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. 2–3, Tues. 9:45–10:45, Wed. 9–10, Thurs 1–2, Fri. 10:30–11:30, or by appointment in RMS 405

**Help sessions:** Tues. 7–8pm and Sat. 3–4pm in RNS 206

Jump to today

Do the following before next class:

- Complete the Syllabus Quiz.
- Complete the Computational Assessment.
- Watch this video:
*What are differential equations?* - Read §1.1 in the textbook, and do exercises 3, 5, and 17. (Solutions due 4pm Monday in the homework box.)
- Read §1.2, up to the heading "Missing Solutions" on page 27. Come to class on Monday knowing what is a
*separable differential equation*.

Do the following before next class:

- Finish reading §1.2. Then do exercises #1, 3, 5, 8, 15, 17, 25, 28.
- Read this article and answer the following question:
*What are three ways that students with a growth mind-set approach challenges differently than students with a fixed mind-set?* - Your answers to the two items above are due 4pm Wednesday in the homework box.
- Read §1.3. Come to class knowing how to interpret a
*slope field*.

Do the following before next class:

- Do §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. - Read this article and answer the following questions:
*According to Devlin, what is the secret to doing mathematics? How does this relate to the growth mind-set from the article you read last week? How might Devlin’s secret be relevant in this course?* - Your answers to the two items above are due 4pm Friday in the homework box.
- Read §1.4, up to the middle of page 59.
- If possible, bring a computer with Mathematica to class on Friday. (Instructions for installing Mathematica at St. Olaf.)

Do the following before next class:

- Finish reading §1.4, then do exercises #1, 3, 5, 6, 11. (Solutions due 4pm Monday.)
- Watch the video Existence and Uniqueness. Also read §1.5, at least through page 67.

Do the following before next class:

- Finish reading §1.5, then do #2, 3, 5–8, 9ab, 11, 13. (Solutions due 4pm Wednesday.)
- Read from the beginning of §1.6 through page 79. Take note of the definition of
*autonomous differential equation*and pay special attention to how a*phase line*can be used to sketch solutions.

Do the following before next class:

- Finish reading §1.6, then do #1, 4, 7, 10, 13, 16, 19, 31, 32, 33, 34. (Solutions due 4pm Friday.)
- Read from the beginning of §1.7 through page 99. Take special note of the definition of a
*bifurcation*.

Do the following before next class:

- Finish reading §1.7, then do #4, 8, 9, 11, 12, 13, 16. (Solutions due 4pm Monday.)
- Read §1.8. Take note of the
*Linearity Principle*and the*Extended Linearity Principle*, and how they are used in solving linear differential equations.

Do the following before next class:

- Watch the video The Integrating Factor Method. Then read §1.9 through page 128.
- Finish Lab 1 (Euler's Method)
- The next homework includes §1.8, exercises 1, 4, 5, 8, 10, 17, 19, 23. Because the lab is due Wednesday, this homework is due Friday.

Do the following before next class:

- Finish reading §1.9, especially the subsection
*Comparing the Methods of Solution for Linear Equations*(p. 131–132). - Do §1.8, #1, 4, 5, 8, 10, 17, 19, 23 and §1.9 #1, 4, 5, 15, 19, 23. (Due 4pm Friday.)
- Read §2.1, through the end of the predator-prey discussion on page 156. Take special note of how the
*R*(*t*) and*F*(*t*) graphs relate to the solution curves in the*phase portrait*.

Do the following before next class:

- Do §2.1 exercises 1–4, 7a, 8ab, 15. (Due 4pm Monday.)
- Read the spring-mass discussion in §2.1 (pages 156–160). Also read §2.2 and note how direction fields can be used to understand phase portraits.

Do the following before next class:

- Do §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.

Do the following before next class:

- Do §2.3 exercises 1, 2, 5, 6, 7. (Due 4pm Friday. You may use
*Mathematica*or other technolgy instead of HPGSystemSolver.) - Read §2.4. Note how a
*decoupled*system can be solved by solving each differential equation separately.

Do the following before next class:

- Do §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.

Study for the exam! Consider the following problems for review (not to be collected):

- 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

Wednesday

Oct. 11

Oct. 11

**Exam 1**

- This exam will cover Chapter 1 and the first four sections of Chapter 2.
- Calculators will be permitted, but probably not very helpful, and certainly not necessary. Computer algebra systems (including the TI-89, TI-92, and TI-Nspire calculators) and internet-capable devices will not be permitted.

