MATB44H3 - Differential Equations I - Fall 2019
Info
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Lectures:
- Tuesdays, 3-5pm in Highland Hall HL B101
- Thursdays from 10-11am in Instructional Centre IC 130
- The first lecture will be on Tuesday, September 3, and the last one on Thursday, November 28.
- Teaching assistants: Navin Chandradat, Qiaoyu Liang, Hussein Nassereddine, CheukHung (Gabriel) Lou, and Angela Zavaleta Bernuy, Xiao Jie
- Angela Zavaleta's Office Hours: Wednesday, 10am-11am in IC 404.
- Xiao Jie's Office Hours: Thursday, 2:30-4:30pm in IC 404.
- Qiaoyu Liang's Office Hours: Wednesday, 12pm-1pm in IC 404.
- Navin Chandradat's Office Hours: Friday, 2-4pm in IC 404.
- CheukHung Lou will be holding pre-exam office hours instead of weekly hours, to be announced later.
- Hussein Nassareddine will be spending time on Piazza answering questions
- Preliminary outline: http://www.math.toronto.edu/ywyu/MATB44H32019/#overview
- Course calendar: http://www.math.toronto.edu/ywyu/MATB44H32019/#schedule
- Syllabus (updated via in class vote Oct 31): http://www.math.toronto.edu/ywyu/MATB44H32019/syllabus.pdf
- Syllabus (old): http://www.math.toronto.edu/ywyu/MATB44H32019/syllabus-old.pdf
- Notes: http://www.math.toronto.edu/ywyu/MATB44H32019/notes
- Notes-old: http://www.math.toronto.edu/ywyu/MATB44H32019/notes-old
- Quercus: https://q.utoronto.ca/courses/106849
- Piazza: https://piazza.com/mail.utoronto.ca/fall2019/matb44h3/
Preliminary Course Outline (subject to revision)
- MAT B44 Introduction
- What is a differential equation?
(Teschl 1.1; Tenenbaum Lesson 1)
- Classification of differential equations
(Teschl 1.2; Tenenbaum Lessons 2-3)
- First-order ODEs
(Teschl 1.3-1.6)
- Separable ODEs
(Tenenbaum Lesson 6)
- Homogenous ODEs
(Tenenbaum Lesson 7)
- ODEs with linear coefficients
(Tenenbaum Lesson 8)
- Exact ODEs
(Tenenbaum Lesson 9)
- Linear first order ODEs (and integrating factors)
(Tenenbaum Lessons 10-11)
- Qualitative analysis (direction fields, isoclines, etc.)
(Tenenbaum Lesson 5)
- Existence and Uniqueness (Picard-Lindelof)
(Teschl 2.1-2.2, 2.7; Tenenbaum Lessons 57-59)
- Second-order ODEs (note that while the books do not always specialize from nth to 2nd order, we shall)
- Complex Numbers and Complex Functions
(Tenenbaum Lesson 18)
- Linear Independence and second-order linear ODEs
(Tenenbaum Lesson 19)
- Homogeneous second-order linear ODEs
(Tenenbaum Lesson 20)
- Nonhomogeneous second-order linear ODEs
(Tenenbaum Lesson 21-23)
- Systems of first-order ODEs
- Matrix exponential
(Teschl 3.1)
- Linear autonomous first-order systems
(Teschl 3.2)
- General linear first-order systems
(Teschl 3.4, Tenenbaum Lesson 31)
- Existence and Uniqueness for system of first-order ODEs
(Tenenbaum Lesson 62)
- Determinants and Wronksians
(Tenenbaum Lessons 63-64)
- Series methods
(Tenenbaum Lessons 37-40)
- Miscellaneous topics (as time permits)
- Dynamical systems
(Teschl 6.1-6.5)
- Discrete dynamical systems
(Teschl 10.1-10.4)
- …
Class calendar and schedule
Lecture 1. Tuesday, September 3
Recommended reading: Teschl 1.1-1.2, Tenenbaum Lessons 1-3
MATB44 introduction. What is an ordinary differential equation? When are they useful? How do we classify them?
Lecture 2. Thursday, September 5
Recommended reading: Teschl 1.3, Tenenbaum Lesson 4
The general solution of a differential equation, and first order-autonomous equations.
If we have time, we might also start getting into some techniques for explicitly solving ODEs, but this topic will be multiple lectures (Tenenbaum Lessons 6-11, Teschl 1.4).
