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AMATH 351 A: Introduction to Differential Equations and Applications

Summer Term: 
Meeting Time: 
MWF 1:10pm - 2:10pm
DEN 258
Ying-Jen Yang
Ying-Jen Yang

Syllabus Description:

See the PDF version of the Syllabus here.
Instructor: Ying-Jen Yang
Classes Time and Location: MWF 1:10~2:10 at DEN 258.
Office Hours: M 2:30-3:30  and Th 1:30-2:30 at the conference room in GRB B054
Teaching Assistant (TA): Matthew Farkas
TA's office hours: Fri 9:50-11:50 at DEM 024

Course Description

This course serves as the first course in differential equations after learning Calculus from MATH 124 and 125. In this course, we will learn how to solve Ordinary Differential Equations (ODEs) with examples drawn from physical, chemical and biological sciences and engineering. We will motivate most of our theoretical discussions from a application point of view and aim to illustrate the fundamental logic hidden behind different solution methods. In other words, tricks are not just tricks! See this Study Guide for more detail.
Warning! For those who love Laplace Transform and hate matrices, we will emphasize more on system of linear ODEs, blend in a modern dynamical system perspective of ODE, and entirely get rid of Laplace Transform in this course. The reasons, in short, are as follow.
  1. Laplace Transform is better learned after complex analysis, especially with other Integral transform method such as Fourier Transform.
  2. Laplace Transform does not help us solve more problems, just more efficient.
  3. System of linear ODEs does help us solve more problems and have a deeper understanding on ODE theories. 
  4. Dynamical system approach allow us to tackle nonlinear high dimensional ODEs!


Proficiency in manipulation of algebraic equations, evaluation of limits, methods of differentiation & integration at the level of MATH 124 & 125 are assumed. We will introduce and use quite a lot of linear algebra, some complex variable, and Power series that are relevant to us. Exposure to linear algebra, complex variable, and Taylor series would definitely be helpful. Our discussion would also motivate us to learn more on these mathematical subjects!

Textbook and Reference

No textbook is required. A self-contained, typed-up lecture note will be provided.
For more references, the following two books are recommended:
  1. W.E. Boyce & R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Chap 1-5 & 7-9)
    It is the standard textbook for undergraduate ODE class. Most of this course would be focussing on the material in it.
  2. Steven H. Strogatz, Nonlinear Dynamics and Chaos (Chap 1, 2, 5, and 6)
    It is an awesome textbook that is used in almost all undergraduate Dynamical Systems class throughout the world! We will only touch a little bit of it. You can take Amath 402/502 for a thorough introduction of the book and the subject.

Assignments, Examinations, and Grades (Tentative)

Homework is  assigned weekly. No late homework is allowed. At the end of the quarter, the lowest homework grade will be dropped.
We will have one in-class midterm one take-home midterm and one final (maybe take home maybe in-class):

Midterm 1: July 17 (Wed), Chap 1 and 2

Midterm 2: August 07 (Wed), Chap 1, 2, 3, and 4

Final Exam: August 23 (Fri), Chap 1 to 5

All examinations are accumulative because later content built on earlier one. Midterm 2 will focus on the new material. In all the exams, students are allowed to have one hand-written, letter-size, two-sided allowed, note sheet. No calculator is allowed. If you find a “really ugly” answer in the exam, there is a high chance you get the answer wrong.

Grades in 100% is computed by 50% Homework; 15% Higher Midterm; 10% Lower Midterm; 25% Final. The conversion formula from 100% to 4.0 GPA would be decided at the end of the quarter.


Chap 1: Differential Equations and Dynamical Systems

Have an overall introduction on the topic, get familiar with the terminology, and know how to classify different problems so that we can know how to solve them.
1.1 Introduction: Importance of DE and DS
1.2 Classification of ODE with examples from physics, chemistry, and biology
1.3 1D Dynamical System and Nonlinearity

Chap 2: 1st Order ODE

Given a 1st order ODE, how to find a suitable solution method and solve it.
2.1 Separable ODE
2.2 Linear: integration factor
2.3 Substitution: Bernoulli and “Homogeneous”
2.4 Exact 1st order ODE
2.5 Integration Factors to make ODE exact

Chap 3: General Theory for linear ODEs

Knowing how to solve 1st order ODE (i.e. 1-D ODE), extend it to solve linear ODEs with higher orders and/or higher dimensions.
3.1 Concept and General Solution
3.2 Matrix, determinant, and inverse matrix
3.3 Eigenvalues, Eigenvectors, and Diagonalization
3.4 Solutions to System of linear ODEs and Wronskian

Chap 4: 2-D, constant-coefficient, linear ODE

Apply and extend Chap. 3 to solve 2-D/2nd order, autonomous, linear ODEs.
4.1 Homogeneous: distinct real eigenvalues
4.2 Homogeneous: complex conjugated eigenvalues
4.3 Homogeneous: repeated eigenvalues and reduction of order
4.4 Summay for Homogeneous cases and Fixed point classification
4.5 Inhomogeneous cases

Chap 5: 2nd order, non-autonomous, linear ODE

Learn how to solve or find power series solutions to 2-D/2nd order, non-autonomous, linear ODE.
5.1 Special cases with closed-form solutions
5.2 Taylor series and Convergence
5.3 Ordinary point and Power Series solution

Chap 6: A glimpse of Nonlinear Dynamics

Have a taste on how to analyze nonlinear ODEs with our theories for linear systems.

6.1 Nonlinear ODEs and a geometrical way of thinking
6.2 Linear Stability Analysis

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: 
August 2, 2019 - 9:00pm