Source: https://www.mlwright.org/teaching/math230f17/
Timestamp: 2019-04-19 15:12:11+00:00

Document:
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.
Watch this video: What are differential equations?
Read §1.2, up to the heading "Missing Solutions" on page 27. Come to class on Monday knowing what is a separable differential equation.
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 §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.
Watch the video Existence and Uniqueness. Also read §1.5, at least through page 67.
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.
Read from the beginning of §1.7 through page 99. Take special note of the definition of a bifurcation.
Read §1.8. Take note of the Linearity Principle and the Extended Linearity Principle, and how they are used in solving linear differential equations.
Watch the video The Integrating Factor Method. Then read §1.9 through page 128.
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.
Finish reading §1.9, especially the subsection Comparing the Methods of Solution for Linear Equations (p. 131–132).
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.
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.
Read §2.3. Note how the "guessing" method is used to solve the differential equation in this section.
Read §2.4. Note how a decoupled system can be solved by solving each differential equation separately.
Read §2.5, and observe how a 2-D version of Euler's method can be used to solve systems of two differential equations.
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.
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.
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?
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 §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.
Read §3.5. Note what types of phase portraits can occur for linear systems with repeated (real) eigenvalue or zero eigenvalues.
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.
Read §3.6. How can we use our knowledge of linear systems to solve second-order differential equations?
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.
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).
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.
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 reading §4.3. Understand that a forcing frequency very close to the natural frequency produces a large-amplitude forced response.
Study for the exam: see below for review problems, and don't forget about the help session Saturday, 3–4pm in RNS 206.
Chapter 3 review (pages 376–380) exercises 1–32.
Chapter 4 review (pages 449–451) exercises 1–4, 10–12, 15–23.
This exam will cover Chapter 3, sections 1 through 7, and the first three sections of Chapter 4.
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.
Read §5.2. Come to class knowing the definition of a nullcline.
Thanksgiving break! No class Wednesday, Nov. 22 or Friday, Nov. 24.
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.
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.
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).
Read §6.2. Come to class able to draw the graph of the Heaviside function ua(t).
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.
Finish Lab 4 (oscillating chemical reactions).
Read §6.4. Pay special attention to the definition of impulse forcin and how it is modeled using the Dirac delta function.
Read the final exam information below, and do some review problems for the final exam.
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.

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