Lecture 3. Tuesday, September 10
Recommended reading: Online notes from Thursday, September 6, Teschl 1.3, Tenenbaum Lesson 6-9
We are going to review and finish the general solution for first-order autonomous equations; i.e. the long complicated proof from Sep 6.
Then, now that we will have seen at least one example of how to prove our solution, we will give heuristics for explicitly solving ODEs.
We will start with separable equations, which are similar to first-order autonomous equations in how they're solved.
Given enough time, we will also cover ODEs with homogeneous coefficients, ODEs with linear coefficients, and exact ODEs.
Extra resource on proof writing
For those without much experience writing mathematical proofs, here's an excellent introduction by Dr. Eugenia Cheng on how to write a mathematical proof.
Lecture 4. Thursday, September 12
Recommended reading: Tenenbaum Lessons 9-12, Tenenbaum Lesson 5, Teschl 1.5-1.6
Exact differentials, integrating factors, linear differential equations of order 1, and other misc. methods.
If we have time, we might also get into graphical/qualitative methods.
Lecture 5. Tuesday, September 17
Recommended reading: Tenenbaum Lesson 5, Tesch 1.5-1.6
More on exact differentials and integrating factors. We'll also go over again how to check a given solution to an ODE.
Futhermore, we will begin looking at graphical/qualitative methods.
Lecture 6. Thursday, September 19
Recommended reading: Teschl 2.1-2.2
Fixed point theorems and the basic existence/uniqueness result. This will be a proof-heavy topic.
Problem set 1. Due date: September 20
You can download a recommended LaTeX template, which will generate PDFs like this. If you are new to LaTeX, you may wish to look at Overleaf and their online introduction.
We strongly recommend you learn LaTeX and typeset your problem sets, as LaTeX is a useful skill to learn, but you may also neatly write your problem sets and scan them to upload into Quercus.
- Teschl Problem 1.3. Parts (i), (ii), (iii), (v). Make sure to say if the equation is linear, autonomous, or homogeneous, and give its order. Note that these classifications only apply to differential equations, so a valid answer is that the equation is not an ODE.
- Teschl Problem 1.4
- Teschl Problem 1.5
- Teschl Problem 1.6. Prove your answer to the question about whether linearity is always preserved.
- Teschl Problem 1.7. Explain why linearity is or is not preserved in this particular problem.
- Teschl Problem 1.8.
- Teschl Problem 1.9. Parts (i), (ii) only.
- Teschl Problem 1.12
- Teschl Problem 1.14
- Teschl Problem 1.15
- Teschl Problem 1.18. Parts (i), (ii) only.
- Teschl Problem 1.22
- Teschl Problem 1.23
Optional bonus problems
- Teschl Problem 1.10 (bonus)
- Teschl Problem 1.13 (bonus)
- Teschl Problem 1.24 (bonus)
Lecture 7. Tuesday, September 24
Recommended reading: Teschl 2.1-2.2, Tenenbaum Lessons 57-58
Continuation of fixed point theorems and the basic existence/uniqueness result. We will cover in depth contractions and Picard iteration.
Lecture 7. Thursday, September 26
Recommended reading: Teschl 2.1-2.2, Tenenbaum Lessons 57-58
Continuation of fixed point theorems and the basic existence/uniqueness result. (It's an important proof!)
Mastery Quiz "Set1". Due date: September 29
This is available through Quercus → Assignments → Mastery quizzes - WebWork. It will open up in a new browser window, and you'll be able to "Take Set1 test".
As mentioned in the syllabus, you may repeat the test as many time as you wish before the September 29 deadline. However, the test will randomize some of the questions. As you are allowed infinite retries, we expect you to get a perfect score on the mastery quiz. If you do not successfully answer all questions, we will not give you credit for the quiz, so be sure to retry it until you do.
Lecture 8. Tuesday, October 1
Recommended reading: Teschl 2.2, Tenenbaum Lesson 18
We will finally complete the basic existence/uniqueness result of Picard-Lindelof. Additionally, we will review some basics on complex numbers in preparation for the next topic, which will be 2nd-order linear ODEs.
In class midterm exam 1. Thursday, October 3
For details on the composition of the midterm, see the study guide. Note that the second problem set will be due around the midterm. This is intentional, as the pset is much shorter and has significant overlap with the midterm material. i.e. I believe that working on the problem set generally will help you do well on the midterm.
