos here so you can decide if you have time today or save them for next time.
  - Video 1: Section 3.1 (Part 1 of 3)
  - Video 2: Section 3.1 (Part 2 of 3)
  - Video 3: Section 3.1 (Part 3 of 3)
  - Video: Linear Operators (Part 1 of 3)
  - Video: Linear Operators (Part 2 of 3)
  - Video: Linear Operators (Part 3 of 3)
  - Homework on Cramer's Rule and Linear Operators
  - Homework: Sect 3.1: 1, 3, 10*, 13, 15, 17, 18*, 19, 22, 24* and handout (Try all, turn in 4(a,c)* and 5(e)*). Homework due on Tuesday, Feb 23.
Week 5
- Monday, Feb 22: 3.2- Theory behind second order equations.
  - Video 1: Review and the Wronskian.
  - Video 2: Finding all solutions; The fundamental set
  - Video 3: Abel's Theorem
  - Video 4: Finish Abel's Theorem and its significance.
  - Video 5: Summary and Exercises
  - Video 6: More exercises
  - Homework: Sect 3.2: 1, 2, 4, 7, 9, 14*, 16, 18*, 28, 29*, 35. Homework due on Thursday, Feb 25.
- Wed, Feb 24: Complex numbers, some useful trig and Section 3.3 (Complex roots to the characteristic equation).
  - Complex Numbers
    - This handout summarizes the info from the videos if you're somewhat familiar with complex numbers. As usual, the videos are 10 minutes each or less.
    - Video 1 of 4
    - Video 2 of 4
    - Video 3 of 4
    - Video 4 of 4
  - Reminder of some useful trigonometry
  - Section 3.3: Complex roots to the characteristic equation (1 of 3).
  - Section 3.3 Exercises 17 and 23 (2 of 3).
  - Section 3.3- Using Wolfram Alpha for #23. (3 of 3).
  - Homework: Section 3.3: 1, 3, 4, 7-15 odd (Turn in 11), 24*, 27*. Due Tuesday, Mar 2d.
- Fri, Feb 26: Finish homogeneous linear second order DEs with constant coefficients, and look at "Reduction of Order" to justify our technique.
  - Before we start, a clarification of something from 3.2 and Abel's Theorem
  - Section 3.4 (Video 1 of 2)
  - Summary of Section 3.4, examples (Video 2 of 2)
  - Homework: Section 3.4: 1, 3, 5, 6, 8, 12*, 14, 16, 18*, 24*, 25, 34, 38. Due Wed, Mar 3d.
Week 6: The non-homogeneous equation. Models.
- Mon, Mar 1- Section 3.5 The Method of Undetermined Coefficients.
  - Slides from the lecture, no pauses, PDF format
  - Video 1: Method of Undetermined Coefficients.
  - Video 2 - Finish the method, do lots of examples
  - Homework: Section 3.5: 1, 3, 5, 7*, 13, 15, 19(a), 20(a)*, 21(a), 22(a)*, 23(a). Homework due on Thursday, Mar 4.
- Wed, Mar 3: Live session. Review/Catch up/Problem session.
- Fri, Mar 5: Section 3.6 and Complexification.
  - Section 3.6 (Variation of Parameters)
  - Section 3.6, examples
  - Slides for 3.6
  - Handout and Summary- Complexification
  - Video 1 of 3 on Complexification
  - Video 2 of 3 on Complexification
  - Video 3 of 3 on Complexification
  - Homework: For 3.6: 1, 6, 9*, 13*
    For the Complexification Handout: Turn in the solutions to 2(c) and 3.
    HW due on Tues Mar 9.
Week 7
- Mon, Mar 8: Spring Break Day- No classes, nothing due today.
- Wed, Mar 10: Section 3.7: The Mass-Spring Model
  - The Mass-Spring model (Video 1 of 3)
  - The Mass-Spring model (Video 2 of 3)
  - Additional Comments (Video 3 of 3)
  - Handout to replace textbooks problems.
  - Homework: Try all the problems on the handout. Turn in solutions to 1(b) (or 3.7.2), and 2(a,b). Due Sat, Mar 13.
- Fri, Mar 12 (mid-semester): Section 3.8: Analysis of the Mass-Spring Model. See the notes and homework write up below that replaces 3.8 in the book.
  - Video: Mass-Spring model with no damping and periodic forcing (Beating and Resonance) (CAUTION- LOUD: The demo at the end is really loud for some reason- Please turn down your audio at about the 11 minute mark).
  - Video: Mass-Spring model with damping and periodic forcing (the full model)
  - PDF of the slides from the video.
  - Notes with the Homework. Similar to the slides; the slides have the numerical examples.
  - Homework solutions
- Optional videos from YouTube about Beating and Resonance (some were seen in the slides from Friday).
  - (Youtube clip) Video from MIT- Beating and Resonance (About 2 minutes)
  - Watch a wine glass shatter. (Less than 1 minute)
  - (Youtube clip) The Tacoma Narrows Bridge Collapse (Or was it resonance?) (About 6 minutes)
