• Welcome to choose this class~!
• The syllabus was created in April 2024 and it is subject to update.

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• Contacts
• Course description
• Textbook
• Coverage
• Prerequisite
• Obligations and tips
• Homework
• Test and exam
• Attendance
• Assessment
• Tentative schedule and slides
• Gradescope
• Honor code
• Accessibility
• Harassment and Discrimination
• Acknowledgment

Lecture Instructor	Dr. Le Chen	lzc0090@auburn.edu
Class Time and Room	TR, 11:00 AM -- 12:15 PM	Parker Hall 352
Office hours	TR, 13:00 PM -- 13:50 PM	Parker Hall 203

• When you send us emails, please do include the keyword Math 7280 or Math 7290 in the subject field of your email to ensure a timely response.
• In case you want to make an appointment with the instructor via Zoom, here is the link:
  https://auburn.zoom.us/j/8141875411

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In this sequence of courses, Math 7280 and Math 7290, we will cover advanced topics on ordinary differential equations (ODEs). Our study will begin with the foundational principles of ODEs, focusing on existence and uniqueness of the linear system of differential equations. We will follow mostly the presentation by Coddington and Levinson's book.

Our studies will then extend to ODEs on the complex plane, demonstrating how numerous classical special functions serve as solutions to these equations. This foundational knowledge will pave the way for Math 7290, the subsequent course in this series, which concentrates on nonlinear ODEs.

The program will culminate with an introduction to the six Painlevé transcendents, a group of special functions as solutions to some nonlinear ODEs. These functions have significant implications across various fields, including mathematical physics, random matrices, and the 2D Ising model, among others. This course sequence is designed to equip students with a thorough understanding of both the theoretical and applied aspects of ODEs, preparing them for advanced research in these dynamic areas.

• [CL] "Theory of ordinary differential equations", by Coddington, Earl A. and Levinson, Norman. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
• [Ince] "Ordinary differential equations", by Ince, E. L. Dover Publications, Inc., New York, 1944 (reprint of 1926).
• [Davis] "Introduction to nonlinear differential and integral equations", by Davis, Harold T. Dover Publications, Inc., New York, 1962.
• [HSD] "Differntial equations, dynamical systems, and an introduction to chaos", by Hirsch, Moris W., Smale, Stephen, and Devaney, Robert L., Elsevier/Academic Press, Amsterdam, 2004.
• 

Math 7280 will cover topics from the following chapters of Coddington and Levinson's book:

• CHAPTER 1 EXISTENCE AND UNIQUENESS OF SOLUTIONS
  1. Existence of Solutions
  2. Uniqueness of Solutions
  3. The Method of Successive Approximations
  4. Continuation of Solutions
  5. Systems of Differential Equations
  6. The nth-order Equation
  7. Dependence of Solutions on Initial Conditions and Parameters
  8. Complex Systems
• CHAPTER 2 EXISTENCE AND UNIQUENESS OF SOLUTIONS (continued)
  1. Extension of the Idea of a Solution, Maximum and Minimum Solutions
  2. Further Uniqueness Results
  3. Uniqueness and Successive Approximations
  4. Variation of Solutions with Respect to Initial Conditions and Parameters
• CHAPTER 3 LINEAR DIFFERENTIAL EQUATIONS
  1. Preliminary Definitions and Notations
  2. Linear Homogeneous Systems
  3. Nonhomogeneous Linear Systems
  4. Linear Systems with Constant Coefficients
  5. Linear Systems with Periodic Coefficients
  6. Linear Differential Equations of Order n
  7. Linear Equations with Analytic Coefficients
  8. Asymptotic Behavior of the Solutions of Certain Linear Systems
• CHAPTER 13 ASYMPTOTIC BEHAVIOR OF NONLINEAR SYSTEMS: STABILITY
  1. Asymptotic Stability
  2. First Variation: Orbital Stability
  3. Asymptotic Behavior of a System
  4. Conditional Stability
  5. Behavior of Solutions off the Stable Manifold
• CHAPTER 14 PERTURBATION OF SYSTEMS HAVING A PERIODIC SOLUTION
  1. Nonautonomous Systems
  2. Autonomous Systems
  3. Perturbation of a Linear System with a Periodic Solution in the Nonautonomous Case
  4. Perturbation of an Autonomous System with a Vanishing Jacobian
• CHAPTER 15 PERTURBATION THEORY OF TWO-DIMENSIONAL REAL AUTONOMOUS SYSTEM
  1. Two-dimensional Linear Systems
  2. Perturbations of Two-dimensional Linear Systems
  3. Proper Nodes and Proper Spiral Points
  4. Centers
  5. Improper Nodes
  6. Saddle Points

Math 7280 will cover topics from the following chapters of Coddington and Levinson's book:

• CHAPTER 16 THE POINCARE-BENDIXSON THEORY OF TWO-DIMENSIONAL AUTONOMOUS SYSTEMS
  1. Limit Sets of an Orbit
  2. The Poincare-Bendixson Theorem
  3. Limit Sets with Critical Points
  4. The Index of an Isolated Critical Point
  5. The Index of Simple Critical Point
• CHAPTER 4 LINEAR SYSTEMS WITH ISOLATED SINGULARITIES: SINGULARITIES OF THE FIRST KIND
  1. Introduction
  2. Classification of Singularities
  3. Formal Solutions
  4. Structure of Fundamental Matrices
  5. The Equation of the nth Order
  6. Singularities at Infinity
  7. An Example: the Second-order Equation
  8. The Frobenius Method
• CHAPTER 5 LINEAR SYSTEMS WITH ISOLATED SINGULARITIES: SINGULARITIES OF THE SECOND KIND
  1. Introduction
  2. Formal Solutions
  3. Asymptotic Series
  4. Existence of Solutions Which Have the Formal Solutions as Asymptotic Expansions—the Real Case
  5. The Asymptotic Nature of the Formal Solutions in the Complex Case
  6. The Case Where A0 Has Multiple Characteristic Roots
  7. Irregular Singular Points of an nth-order Equation
  8. The Laplace Integral and Asymptotic Series
• CHAPTER 6 ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS CONTAINING A LARGE PARAMETER
  1. Introduction
  2. Formal Solutions
  3. Asymptotic Behavior of Solutions
  4. The Case of Equal Characteristic Roots
  5. The nth-order Equation

Additionally, if time permits, we may want to cover materials from the first eight chapters of Davis' book:

1. Introduction
2. ODE of first order
3. Riccati Equation
4. Existence Theorem
5. Introduction to Second order ode -- Conflict and Pursuit
6. Elliptic integrals, elliptic function, and theta function
7. Differential equation of second order
8. Second order ODE of polynomial class

• A good knowledge of linear algebra and complex analysis will be helpful.

This is a demanding course and it requires a great deal of work from your side. In order to successfully master the material and complete the course, you are expected to

• Read the textbook and attend lectures.
• Take the advantage of the office hours, which give you additional chance to interact with the
  instructor.
• Read solutions and any feedback you receive for each problem set.
• Complete and submit weekly homework through Gradescope.
• Complete both midterm test and the final exam.
• Use appropriate etiquette and treat other students with respect in all discussions.
• Do not hesitate to ask for help whenever needed.

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Note: The syllabus was created in April 2024, and it is subject to changes during the semester.

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• There will be about 5 homework assignments in each semester:

  	Due at	Semester
  Homework 1	Sept. 12, 6pm	2024 Fall
  Homework 2	Sept. 26, 6pm
  Homework 3	Oct. 24, 6pm
  Homework 4	Nov. 14, 6pm
  Homework 5	Dec. 3, 6pm
  Homework 6	TBA	2025 Spring
  Homework 7
  Homework 8
  Homework 9
  Homework 10

• No late homework will be accepted.
• You need to write full details of the problems and upload your solutions to gradescope.
  • Go to Canvas --> Find this course --> Click Gradescope on the left panel.
  • See below for more instructions on gradescope.
• We will randomly select a few problems to grade and the rest will be checked only for completion.

• There will be one midterm test during the class session:
• Final exam will be cumulative.

	Date/Time	Coverage	Semester
Midterm Test	10/08	Chapters 1, 3, and 13.	2024 Fall
Final Exam	12/12, Dec., Thursday, 10:00 AM -- 1:00 PM	Chapters 1, 2, 3, 13, 14, 15
Midterm Test	TBA	TBA	2025 Spring
Final Exam	TBA	Comprehensive

• Please note down the above dates. No late exam/test will be given.
• Makeup exams will only be allowed in extreme circumstances. Exams cannot be made up without a
  university-approved excuse. Any excuse must be submitted by the date of exam to be considered.
  Please refer to the Tiger Cub for the list of acceptable reasons for being absent from an exam or
  a test. Makeup exam/test has to be scheduled and made up in a timely manner.
• More details will come during the semester.

• Notify the instructor if you are not able to participate in a lecture due to illness or some other emergency.
• Attendance will not be directly counted into your final score.
• But sufficient attendance will make your eligible for grade curving at the end of semester.

• The final score will be determined as follows:
• 
• Based on the final score (plus potential bonus points), the final letter grade will be
  determined as follows:

  Grade (+)		Grade		Grade (-)
  		A	92%-100%	A-	90%-91.9%
  B+	87%-89.9%	B	82%-86.9%	B-	80%-81.9%
  C+	77%-87.9%	C	72%-76.9%	C-	70%-71.9%
  D+	67%-67.9%	D	67%-67.9%	D-	60%-61.9%
  		F	0%-59.9%

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• Below is the tentative schedule that may change during the semester of 2024 Fall:

  	Tuesday	Slides or coverage	Thursday	Slides or coverage
  Week 1	08/20	Introduction	08/22	Section 1.1
  Week 2	08/27	Sections 1.2 and 1.3	08/29	Sections 1.5 and 1.6
  Week 3	09/03	Section 1.7	09/05	Section 1.8
  Week 4	09/10	Section 2.1	09/12	Section 2.2
  Week 5	09/17	Section 3.1	09/19	Section 3.2
  Week 6	09/24	Sections 3.3 and 3.4	09/26	Sections 3.6 and 3.7
  Week 7	10/01	Section 13.1	10/04	Section 13.2
  Week 8	(10/08)	Midterm Exam	(10/11)	--
  Week 9	10/15	Section 13.3	10/17	Section 13.4
  Week 10	10/22	Section 13.4	10/24	Section 13.4
  Week 11	10/29	Section 13.5	10/31	Section 3.8
  Week 12	11/05	Section 3.8	11/07	Section 14.1
  Week 13	11/12	Section 14.2	11/14	Sections 15.1 and 15.2
  Week 14	11/19	Section 15.3	11/21	Section 15.4
  Week 15	(11/26)	--	(11/28)	--
  Week 16	11/33	Sections 15.5 and 15.6	12/05	Review

• Below is the tentative schedule that may change during the semester of 2025 Spring:

  	Tuesday	Slides or coverage	Thursday	Slides or coverage
  Week 1	01/14	Section 15.1	01/16	Section 15.2
  Week 2	01/21	Section 15.3	01/23	Sections 15.4 and 15.5
  Week 3	01/28	Section 4.1	01/30	Section 4.2
  Week 4	02/04	Section 4.3	02/06	Section 4.4
  Week 5	02/11	Section 4.5	02/13	Section 4.6
  Week 6	02/18	Sections 4.7 and 4.8	02/20	Section 5.1
  Week 7	02/25	Section 5.2	02/27	Section 5.3
  Week 8	03/04	Section 5.4	03/06	Midterm Exam
  Week 9	(03/11)	Spring break	(03/13)	Spring break
  Week 10	03/18	Section 5.5	03/20	Section 5.6
  Week 11	03/25	Sections 5.7 and 5.8	03/27	Sections 6.1 and 6.2
  Week 12	04/01	Section 6.3	04/04	Section 6.4
  Week 13	04/08	Section	04/11	Section
  Week 14	04/15	Section	04/18	Section
  Week 15	04/22	Section	04/25	Section
  Week 16	04/29	Reviewing Session	--

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• We will use gradescope to handle submissions of homework, which allows
  us to provide fast and accurate feedback on your work.
• As soon as grades are posted, you will be notified immediately so that you can log in and see your
  grades and feedback.
• If you have any questions regarding Gradescope, please send your message to
  • help@gradescope.com
• Printer+scanner or tablet
  • The easiest way to submit the homework/tests/exams is the following steps:
    1. print the given template;
    2. complete the problem sets;
    3. scan the resulting paper (make sure it is legible);
    4. upload the scanned file to gradescope.
  • Alternatively, if you have a tablet that you can write on it, you may simply write on the
    template pdf file and upload the resulting file.
  • Make sure that you make the correct association of your solutions to the problems.
  • Double check your scan quality and make sure your solutions are legible.

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• Students should familiarize themselves with Auburn honor code here
  • https://www.auburn.edu/academic/provost/academic-honesty/.
• Students are encouraged to share ideas and solutions on problem sets and labs, but must
  express those ideas in their own words in their submitted work.
• Students are not authorized to view or use the work of another student during exams.

Your success in this class is important to me. We will all need accommodations because we all learn differently. If there are aspects of this course that prevent you from learning or exclude you, please let me know as soon as possible. Together we will develop strategies to meet both your needs and the requirements of the course.

I encourage you to visit the Office of Accessibility to determine how you could improve your learning as well. You can register and make a request for services from the Office of Accessibility. In this case, please do inform me of such requests. See the following link for more information:

• https://accessibility.auburn.edu/

• According to Auburn University policies: http://auburn.edu/administration/aaeeo/H&D.php

  
  Auburn University is committed to providing a working and academic environment free from prohibited
  discrimination and harassment and to fostering a nurturing and vibrant community founded upon the
  fundamental dignity and worth of all its members. Auburn University prohibits harassment of its
  students and employees based on protected classes and works to eliminate prohibited behavior from
  its academics and employment through corrective measures and education.
  
  The Office of AA/EEO oversees compliance with the Policy Prohibiting Harassment of Students, the
  Policy Prohibiting Harassment of Employees, and the Policy on Sexual and Gender-Based Harassment and
  Other Forms of Interpersonal Violence.
  
  Protected classes are race, color, sex (which includes sexual orientation, gender identity, and
  gender expression), religion, national origin, age, disability, protected veteran status, or genetic
  information.
  
  Auburn University also prohibits retaliation against any individual for opposing a practice he/she
  reasonably believed to be discriminatory; for filing an internal or external complaint, grievance,
  or charge; or for participating in any investigation or proceeding, in accordance with Auburn
  University's policies.
  

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