Welcome to Differential Equations! For course info and policies, please see the syllabus. For grades, log into Moodle. If you need help or have questions, please contact Prof. Wright.

**Prof. Wright's office hours:** Mon. 1–2, Tues. 10–11, Wed. 2–3, Thurs 10–11, Fri. 1–2, whenever the door is open, or by appointment in RMS 405

**Help sessions:** Tuesdays 9–10pm and Thursdays 7:30–8:30pm in Tomson 186

- Complete the Syllabus Quiz.
- Complete the Computational Assessment.
- Watch this video:
*What are differential equations?* - From the textbook, read §1.1 and §1.2, up to the heading "Missing Solutions" on page 27. Complete the reading questions on Moodle before class on Monday.

- Finish reading §1.2.
- Homework 1: §1.1 exercises 3, 5, 17; and §1.2 exercises #1, 3, 5, 8, 15, 17, 25, 28. This is due in the homework box (RMS 3rd floor, near the fireplace) at 4pm Wednesday.
- Read this article and §1.3 in the textbook. Then complete the reading questions on Moodle.
- If possible, bring a computer with Mathematica to class on Wednesday.

- Homework 2: §1.2 exercise 33 and §1.3 exercises #1, 3, 8, 11, 13, 14, 16, 17.
*Note*: You do not need to use HPGSolver; instead, you may use Mathematica, Desmos, GeoGebra, or other technology. (Due 4pm Friday in the homework box.) - Read §1.4 and complete the reading questions on Moodle.
- If possible, bring a computer with Mathematica to class on Friday. (Instructions for installing Mathematica at St. Olaf.)

- Homework 3: §1.3 exercises 18, 19 and §1.4 exercises #1, 3, 5, 6, 11. (Due 4pm Monday in the homework box.)
- Watch the video Existence and Uniqueness. Also read §1.5, at least through page 67.

- Read this article and §1.6 in the tetbook. Then complete the reading questions on Moodle.
- Homework 4: §1.5 exercises #2, 3, 5–8, 9ab, 11, 13. (Due 4pm Wednesday.)
*This week:*begin work on Lab 1.

- Read §1.7 and complete the reading questions on Moodle..
- Homework 5: §1.5 exercise 12 and §1.6 exercises 1, 4, 7, 10, 13, 16, 19, 31, 32, 33, 34. (Due 4pm Friday.)
- Work on Lab 1.

- Read §1.8. Especially note the
*Linearity Principle*and the*Extended Linearity Principle.*. - Homework 6: §1.6 exercises 28, 37 and §1.7, exercises 4, 8, 9, 11, 12, 13. (Due 4pm Monday.)
- Work on Lab 1.

- Watch the video The Integrating Factor Method. Then read §1.9.
- Finish Lab 1. You may submit your lab report on Moodle in PDF format or print it and place it in the homework box by 4pm Wednesday.
- The next homework appears below. Because the lab is due Wednesday, this homework is due Friday.

- Re-read the subsection
*Comparing the Methods of Solution for Linear Equations*(p. 131–132). Then read §2.1, and complete the reading questions on Moodle. - Homework 7: §1.7 exercises 14, 16; §1.8 exercises 1, 4, 5, 8, 10, 17; and §1.9 exercises 1, 4, 5, 15. (Due 4pm Friday.)
- If possible, bring a computer with Mathematica to class on Friday.

September 28

Predator-prey systems

- Read §2.2. Take note of how vector fields can be used to visualize the behavior of solutions to systems of differential equations.
- Homework 8: §1.8 exercises 19, 23; §1.9 exercises 19, 23; and §2.1 exercises 1–4, 7a, 8ab, 15. (Due 4pm Monday.)
- If possible, bring a computer with Mathematica to class on Monday.

- Homework 9: §2.1 exercises 20, 21, 22 and §2.2 exercises 5, 9, 11, 14, 21. (Due 4pm Wednesday. You may use
*Mathematica*or other technolgy instead of HPGSystemSolver.) - Read §2.3. Note how the "guessing" method is used to solve the differential equation in this section.

- Homework 10: §2.3 exercises 1, 2, 5, 6, 7. (Due 4pm Friday. You may use
*Mathematica*or other technolgy instead of HPGSystemSolver.) - Read §2.4 and complete the reading questions on Moodle.
- Familiarize yourself with Lab 2: Bifurcation Plane, which is due on October 19.

