Matthew L. Wright
Assistant Professor, St. Olaf College

# Differential Equations

## Math 230 ⋅ Fall 2017

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

Friday
September 8
Introduction
Modeling with differential equations
Do the following before next class:
Monday
September 11
Separation of variables
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.
Wednesday
September 13
Slope fields
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.)
Friday
September 15
Euler's method
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.
Monday
September 18
Existence and uniqueness
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.
Wednesday
September 20
Phase line
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.
Friday
September 22
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.
Monday
September 25
Linear equations
Do the following before next class:
Wednesday
September 27
Integrating factor
Lab 1
due today
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.
Friday
September 29
Systems of differential equations
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.
Monday
October 2
Geometry of systems
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.
Wednesday
October 4
Damped harmonic oscillation
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.
Friday
October 6
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.
Monday
October 9
topics in Chapter 2
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
October 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.
Friday
October 13
Linear systems, linearity principle
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?
Wednesday
October 18
Linear systems and straight-line solutions
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.
Friday
October 20
Linear systems with real eigenvalues
Lab 2
due today
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.
Monday
October 23
Linear systems with complex eigenvalues
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.
Wednesday
October 25
Linear systems with repeated 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
October 27
Linear systems with zero eigenvalues
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?
Monday
October 30
Second-order linear systems
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.
Wednesday
November 1
Trace-determinant plane
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).
Friday
November 3
Forced harmonic oscillation
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.
Monday
November 6
Sinusoidal forcing
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.
Wednesday
November 8
Undamped forcing
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.
Friday
November 10
Resonance and beats
Lab 3
due today
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.
Monday
November 13
Review
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
November 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.
Friday
November 17
Nonlinear systems: equilibrium point analysis
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.
Monday
November 20
Qualitative analysis
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
November 27
Qualitative analysis
Do the following before next class:
• Do §5.2 #17, 18, 21, 22, 23, and Chapter 5 review exercises (page 555) #9–12. (Due 4pm Wednesday.)
• Read §5.3. Pay special attention to the story on pages 490–493. Come to class knowing what is a conserved quantity and a Hamiltonian system.
Wednesday
November 29
Hamiltonian systems
Do the following before next class:
• Do §5.3 #1, 3, 9, 10, 12, 14, 15. (Due 4pm Friday.)
• Read §6.1. Note the definition of the Laplace transform and how it can be used to solve differential equations.
• To learn more about William Rowan Hamilton, watch this music video (a parody by acapellascience of the Alexander Hamilton song from the Hamilton musical).
Friday
December 1
Laplace transforms
Do the following before next class:
• Do §6.1 #1, 2, 6–10, 15, 16. (Due 4pm Monday.)
• Read §6.2. Come to class able to draw the graph of the Heaviside function ua(t).
Monday
December 4
Laplace tranforms, discontinuous functions
Do the following before next class:
• Do §6.2 #1, 2, 3, 5, 7, 9, 11. (Due 4pm Wednesday.)
• Read §6.3. Note the technique used to compute the Laplace transform of the sine function, and how the Laplace transform can be used to solve second-order differential equations.
Wednesday
December 6
Laplace transforms of second-order equations
Do the following before next class:
Friday
December 8
Delta functions and impulse forcing
Lab 4
due today
Do the following before next class:
• Do §6.3 #5, 11, 15, 27, 30, and §6.4 #2, 3, 5, 7. (Due 4pm Monday.)
• Read the final exam information below, and do some review problems for the final exam.
Monday
December 11
Review
Final Exam Information: The final exam will consist of a take-home problem and an in-class exam.
• The take-home problem will be distributed on the last day of class and due at the final exam period. You may use technology and other course resources to solve the problem, but you may not talk to people (other than the professor) about the problem.
• The exam will cover the sections we have studied from Chapters 1 through 6, with emphasis on Chapters 5 and 6.
• For the in-class exam, 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.
• A Laplace Transform Reference will be provided during the exam.
• 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.
Thursday
December 14
Final exam for Math 230A
2:00 – 4:00pm
Tuesday
December 19
Final exam for Math 230B
2:00 – 4:00pm