  - (Youtube) Film from 1969 with Prof. Miller (About 14 minutes total)
  - (Youtube) - Just for fun.
Week 8
- Mon, Mar 15: Section 3.8, Review
- Wed, Mar 17: Exam 2 (Chapter 3)
- Fri, Mar 19: Today we start Chapter 6, The Laplace Transform. We'll start 6.1 today and finish it on Monday, then start 6.2 on Monday as well.
  - The video mentions a table of transforms. This is available in Section 6.2, pg 317. You might take a photo and convert it to a PDF for yourself to make it convenient when you're working on HW.
  - Section 6.1, part 1
  - Section 6.1, part 2
  - Overhead slides for the previous two videos
  - Section 6.1, Examples part 1
  - Notes
  - Section 6.1, Examples part 2
  - Notes
  - Homework for 6.1 (Due Mar 23): 1, 3, 4*, 5(a)* (from the definition, not the table), 15, 22*.
Week 9
- Mon, Mar 22:
  - Section 6.2 Solving DEs
  - Notes from the Video
  - Section 6.2 - Examples
  - Notes from the Video
  - Homework for 6.2 (Due Thur, Mar 25): 1, 3, 5, 10*, 13, 15, 20, 21*
- Wed, Mar 24: Sections 6.3 and 6.4 (The Heaviside Function).
  Sections 6.3 and 6.4 really go together, so I'll put all the videos here. We'll take all day today and Friday will be nothing new, so you can finish this material up.
  - Section 6.3 - Heaviside
  - Section 6.3-4 - Examples, part 1
  - Section 6.3-4 - Examples, part 2
  - Section 6.3-4 - Examples, part 3
  - Section 6.3-4 - Examples, part 4
  - Homework for 6.3: 1, 5, 7, 21, 23*, 33
  - Homework for 6.4: 1(a), 3(a), 8(a)*, 15, 21(b)*
  - Homework for 6.3-6.4 will be due on Tuesday Mar 30.
- Fri, Mar 26: Nothing new today. Finish up Section 6.4.
Week 10
- Mon, Mar 29: Section 6.5- A new function called the Dirac delta-function.
  - Section 6.5 Dirac
  - Section 6.5 Examples
  - Homework for 6.5: 1(a), 3(a)*, 14(a)-just solve, 17(b), 18(b)*. HW due on Thursday Apr 1.
- Wed, Mar 31: Section 6.6: The Convolution (The last section of Chapter 6!)
  - Section 6.6 Convolution
  - Section 6.6 Examples
  - Homework for 6.6: 5, 6*, 9, 10*, 17, 23, 24*. Homework due on Monday, Apr 5.
- Fri, Apr 02
  - (Video) Overview of Ch 6 (Part 1)
  - (Video) Overview of Ch 6 (Part 2)
Week 11
- Mon, Apr 5: Section 5.1
  - Review of Series, Power Series, Radius of Convergence (About 26 minutes- sorry!)
  - Review of Taylor Series (About 18 minutes)
  - Algebra on the index of summation (About 13 minutes)
  - Homework Section 5.1: 1, 2, 4*, 6, 10, 12, 14, 16*, 21, 22, 27*. Due Thur, Apr 8.
- Wed Apr 7: Spring Break Day (no classes).
- Fri Apr 9: Section 5.2
  - Power series solutions, idea and example
  - Power series solutions, solve using derivatives
  - Power series solutions, find recurrence relation (example)
  - Homework, Section 5.2, p. 259. We'll only be finding the recurrence relation for each problem: 1, 3, 5, 8. Use "the derivative method" to determine the first five terms of the Taylor series for each IVP: p. 265- 1, 2, 3. No homework is due from today, but it is important to try them for the exam!
Week 12: Apr 12 - 16
- Mon, Apr 12: Review day.
- Wed, Apr 14: Exam 3 will be available on Wed/Thur.
- Fri, Apr 16: Materials today replace section 7.1
  - Notes (in PDF) for today
  - Video notes 1 of 2
  - Video notes 2 of 2
  - Homework Handout. Do all the problems, but turn in 1(a), 3(a), and 6. Due on Tuesday Apr 20.
Week 13: Apr 19 - 23
- Mon, Apr 19: Day 2, Linear Systems (Notes below).
  - Notes for this material (in PDF), also with homework on the last page
  - Video discussion 1 of 2
  - Video discussion 2 of 2
  - Homework: Try all of the problems on the handout; turn in #1, 2, 4 for Thursday (Apr 22). CHANGED to Friday (Thursday is Spring Break Day).
- Wed, Apr 21: Day 3, Linear Systems
  - Section 7.3-7.5, Part 1
  - Section 7.3-7.5, Part 2
  - Section 7.3-7.5, Part 3
  - PDF notes for Section 7.3 (for the above videos)
  - Homework problems. Work on them all, but turn in solutions to 2(a), 3(a) and 4(a) on Tuesday.
Fri, Apr 23: Live session today. I created new videos that don't take as long as the live session did- This goes through all the examples from the live video.
- Video 1: Summary so far
- Video 2: Sketching solutions (Sorry for running off the edge- It does come back).
- Video 3: More sketching solutions
- No extra homework for tonight.

