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AMATH 351 A: Introduction To Differential Equations And Applications

Meeting Time: 
MWF 10:30am - 11:20am
CSE2 G10
Eli Shlizerman
Eli Shlizerman

Syllabus Description:

General information 

Instructor: Eli Shlizerman,

Lectures: MWF 10:30-11:20 in CSE2 G10

Office hours (LEW  230D):

M 1:00-2:00

T 4:00-5:00


TA: Bea Stollnitz,
TA office hours (LEW 115):

M 2:00-3:00

W 2:00-3:00



Update: LAST WEEK of Class

According to UW guidance

Activities will continue as planned but are shifted to exclusively ONLINE for the remaining duration of the course. In particular:

Lectures will be posted online through Panopto Recordings 

Office Hours will be conducted through ZOOM

Office Hour Schedule (Updated):


Monday 1pm - 2pm:

Tuesday 4pm-5pm:

Thursday 1pm-2pm:


Monday 2pm - 4pm: (Links to an external site.)

Wednesday 2pm-3pm:

Thursday 2pm-3pm:


Tests will be conducted online through ZOOM as well. 

We are planning to keep a similar schedule as with previous weeks. We will announce any changes and updates here. 


Course description

An introductory survey of ordinary differential equations, linear and nonlinear equations, Taylor series, Laplace transforms, and linear systems of ODEs. Emphasizes formulation, solution, and interpretation of results. Examples drawn from physical and biological sciences and engineering.



Proficiency in manipulation of algebraic equations and methods of differentiation and integration at the level of MATH 124 & 125 are required.



W.E. Boyce & R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems.

The text is not required, but obtaining some edition of it is to use as a reference is strongly recommended.

Additional: Lecture Notes from a previous iteration of the course available here.

I will post my written notes after each lecture, see Course Schedule page.


Suggestion box

If you have a suggestion or any comments on anything related to the course that you do not feel comfortable bringing up with me in person, I have created an anonymous suggestion survey. I urge you to bring issues to my attention sooner rather than later. I'll do my best to address your concerns in a timely manner.

The survey can be accessed here.



There will be three in-class examinations and a final, tentatively scheduled as follows:

  • Exam 1: Friday, January 24  (week 3)
  • Exam 2: Monday, February 10 (week 6)
  • Exam 3: Friday, February 28 (week 8)
  • Final: Monday, March 16, 8:30-10:20AM (finals week)

Each in-class exam will contain questions from every key skill previously covered in class, giving you multiple opportunities to demonstrate mastery. For example, if exam 1 covered skills 2-4, then exam 2 could have questions from skill  3-10 (skill 2 is only available through exam 1). The final will include questions from all key skills apart from skill 2.

In addition to the in-class examinations, I will also schedule weekly one-hour sessions out-of-class for you to attempt to demonstrate mastery. At the end of each week I will put up an optional poll where you can specify which key skills you would like to be tested on and choose from a list of possible times. I will select the time that works for most people and a room is available. There will be no testing sessions on the weeks when we already have in-class exams (weeks 3, 6, and 8).

No calculators, computers, collaboration, or cheat sheets will be allowed on exams.



In this course, we will use a mastery-based approach. Associated with the class are 14 key skills. Successful students should demonstrate mastery of all of them by the end of the quarter. Your grade is determined by the number of key skills you master during the course.

Number of mastered skills Grade


13 3.9
12 3.8
11 3.7
10 3.6
9 3.4
8 3.2
7 3.0
6 2.5
5 2.0
4 and below 0.0

The examinations are your opportunity to demonstrate mastery of the skills. Problems on the examinations will be labeled with corresponding key skills. You need only demonstrate mastery in a particular area once. You may attempt to demonstrate mastery multiple times. You will not, in general, be penalized for when a skill is mastered.

Key skills

  1. Homework

    • Demonstrating mastery in this category requires earning a satisfactory grade on all homework assignments (see below for what constitutes a satisfactory grade).
  2. Prerequisite knowledge (must be satisfied by or on the first exam)

    • Derivatives: Students are able to differentiate common functions and combinations of common functions using the following techniques.

      • Derivatives of common functions
      • Product/quotient rule
      • Chain rule
    • Integration techniques: Given an integral, student is able to both identify the appropriate method and correctly apply the method to solve it. Students are expected to be familiar with the following methods.
      • Anti-derivatives of common functions
      • (U-) Substitution
      • Integration by parts
      • Partial fractions
    • Students can perform algebraic manipulations (this includes properties of fractions, exponentials, radicals, and logarithms)
  3. Fundamental ODE knowledge (mandatory)

    • Student is able to identify ODEs and initial value problems
    • Student is able to determine whether a given function is a solution of a differential equation
    • Student can classify differential equations based on their linearity, coefficients, dimension, order, type (ordinary vs. partial), and homogeneity
    • Student can distinguish between explicit and implicit solutions to ODEs
  4. Direction field analysis