Problem set 2. Due date: Monday, October 7
This problem set is much shorter than the last one. It is intended partially as midterm 1 prep. Problem set here. Again, the LaTeX template is available.
Lecture 9. Tuesday, October 8
Recommended reading: Tenenbaum Lesson 19-21, 28
We will begin discussion of higher-order linear equations, and use the example of simple harmonic motion.
Lecture 10. Thursday, October 10
Special guest lecture from Read Jones Christoffersen Ltd.
A structural engineer from RJC will be coming to give a presentation on their work and some of the differential equations that they use regularly. This will be a good opportunity to see real-world applications. Please welcome our speaker in class if at all possible! This will also be a good opportunity to get a sense of a more general presentation.
Lecture 11. Tuesday, October 22
Recommended reading: Tenenbaum Lesson 21-23
Recommended post-lecture exercises (unmarked): Exercise 21, problems 3-33, Exercise 22, problems 1-15, Exercise 23, problems 1-17
We will continue discussion of higher-order linear equations, and spend time on how to solve nonhomogeneous linear differential equations
Lecture 12. Thursday, October 24
Recommended reading: Tenenbaum Lesson 22-23, Teschl 3.3 (don't worry if you don't understand the matrix style proofs yet; Tenenbaum and Teschl do things in different order)
Recommend post-lecture exercises (unmarked): Exercise 23, problems 1-17
Variation of parameters and reduction of order methods
Lecture 13. Tuesday, October 29
Recommended reading: Tenenbaum Lesson 28-30, Teschl 3.3
Recommend post-lecture exercises (unmarked): Exercise 28D, problems 6-9, Exercise 30, problems 5-10
Detailed study of simple harmonic motion (including forcing and damping terms), as well as electrical RLC circuits.
Lecture 14. Thursday, October 31
Recommended reading: Tenenbaum Lesson 37
Review of power series, and discussion of using power series methods to (approximately) solve ODEs
Problem set 3. Due date: November 6
Note that due to TA preference, we will be using Crowdmark instead of Quercus for this assignment. (as an aside, you can view comments on your midterm on Crowdmark as well)
If you have not received an email with instructions by Thursday, 2019-Oct-24 at 12pm, please let me know.
- Tenenbaum Exercise 21, Problem 1
- Teschl Problem 3.17
- Tenenbaum Exercise 22, Problem 20 (note that the solution is given in the book, but you must show your work.)
- Teschl Problem 3.18
- Teschl Problem 3.19 (You may want to look at the hint for Tenenbaum, Exercise 23, Problem 18)
- Teschl Problem 3.21 (note: we have not yet covered how to directly reduce to it using Teschl 3.48, so you should use direct verification instead; I will however accept both answers)
Lecture 15. Tuesday, November 5
Recommended reading: Tenenbaum Lessons 63-64, Teschl 3.1-3.2
Determinants, Wronskians, linear independence of functions (redux), systems of autonomous first-order ODEs, and matrix exponentials
In class midterm exam 2. Thursday, November 7
Details, practice midterm, etc. on Quercus announcements
Lecture 15. Tuesday, November 12
Recommended reading: Teschl 3.2
Systems of autonomous first-order ODEs and phase plane diagrams.
Lecture 16. Thursday, November 14
Recommended reading: Teschl 3.2
How to guess. When Ansätze go wrong.
Problem set 4. Due date: Monday, November 18
Problems here: Problem set here (4 problems). The LaTeX template is available.
As with the last problem set, we will be using Crowdmark, so look out for an email from them.
Lecture 17. Tuesday, November 19
Recommended reading: Strogatz, Nonlinear Dynamics and Chaos 6.4. Boyce & DiPrima, Elementary Differential Equations and Boundary Value Problems 2.12
Linearization of nonlinear 2D systems with phase portraits. First order difference equations.
Lecture 18. Thursday, November 21
Generating functions for difference equations.
Lecture 19. Tuesday, November 26
Review for final.
Lecture 20. Thursday, November 28
In class presentations. Volunteers requested for real audience feedback!
Problem set 5. Due date: Monday, December 02
Problems here: Problem set here (4 problems + 1 unfair super bonus). The LaTeX template is available.
As with the last problem set, we will be using Crowdmark, so look out for an email from them.