Week 14: Apr 26 - 30

Mon Apr 26: Complex Eigenvalues/vectors
- Complex Eigenvalues, Part 1.
- Complex Eigenvalues, Part 2
- Notes about Complex Eigenvalues
- Homework for Complex Eigenvalues
- Homework: From the PDF above, do 1(a) and 2(a,b,c) to turn in (you should try them all!). Due Thursday.
Wed Apr 28: Double Root. Poincare Diagram.
- Double Root:
  - (Video) A Single Eigenvalue (and summary) (An error was made and then repeated about 0:30- The trace is 2 for both matrices).
  - (PDF) Notes
  - (PDF) Homework
- Poincare Diagram:
- Homework: From the single eigenvalue, Try all, but turn in 1(a,b) and 2(a). From Poincare, try all, but turn in 1, 2(c) and 4(b). Due Tues May 4.
Fri Apr 30: The notes for the remaining classes will be found in Canvas (under "Files"). We'll be looking at Sections 5.1-5.3 of the new notes. Also, here is the link for the nice online plotter:
Here's a link to an online plotter for systems of differential equations
- Equilibrium Point Analysis, part 1 (Set up and Example)
- Equilibrium Point Analysis, part 2 (Example)
- Equilibrium Point Analysis, part 3 (continuing the last example)
- Equilibrium Point Analysis, part 4
- Homework: From the new notes, 5.1, turn in #3, 6, 8 (use the online plotter linked above instead of the software suggested), and try (but not due) 17-19. HW Due Wednesday.

Week 15: May 3 - May 7:

Mon May 3: Method of Nullclines
- Video 1 of 3
- Video 2 of 3
- Video 3 of 3
- Homework: 5.2 of the notes, #1, 3, 8. Be sure to do these for the exam; won't be collected.
Wed May 5: Catch up/Review day
Fri May 7: Spring Break Day

Week 16: Review day on Monday. See Final Exam Materials below.

Exam Links

Exam 1
- Notes about the exam.
- Link to the online calculator (if you need one for the exam).
- Summary of the material for the first exam
- Review questions
- Review solutions
- Video Review 1 An overview of the class so far.
- Video Review 2 An overview of graphical analysis.
Exam 2 (over Chapter 3)
- The exam format is exactly the same as the first exam (See the first link under Exam 1 for more details).
- Summary of the topics
- Review questions
- Review solutions
- (Video) Solutions to 3.6 HW
Exam 3 (Laplace Transforms, Power Series Solutions)
Final Exam Links
- Videos:
- (PDF) Written Reviews
  - Linear Systems summary
  - Nonlinear Systems summary
  - General review questions. (Typo corrected on problem 39, which was a duplicate of 38).
  - Solutions to the general review questions

Online Links

An online direction field plotter. You'll need it for 1.1, and in general for plotting the direction fields for first order DEs.

How Will I Upload Materials to Canvas?

There are several apps for cell phones to convert images of your work to PDF, which you can then upload to Canvas. Below, Prof. Schueller demonstrates one such app, Adobe Scan. Unfortunately, you will be asked to set up an account, but otherwise it seems like a good app (if you want to use other apps, that's fine, too- but please upload files as PDF).

Differential EquationsMath 244, Spring 2021