- Homework 11: §2.4 exercises 1, 2, 5, 6, 7, 10, 13. (Due 4pm Monday.)
- Read §2.5, and observe how a 2-D version of Euler's method can be used to solve systems of two differential equations.

- Chapter 1 review (pages 136–141) exercises 1–39, 41–46, 49, 51, 52
- Chapter 2 review (pages 224–226) exercises 1–9, 11, 13, 14–28, 31–34, 35, 36

October 10

**Exam 1**

- This exam will cover Chapter 1 and the first four sections of Chapter 2.
- The exam will consist of a short take-home portion and an in-class portion.
- For the take-home portion, you may (and should) use Mathematica or other technology.
- You may not use Mathematica or similar technology on the in-class exam. Calculators will be permitted, but probably not very helpful. The in-class exam will focus on conceptual understanding. It will involve basic calculus and some arithmetic, but not tedious arithmetic.

- Read §3.1, and complete the reading questions on Moodle.

- Homework 12: §3.1 exercises 5, 9, 14, 24, 27, 29. (Due 4pm Wednesday.)
- Read §3.2. Look for the answer to the question:
*How do straight-line solutions of a linear system connect to eigenvectors of a matrix?*

- Finish Lab 2 (bifurcation plane). You may either submit your lab report on Moodle in PDF format or place it in the homework box by 4pm Friday.
- Read §3.3. What types of phase portraits that are possible for linear systems with real eigenvalues?
- The next homework includes exercises from §3.1 and §3.2. Because the lab is due Friday, the next homework is due Monday.

- Homework 13: §3.1 exercise 16; §3.2 exercises 1, 4, 5, 11, 12, 21; and §3.3 exercises 17, 18. (Due 4pm Monday.)
- Read §3.4, and complete the reading questions on Moodle.
- If you want to know more about Euler's formula, watch this video by 3Blue1Brown.

- Homework 14: §3.4 exercises 1, 2, 4, 5, 10, 11, 15, 16. (Due 4pm Wednesday.)
- Read §3.5. Note what types of phase portraits can occur for linear systems with repeated (real) eigenvalue or zero eigenvalues.

- Homework 15: §3.5 exercises 1, 3, 5, 7, 9, 10, 11, 13. (Due 4pm Friday.)
- Review §3.3 through §3.5. Note the different types of phase portraits that can occur for linear systems, and how they are determined by the eigenvalues of the matrix of coefficients.

- Homework 16: §3.4 exercise 23 and §3.5 exercises 17, 18, 21, 22, 23. (Due 4pm Monday.)
- For review of linear systems, complete the Linear System Summary worksheet. This will give you a catalog containing the form of the solution and the phase portrait for
*all*2x2 linear systems of differential equations. - Read §3.6. How can we use our knowledge of linear systems to solve second-order differential equations?

Extra credit opportunity: Attend the MSCS Colloquium by Minah Oh (Monday, Oct. 29, 3:30pm, in RNS 310) and answer these two questions on Moodle.

- Homework 17: §3.6 exercises 1, 6, 7, 10, 13, 16, 21, 24, 33. (Due 4pm Wednesday.)
- Start working on Lab 3.
- Read §3.7 and complete the reading questions on Moodle.

- Homework 18: §3.7 exercises 2, 3, 4, 5, 9, 10, 11, 12. (For these problems, a "brief essay" can be a sentence or two. Due 4pm Friday.)
- Start working on Lab 3.
- Read §4.1. Come to class knowing the
*Extended Linearity Principle*on page 390. Note that this is the same principle that we previously encountered in Section 1.8 (page 114).

- Homework 19: Ch. 3 review exercises 11, 12, 13, 14; §4.1 exercises 1, 5, 9, 13, 16, 22. (Due 4pm Monday.)
- Read §4.2. Focus on the qualitative analysis and phase portraits. We will discuss "complexification" in class.
- Begin Lab 3 (linear systems), if you haven't already.

- Homework 20: §4.1 exercises 26, 33, §4.2 exercises 1, 3, 5, 11, 16, 19. (Due 4pm Wednesday.)
- Read §4.3, pages 415–420. Pay special attention to the graphs of solutions that can occur when the forcing function is a sine or cosine.

