### MATH 6307 Ordinary Differential Equations 1

### Fall 2015

### MWF 10:05-10:55, Skiles 270

### Office Hours: MWF 11-12, Skiles 270.

### Textbook

The class text book is:

Nonlinear Differential Equation and Dynamical System

Ferdinand Verhulst

Springer, 2nd edition

I'll also use the lecture notes by prof. Jack Hale. The notes are linked below.

The notes are an update and extension of the book:

Ordinary Differential Equations.

Jack K. Hale

Dover

### Syllabus

See th online syllabus.

### Grading

There
will be two midterms and one final. The first midterm will be September
25 and the second October 30. Depending on the results of the first
midterm, the second midterm can be in the form of a take home
assignemet. During the class we will discuss the possibility of a
project based final exam. I will assign HW and collect them every 2 or
3 weeks. The final
grade will be based on HW (15%), midterms (40%) and final (45%).

**First week**

- 1.1 Definition and Notation.
- 1.2 Existence and Uniqueness. (Notes: 1.1, 1.2 and 1.3)

- 1.3 Gronwall's Inequalities. (Notes: 1.4)

- 2.1 Phase Space, Orbits.
- 2.2 Critical Points and Linearization.

Second Week

- 2.3 Periodic Solution

- 2.4 First Integral and Integral of Motion

- 2.5 Evolution of a Volume Element, Liouville Theorem

Third Week

- 3.1 Two Dimensional Linear System

- 3.2 Remark on Three-dimensional linea systems
- 3.3 Critical Point of Nonlinear System

First Homework: from the textbook ex: 2.2, 2.4, 2.6, 3.2, 3.5. Homework due September 14.

Fourth-Seventh Week

- 4.1 Bendixon's criterion

- 4.2 Geometric auxiliaries, preparation for the Poincare-Bendixon theorem.

- 4.3 The Poincare-Bendixon theorem.
- 4.4 Application of the Poincare-Bendixon theorem.
- 4.5 Periodic solution in Rn

Second Homework: from the textbook ex: 4.2, 4.5, 4.6, 4.8. Homework due October 23..