Math 53
Spring 2025

Course Description

Differential equations arise in quantitive modeling throughout the natural sciences, engineering, finance, computer graphics, machine learning, and much more. This course explores many approaches to understanding the behavior of solutions to such equations. In some cases there are explicit solution formulas, but usually there are no formulas for solutions yet we want to understand what happens (e.g., do solutions approach some limiting value as time evolves, or does "chaos" emerge?).

There are many visual and computational tools that can be brought to bear on this topic, such as linear algebra (e.g., eigenvectors and matrices), power series, numerical methods, and derivative matrices. We will learn about all of these methods, and then see how some ideas adapt to multivariable settings (with the aid of Fourier methods) to unlock the mysteries of partial differential equations. Along the way, we will encounter a vast array of sources of real-world motivation and utility for the results and concepts that are discussed.

By the end of this course, you should be able to:

• use visual tools to analyze solutions of low-order differential equations,
• apply insights from linear algebra to describe solutions to both linear and non-linear systems of differential equations,
• carry out a variety of numerical methods and be aware of their advantages and disadvantages, and
• compute Fourier series and Fourier transforms in many cases and use such computations to solve some important partial differential equations.

For a more detailed list of topics see the Schedule page.

First Day Checklist

Per University policy, your decision to take the course implies that you agree to these requirements and to the grading policies spelled out here; so be sure to read everything on these pages.

• Enrollment in lectures and sections: Math 53 students attend lectures on MWF, starting on the first Monday of the quarter, and discussion sections on TuTh, starting on the first Tuesday of the quarter.
  
  
• Required materials:
  • The textbook has been specially created by the Stanford Math Department in consultation with colleagues in many other departments; it is free and electronic-only. To get the book using your SUNet ID, visit the textbook page by clicking here or selecting the Textbook menu item at the top of this page.
    The book contains much more than is covered in the course. It also includes many fully worked examples, helpful for studying. We hope it will be a useful resource for topics that you may encounter in later coursework. On the second page of the introduction, you will find the e-mail address for reporting any corrections, typos, etc. The authors of the text are very eager to hear from you.
    
    
  • Calculators are neither required nor permitted for any exams in Math 53 (numbers are kept simple on exams). No coding is required in this course (but we will provide some software widgets to explore examples). Occasionally, homework problems may call for the use of a scientific or graphing calculator, and it is fine to use them for this purpose.
  
  
• Check for exam conflicts right away and contact us: Except in case of emergency, you must inform us of exam conflicts at least two weeks prior to the exam, together with a valid reason for the conflict. The allowable reasons are course-related or competition-related schedule.

  Exam 1 will be on Thursday, April 24, 08:00pm-10:00pm.

  Exam 2 will be on Thursday, May 15, 08:00pm-10:00pm.

  Final Exam will be on Saturday, June 7, 12:15pm-03:15pm.

  See the exam details and policies here.

  
  
• Access and Accomodations

  Stanford is committed to providing equal educational opportunities for disabled students. Disabled students are a valued and essential part of the Stanford community. We welcome you to our class. If you experience disability, please register with the Office of Accessible Education (OAE). Professional staff will evaluate your needs, support appropriate and reasonable accommodations, and prepare an Academic Accommodation Letter for faculty. To get started, or to re-initiate services, please visit oae.stanford.edu.

  If you already have an Academic Accommodation Letter, please use this form form to upload it and detail the specific accommodations you will need in this course. Letters are preferred by the end of week 2, and at least two weeks in advance of any exam, so we may partner with you and OAE to identify any barriers to access and inclusion that might be encountered in your experience of this course. New accommodation letters, or revised letters, are welcome throughout the quarter; please note that there may be constraints in fulfilling last-minute requests.

Class Structure and Assessment

Math 53 has an active learning structure; research has shown that pre-class reading, combined with daily participation in class activities targeted to specific learning goals, improves student learning outcomes in math and science courses. Furthermore, active learning increases student performances and narrows achievement gaps for historically underserved students. Here's what this means for us:

Both MWF class sessions and TuTh discussion sections are more interactive than traditional math classes:

• Twice each week (exception on weeks with exams) there will be a modest amount of reading in the course text to introduce some of the motivation behind the topic(s) to be discussed in class, along with an associated questionnaire on Canvas to be completed before class. In addition, there will be one check-in question on material from the previous chapter and lecture to re-enforce your learning; this check-in question will be graded for accuracy, you can think of it as a practice exam question. The responses to the rest of the questionnaire are not graded for correctness, just for a good-faith effort, to inform how the instructor organizes the classroom time around the learning goals for that day.
• The TuTh discussion sections focus on small-group collaboration with worksheets consisting of problems designed around the learning goals and themes in the homework and exam questions. The goal is to engage with the new skills and concepts, and to learn from your peers as well as from the guidance of a graduate student who answers questions. The work in discussion sections is aimed at giving practice with the material recently learned in the course; it is not graded, and complete solutions are provided later in the day for each TuTh worksheet.

