- Summer 2019

### Syllabus Description:

### Course Description

- Laplace Transform is better learned after complex analysis, especially with other Integral transform method such as Fourier Transform.
- Laplace Transform does not help us solve more problems, just more efficient.
- System of linear ODEs does help us solve more problems and have a deeper understanding on ODE theories.
- Dynamical system approach allow us to tackle nonlinear high dimensional ODEs!

### Prerequisites

### Textbook and Reference

- 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. - 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)

**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.

### Outline

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.*

*Given a 1st order ODE, how to find a suitable solution method and solve it.*

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.*

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

*Apply and extend Chap. 3 to solve 2-D/2nd order, autonomous, linear ODEs.*

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.*

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