- Finish Lab 3 (linear systems).
- Re-read §4.3. Understand that a forcing frequency very close to the natural frequency produces a large-amplitude forced response.

- Homework 21: §4.1 #38; §4.2 #17, 20; and §4.3 #5, 15, 17, 21. (Due 4pm Monday.)
- Study for the exam: see below for suggested review problems.

- Chapter 3 review (pages 376–380) exercises 1–32.
- Chapter 4 review (pages 449–451) exercises 1–4, 10–12, 15–23.

November 14

**Exam 2**

- This exam will cover Chapter 3, sections 1 through 7, and the first three sections of Chapter 4.
- The exam will consist of a short take-home portion and an in-class portion.
- For the take-home portion, you may use Mathematica or other technology.
- You may not use Mathematica or similar technology on the in-class exam. Calculators will be permitted, but probably not very helpful. The in-class exam will focus on conceptual understanding. It will involve basic calculus and some arithmetic, but not tedious arithmetic.

- Read §5.1. Observe how
*linearization*allows one to approximate a nonlinear system near an equilibrium point by a linear system. Come to class knowing what is a*Jacobian matrix*.

Extra credit opportunity: Attend the MSCS Research Seminar by Jasper Weinburd (Friday, Nov. 16, 3:40pm, in RNS 204) and answer these two questions on Moodle.

- Homework 22: §5.1 #1, 4, 5, 9ab, 18, 21. (Due 4pm Monday.)
- Read §5.2. Come to class knowing the definition of a
*nullcline*.

Extra credit opportunity: Attend the MSCS Colloquium by Wako Bungula (Monday, Nov. 19, 3:30pm, in RNS 310) and answer these two questions on Moodle.

- Homework 23: §5.1 #7a, 8a, 11a, and §5.2 #3, 4, 5, 6, 9. (Due 4pm next Monday.)
- Review §5.1 and §5.2. Notice how analysis of equilibrium points and nullclines can provide a lot of qualitative information about solutions to systems of differential equations, even if you can't write down formulas for the solutions.

- Homework 24: §5.2 #17, 18, 21, 22, 23, and Chapter 5 review exercises (page 555) #9–12. (Due 4pm Wednesday.)
- Read §5.3 and complete the reading questions on Moodle. Pay special attention to the story on pages 490–493. Come to class knowing what is a
*conserved quantity*and a*Hamiltonian system*. - Take a look at Lab 4.

- Homework 25: §5.3 #1, 3, 9, 10, 12, 14, 15. (Due 4pm Friday.)
- Read §7.1. Note how we can quantify the error in approximating a solution using Euler's method.
- To learn more about William Rowan Hamilton, watch this music video (a parody by acapellascience of the Alexander Hamilton song from the Hamilton musical).

November 30

Mathematica notebook

- Work on Lab 4.
- Read §7.2. How can Euler's method be improved?

- Homework 26: §7.2 #1, 3, 9, 11, 13 (Due 4pm Wednesday.)
- Read §7.3. Note that the Runge-Kutta is more sophisticated than Improved Euler's method.
- Bring a computer with
*Mathematica*to class next time, if possible.

- Homework 27: §7.3 #3, 6, and Chapter 7 review exercises #1, 2, 3, 4, 5, 6. (Due 4pm Friday)
- Read Appendix B: Power Series (pages 742–748). Pay special attention to the examples, observing how power series can be used to find solutions to differential equations.
- Work on Lab 4.

- Finish on Lab 4.
- Read the final exam information below, and do some review problems for the final exam.

- Homework 28: Appendix B, #1, 2, 5, 6, 9, 10, 16, 17. (Due 4pm Wednesday.)
- Read the final exam information below, and do some review problems for the final exam.

- The take-home problems will be distributed on the last day of class and due at the final exam period. You may use technology and other course resources, but you may not talk to people (other than the professor) about these problems.
- The exam will cover the all sections of the textbook that we have studied, with emphasis on the last third of the course.
- For the in-class exam, calculators will be permitted, but probably not very helpful, and certainly not necessary. Mathematica and internet-capable devices will not be permitted.
- As you study, consider working the problems from these old exams by Bob Devaney, one of the authors of our textbook.
- Also consider problems from the chapter review sections in the text.
- Lastly, make sure you are familiar with the St. Olaf final exam policies.

December 14

**A**

9:00 – 11:00am

December 18

**B**

9:00 – 11:00am