Canvas questionnaire assignments on the twice-weekly pre-class reading: a typical questionnaire consists of 1 check-in question (always the first question) and 3 to 5 "low-stress" questions. Except for the check-in question, your answer do not have to be more than one or two sentences per question, and you get full marks for ANY good-faith answer. These assignments are intended to give the instructor feedback on how the reading went and how the course is going; think of them as surveys in which students are voting for which topics need more motivation in class (and which need less or none). Because we will have to review your feedback in a limited time period, the firm deadlines are:

• Mondays at 08:00am (delayed 24 hours if Monday is a University holiday).
• Fridays at 08:00am (except on exam weeks).
• First week Wednesday: As a "warm-up", in the first week we'll have an ADDITIONAL graded pre-class reading questionnaire, due on the first Wednesday at 08:00am.

• 20% Exam 1, 20% Exam 2, 30% Final Exam (see exams page for more details);
• 20% for weekly written homework assignments (total points earned divided by 80% of total possible points, not to exceed 100%);
• 10% for pre-class reading questionnaires on Canvas (total points earned divided by 80% of total possible points, not to exceed 100%).

Honor Code Policy

• Collaborate with another student or any other person on an exam.
• Copy from another's homework or exam, or allow another student to copy your work.
• Communicate with a person other than the teaching staff via email, text messaging, Google, any form of social media, messenger, chat rooms, message boards, etc., about anything related to the exam.
• Plagiarize material that you did not create, such as copying parts of posted solutions or text wholesale from anywhere, including the internet. The work that you submit must be your own.
• Share the exam questions or anything in your solutions with any other person for any reason. The restrictions on sharing exam content applies until 11:59pm on the exam date.

Generative AI Policy

''Absent a clear statement from a course instructor, use of or consultation with generative AI shall be treated analogously to assistance from another person. In particular, using generative AI tools to substantially complete an assignment or exam (e.g. by entering exam or assignment questions) is not permitted. Students should acknowledge the use of generative AI (other than incidental use) and default to disclosing such assistance when in doubt.

Individual course instructors are free to set their own policies regulating the use of generative AI tools in their courses, including allowing or disallowing some or all uses of such tools. Course instructors should set such policies in their course syllabi and clearly communicate such policies to students. Students who are unsure of policies regarding generative AI tools are encouraged to ask their instructors for clarification.''

Exam Proctoring

Instructor

• Dr. Lernik Asserian
  Email: lernik(at)stanford(dot)edu
  

Teaching Assistants

• Hongrui Chen
  Email: hongrui(at)stanford(dot)edu
  


• Milo Marsden
  Email: mmarsden(at)stanford(dot)edu
  


• Lucca Borges Prado
  Email: luccabp(at)stanford(dot)edu
  


• Yuefeng Song
  Email: songyf(at)stanford(dot)edu
  

Course Assistants

• Tanon Chokkhanchitchai
  Email: tchokkha(at)stanford(dot)edu
  


• Nikhil Pandit
  Email: npandit0(at)stanford(dot)edu
  

Office Hours and Other Resources for Help

You are encouraged to attend anyone's office hours, regardless of what section you are enrolled in. No appointment is necessary, just drop in at the scheduled office hours with your questions! Check Canvas for office hours and locations.

The following are additional resources available to you:

• SUMO Tutoring: For more information about SUMO tutoring and this quarter's schedule click here.
  
• CTL Tutoring: CTL offers free drop-in and/or by appointment tutoring for this course and a variety of other courses. For more information, the drop-in schedule, or to schedule an appointment click here.
  

Affordability of course materials: All students should retain receipts for books and other course-related expenses, as these may be qualified educational expenses for tax purposes. If you are an undergraduate receiving financial aid, you may be eligible for additional financial aid for required books and course materials if these expenses exceed the aid amount in your award letter. For more information, review your award letter or visit the Student Budget website.

Attendance

Attendance is not required at lecture, but regular attendance is important to your success in this class. A student who misses class is responsible for finding out what was discussed and learning the material that was covered on that day.

Discussion sessions are a great additional resource we have in Math 53. Held at various times on Tuesdays and Thursdays, they will provide opportunities to see more guided examples and try your hand at exercises with a member of the teaching team present. More exposure to and practice with the material will greatly aid to your learning.

Regarding covid adaptations, follow the university guidelines on classroom and course policies.

Students should attend the in-person lecture and discussion section to which they are officially assigned. This ensures that classrooms do not exceed official room capacity, and supports prompt notification should a specific section need to move online at short notice. Course messages are sent out via Canvas, please ensure that your Canvas notifications are on so that you receive any announcements promptly.

As standard practice, lectures and discussion sections in Mathematics courses are taught in-person. As such, Zoom links will not be provided. Additionally, in-person lectures and discussion sections will not be recorded.

Students who miss class due to illness should make arrangements to obtain lecture notes from other students in the class. As standard practice, there are no make-up exams or remote exams. If you will miss an exam due to illness, please reach out to your instructor for more information.

Stanford as an institution is committed to the highest quality education, and as your teaching team, our first priority is to uphold your educational experience. To that end we are committed to following the syllabus as written here, including through short- or long-term disruptions, such as public health emergencies, natural disasters, or protests and demonstrations. However, there may be extenuating circumstances that necessitate some changes. Should adjustments be necessary, we will communicate clearly and promptly to ensure you understand the expectations and are positioned for successful learning.