Matthew L. Wright
Assistant Professor, St. Olaf College

# Differential Equations

## Math 230 ⋅ Fall 2018

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

Friday
September 7
Introduction
Modeling with differential equations
Do the following before next class:
Monday
September 10
Separation of variables
Do the following before next class:
• 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.
• If possible, bring a computer with Mathematica to class on Wednesday.
Wednesday
September 12
Slope fields
Do the following before next class:
Friday
September 14
Euler's method
Do the following before next class:
• 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.
Monday
September 17
Existence and uniqueness
Do the following before next class:
• Homework 4: §1.5 exercises #2, 3, 5–8, 9ab, 11, 13. (Due 4pm Wednesday.)
• This week: begin work on Lab 1.
Wednesday
September 19
Phase line
Do the following before next class:
• 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.
Friday
September 21
Bifurcations
Do the following before next class:
• 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.
Monday
September 24
Linear equations
Do the following before next class:
Wednesday
September 26
Integrating factor
Lab 1
due today
Do the following before next class:
• 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.
Friday
September 28
Systems of differential equations
Predator-prey systems
Do the following before next class:
• 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.
Monday
October 1
Geometry of systems
Spring-mass systems
Do the following before next class:
• 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.
Wednesday
October 3
Damped harmonic oscillation
Do the following before next class:
• Homework 10: §2.3 exercises 1, 2, 5, 6, 7. (Due 4pm Friday. You may use Mathematica or other technolgy instead of HPGSystemSolver.)
• Familiarize yourself with Lab 2: Bifurcation Plane, which is due on October 19.
Friday
October 5
Do the following before next class:
• 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.
Monday
October 8
topics in Chapter 2
Study for the exam! Do the take-home problems (PDF here), distributed in class. For review, consider doing some of the following problems (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 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.
Do the following before next class:
Friday
October 12
Linear systems, linearity principle
Fall break! No class Monday, October 15.
Do the following before next class:
• 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?
Wednesday
October 17
Linear systems and straight-line solutions
Do the following before next class:
• 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.
Friday
October 19
Linear systems with real eigenvalues
Lab 2
due today
Do the following before next class:
Monday
October 22
Linear systems with complex eigenvalues
Do the following before next class:
• 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.
Wednesday
October 24
Linear systems with repeated eigenvalues
Do the following before next class:
• 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.
Friday
October 26
Linear systems with zero eigenvalues
Linear system summary
Do the following before next class:
• 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?
Monday
October 29
Second-order linear systems

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.

Do the following before next class:
• Homework 17: §3.6 exercises 1, 6, 7, 10, 13, 16, 21, 24, 33. (Due 4pm Wednesday.)
• Start working on Lab 3.
Wednesday
October 31
Trace-determinant plane
Do the following before next class:
• 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).
Friday
November 2
Forced harmonic oscillation
Do the following before next class:
• 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.
Monday
November 5
Sinusoidal forcing
Do the following before next class:
• 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.
Wednesday
November 7
Undamped forcing
Do the following before next class:
• 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.
Friday
November 9
Resonance and beats
Lab 3
due today
Do the following before next class:
• 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.
Monday
November 12
Review
Study for the exam! Do the take-home problems (PDF here), distributed in class. Consider the following problems for review (not to be collected):
• Chapter 3 review (pages 376–380) exercises 1–32.
• Chapter 4 review (pages 449–451) exercises 1–4, 10–12, 15–23.
Wednesday
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.
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 16
Nonlinear systems: equilibrium point analysis

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.

Do the following before next class:
• 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.
Monday
November 19
Qualitative analysis

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.

Thanksgiving break! No class Wednesday, Nov. 22 or Friday, Nov. 24.
Do the following before next class:
• 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.
Monday
November 26
Qualitative analysis
Do the following before next class:
• 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.
Wednesday
November 28
Hamiltonian systems
Do the following before next class:
• 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).
Friday
November 30
Revisiting Euler's Method: Quantifying Numerical Error
Mathematica notebook
Do the following before next class:
• Work on Lab 4.
• Read §7.2. How can Euler's method be improved?
Monday
December 3
Improving Euler's Method
Do the following before next class:
• 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.
Wednesday
December 5
Runge-Kutta Method
Do the following before next class:
• 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.
Friday
December 7
Power Series Solutions
Do the following before next class:
• Finish on Lab 4.
• Read the final exam information below, and do some review problems for the final exam.
Monday
December 10
Power Series Solutions
Lab 4
due today
Do the following before next class:
• 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.
Wednesday
December 12
Review
Final Exam Information: The final exam will consist of a short take-home portion and an in-class 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.
Friday
December 14
Final exam for Math 230A
9:00 – 11:00am
Tuesday
December 18
Final exam for Math 230B
9:00 – 11:00am