    • Student is able to derive the direction filed formula for a first order ODE
    • Student is able to compute the direction field for values of interest
    • Student can interpret the direction field of a first order ODE
    • Student can identify equilibrium points of a first-order autonomous ODE and classify their stability
    • Student is able to draw a qualitatively correct phase line diagram for first-order autonomous ODE
  5. Phase plane analysis

    • Student is able to compute nullclines and equilibrium points for first-order 2D autonomous ODE
    • Student is able to draw a qualitatively correct phase plane diagram for a given 1O 2D ODE
    • Student can identify directions of nullclines
    • Student can qualitatively interpret the solution of 1O 2D ODE based on the phase plane
  6. Solving first-order ODEs

    • Student can solve a first order ODE using one of the methods from class:  method for separable equations, integrating factor, method for exact equations, or substitution
  7. Identifying appropriate solution technique for first order ODEs

    • Given a first order ODE, student is able to identify an applicable analytic solution technique or recognize that none of the techniques discussed will work
  8. Systems of ODEs and the method of Ansatz

    • Student can express high-order linear differential equations as a system of first-order differential equations
    • Student can use an ansatz to solve a differential equation.
  9. Second-order linear equation theory

    • Student is able to distinguish between homogeneous, particular, and general solutions to an ODE and explain the differences between them
    • Student is able to explain the superposition principle and use it to find a general solution
    • Student can determine whether or not two functions are linearly independent using the Wronskian or another technique
    • Given one solution to a second order linear ODE, student is able to use Abel's theorem/reduction of order to obtain a second linearly independent solution
  10. Variation of parameters

    • Student can use variation of parameters to solve second order linear constant coefficient ODEs
  11. Series solutions

    • Student can identify ordinary points of an ODE
    • Student can find a series solution for a given ODE
  12. Laplace transform

    • Student can use the Laplace transform and its fundamental properties to solve second order linear constant coefficient differential equations
  13. Mathematical modeling

    • Given a description of a physical system or phenomenon, student is able to construct a differential equation which models it
  14. Identifying appropriate solution technique for ODEs

    • Given an ODE, student is able to identify an applicable solution technique or recognize that none of the techniques discussed will work

The fine print

  • In order to obtain a passing grade, students must demonstrate mastery in skill 3, i.e. skill 3 is mandatory.
  • Students will only be allowed to demonstrate mastery in skill 2 during the first three weeks of class (up to and including exam 1). Also homework 1, 2 that primarily focus on skill 2 will have a passing grade of 16/18pts. If you have difficulties with these assignments consider to fulfill the prerequisites first and take the course in the future.



Weekly homework assignments will allow students to practice the skills presented in class. Homework assignments will typically be assigned on Wednesdays and will be due a week later. 

Homework must be submitted as a single PDF file , legible writing, and portrait mode on canvas.

Collaboration: Collaboration is encouraged, but every student must submit their own assignment consisting of their own work. Below is a reminder of the UW policy on academic misconduct. Please discuss with me or one of the TAs if you have concerns about what constitutes misconduct. 

Late assignments: No late assignments will be accepted for any reason. In the rare event of an emergency, with sufficient documentation, a homework will be dropped.

Scoring: Each homework assignment will be graded for completeness, with a few problems selected at random to be graded for correctness. Students are urged to treat each problem seriously, as it is possible that any problem will be graded for correctness. Each homework is worth 18 points, and you need 132 points to earn skill 1, which is an average of 14.67 per homework.The TA will take completeness, presentation (showing work, clarity), approach and correctness of the final answer into account in grading your assignments. 


Attendance is not recorded nor taken into consideration in grades. However, it is highly recommended that students attend lecture to keep up with the material.


Policy on Academic Misconduct
It is essential that students in fulfillment of their academic requirements and in preparation to enter their profession shall adhere to the University of Washington’s Student Code of Conduct. Any student in this course suspected of academic misconduct (e.g., cheating, plagiarism, or falsification) will be reported to the University’s Office of Community Standards and Student conduct. 


University of Washington Religious Accommodations Policy

Washington state law requires that UW develop a policy for accommodation of student absences or significant hardship due to reasons of faith or conscience, or for organized religious activities. The UW’s policy, including more information about how to request an accommodation, is available at Religious Accommodations Policy ( Accommodations must be requested within the first two weeks of this course using the Religious Accommodations Request form (


Catalog Description: 
Introductory survey of ordinary differential equations; linear and nonlinear equations; Taylor series; and. Laplace transforms. Emphasizes on formulation, solution, and interpretation of results. Examples drawn from physical and biological sciences and engineering. Prerequisite: MATH 125 or MATH 135. Offered: AWSpS.
GE Requirements: 
Natural World (NW)
Last updated: 
December 18, 2019 - 9:49am