# Ordinary Differential Equations

Warren Weaver Hall, room 202, Tuesdays and Thursdays, 11am - 12:15pm
Courant Institute of Mathematical Sciences
New York University
Fall Semester, 2012

## Lecture materials

I will post reading assignments here both before and after class. Even before the first class, review background calculus material in chapter 0 (Preliminaries) of Coddington.

### 1. (Sept 4th) Intro and first-order ODEs

Based on sections 1-3 of chapter 1 of Coddington. First assignment posted also.

### 2. (Sept 6th) First and second-order ODEs

Finish all sections of chapter 1 and sections 1 and 2 of chapter 2.

### 3. (Sept 11th) Constant-coefficient second-order ODEs

Chapter 2 of Coddington, sections 1, 2, first half of 3, 4, and 6. The sections on higher-order equations will not be covered. Second assignment posted.

### 4. (Sept 13th) Variable-coefficient second-order ODEs

Chapter 3 of Coddington, sections 1, first part of 2, 5, and 6, and generalization/review of results for constant-coefficient equations. We will focus on second-order equations unlike the book.

### 5. (Sept 18th) Euler's equation

Explan use of complex numbers at end of section 2.2. Solve Euler's equation including use of complex numbers, section 4.2 in book. We will return to ch 4 and the rest of ch 3 later on.

### 6+7. (Sept 25th and 27th) Separable and exact nonlinear ODEs

Chapter 5 of Coddington, sections 1, 2 and 3, and a little bit on existence/uniqueness.

### 8. (Oct 2nd) Analytical series solutions

Chapter 3, section 7.

### 9. (Oct 4th) Existence and uniqueness

Proof of existence/uniqueness (not on midterm, more advanced).

### 10. (Oct 9th) Review

Review for midterm.

### 11. (Oct 11th) Midterm

Problems will be similar to homework problems, nothing tricky, just a check that you are not falling behind.

### 13. (Oct 23rd) Intro to Linear Systems of ODEs

We will review briefly key concepts from Linear Algebra and start discussing systems of linear first-order ODEs. Here are some lecture notes since this is not covered well in either book. The corresponding chapter in Miller & Michel is sections 3.1-3.3, but they are at an advanced level. My notes are mostly based on "The Qualitative Theory of ODEs: An Introduction" by Fred Brauer and John A. Nohel.

### 16. (Nov 6th) Constant-Coefficient Linear Systems: General Theory

Here is a summary Review of Linear Systems.

### 17. (Nov 8th) Introduction to Phase-Space Analysis

The next few lectures will focus on phase-space analysis of the qualitative behavior of the solutions to systems of ODEs. I will rely mostly on lecture notes posted here, based in part on the (optional) book by Fred Brauer and John A. Nohel. In the book by Miller & Michel, the material is in Section 5.6.

### 18. (Nov 13th) Linear Stability

I will rely mostly on lecture notes posted here, based in part on the (optional) book by Fred Brauer and John A. Nohel. In the book by Miller & Michel, the material is in Sections 5.1-5.5.

### 16. (Nov 15th) Nonlinear Stability

In the book by Miller & Michel, the material is in Sections 6.1-6.2.

### 19. (Nov 20th) Lyapunov Functions

Based on Sections 5.1 and 5.2 (optional) book by Fred Brauer and John A. Nohel.

### 20. (Nov 27th) Fun with ODEs: Fibonacci sequence

Guest lecture by Prof. Jonathan Goodman

### 21. (Nov 29th) Introduction to Numerical solution of ODEs

Guest lecture by Prof. Jonathan Goodman. Here are the Matlab codes (download them into one directory and then run 'matlab' and execute command 'ssim', Ctrl+C to stop the animation): fsim.m, ssim.m

### 22. (Dec 4th) Solving ODEs using Maple

Here is a Maple script (execute 'xmaple ODE_Maple.mw'). Here is a PDF and an html version.

### 23. (Dec 6th) Solving ODEs using Matlab

And here is a related Matlab script ODE_Matlab.m which solves the pendulum equation using Euler's method. And here is the script PredatorPrey.m to solve the Lotka-Volterra equations from Travis Askham's recitation.