Do the following before next class:

- Read §3.1. Note how the concepts of
*determinate*,*linear combination*, and*linear independence*from linear algebra can be applied to systems of differential equations.

Fall break! No class Monday, October 16.

Do the following before next class:

- Do §3.1 exercises 5, 9, 14, 16, 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?*

Do the following before next class:

- Finish Lab 2 (bifurcation plane)
- Read §3.3. What types of phase portraits that are possible for linear systems with real eigenvalues?
- The next homework includes §3.2, exercises 1, 4, 5, 11, 12, 21. Because the lab is due Friday, the next homework is due Monday.

Do the following before next class:

- Do §3.2 exercises 1, 4, 5, 11, 12, 21 and §3.3 exercises 17, 18.
*For each of these problems, identify the type of equilibrium point that you find.* - Read §3.4, at least through the box at the top of page 305. Pay attention to the how the authors solve the example linear system, especially to how two linearly-independent real solutions are obtained from the complex solution.
- If you want to know more about Euler's formula, watch this video by 3Blue1Brown.

This weekend, the help session will be Sunday, 1–2pm in RNS 206.

Do the following before next class:

- Do §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.

Do the following before next class:

- Do §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.

Friday

Oct. 27

Oct. 27

Linear systems with zero eigenvalues

Linear system summary

Linear system summary

Do the following before next class:

- Do §3.4 exercise 23 and §3.5 exercises 17, 18, 21, 22, 23. (Due 4pm Monday.)
- Read §3.6. How can we use our knowledge of linear systems to solve second-order differential equations?

Do the following before next class:

- Do §3.6 exercises 1, 6, 7, 10, 13, 16, 21, 24, 33. (Due 4pm Wednesday.)
- Read §3.7. Observe how the type of phase plane of a linear system can be found from the trace and determinant of the matrix.

Do the following before next class:

- Do §3.7 exercises 2, 3, 4, 5, 11, 12. (For these problems, a "brief essay" can be a sentence or two. Due 4pm Friday.)
- 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).

Do the following before next class:

- Do §4.1 exercises 1, 5, 9, 13, 16, 22, 26, 33. (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.

Do the following before next class:

- Re-read §4.2. Then do §4.2 exercises 1, 3, 5, 11, 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.

Do the following before next class:

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

Do the following before next class:

- Do §4.1 #38, §4.2 #17, 20, and §4.3 #5, 15, 17, 21. (Due 4pm Monday.)
- Study for the exam: see below for review problems, and don't forget about the help session Saturday, 3–4pm in RNS 206.

Study for the exam! Consider the following problems for review (not to be collected):

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

Wednesday

Nov. 15

Nov. 15

**Exam 2**

- This exam will cover Chapter 3, sections 1 through 7, and the first three sections of Chapter 4.
- Calculators will be permitted, but probably not very helpful, and certainly not necessary. Computer algebra systems (including the TI-89, TI-92, and TI-Nspire calculators) and internet-capable devices will not be permitted.

Do the following before next class:

- 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*.

Do the following before next class:

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

Thanksgiving break! No class Wednesday, Nov. 22 or Friday, Nov. 24.

Do the following before next class:

- Do §5.1 #7a, 8a, 11a, and §5.2 #3, 4, 5, 6, 9. (Due 4pm 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.

Monday

Nov. 27

Nov. 27

Qualitative analysis

Do the following before next class:

- TBA...

Wednesday

Nov. 29

Nov. 29

Hamiltonian systems

Do the following before next class:

- TBA...

Friday

Dec. 1

Dec. 1

Laplace transforms

Do the following before next class:

- TBA...

Monday

Dec. 4

Dec. 4

Laplace tranforms, discontinuous functions

Do the following before next class:

- TBA...

Wednesday

Dec. 6

Dec. 6

Laplace transforms of second-order equations

Do the following before next class:

- TBA...

Do the following before next class:

- TBA...

Monday

Dec. 11

Dec. 11

Review

Study for the final exam!

Thursday

Dec. 14

Dec. 14

Final exam for Math 230

2:00 – 4:00pm

**A**2:00 – 4:00pm

Tuesday

Dec. 19

Dec. 19

Final exam for Math 230

2:00 – 4:00pm

**B**2:00 – 4:00pm