Syllabus for Math 53, Winter 1999

Introduction

The subject of this course is ordinary differential equations (ODE), with a bit of linear algebra and some applications. A differential equation is an equation for an unknown function or functions, involving derivatives. The word "ordinary" indicates that we consider functions of only one variable, usually called "time". (The alternative is partial differential equations (PDE), which involve functions of more than one variable and their partial derivatives. PDE's are the subject of Math 131.) A bit of linear algebra is very helpful for understanding differential equations. We will briefly review some basic linear algebra which you should have seen in Math 51 or the equivalent, and we will also introduce some new linear algebra, particularly eigenvalues.

There are three basic approaches to studying differential equations:

Analytic: Find an explicit formula for the solution. This is great if it works, but it is sometimes difficult or impossible. There is no single general procedure to try; rather there is a "grab bag" of different tricks that work in different situations. Some of these tricks have beautiful theory underlying them.

Numerical: find an approximate solution using a computer. This is necessary in real world applications (such as predicting the weather) when an answer is required and the equations cannot be solved analytically. The computer can also give useful "experimental" information about complicated or difficult equations and produce pretty pictures. However the computer does not tell us "why" things work the way they do.

Qualitative: If the analytical and numerical approaches fail, or if we don't care about exact answers, we can still ask some general questions about how solutions behave. For example: Does a solution exist? Is it unique? Does it "blow up" and go to infinity, or does it stay bounded for all time? Does it converge to a steady state, or does it oscillate forever? Is the solution stable under small changes in the initial conditions? One can sometimes answer questions like these without knowing exact formulas for the solutions. On the other hand, this abstract style of reasoning may not always help one understand particular examples.

In this course we will emphasize the analytic and qualitative approaches. Regarding numerical methods, we will play with calculators a little bit, and there will be at least one extra credit computer project for those who are interested.

Course outline

Don't worry, this isn't quite as much material as it may appear. We won't go too fast, and we might not cover every single topic in this outline.

First-order ODE. (Edwards and Penney, most of Chapter 1 and part of Chapter 2)
- Analytical techniques:
  - Separable equations.
  - Linear equations.
  - Substitution.
  - Exact equations and level curves.
- Qualitative concepts:
  - Slope fields and solution curves.
  - Existence and uniqueness of solutions. Subtleties.
  - Equilibrium and stability.
- Numerical technique: Euler's method.
Linear ODE. (E&P, most of Chapter 3)
- Brief review of some linear algebra.
  - Vector spaces, particularly spaces of functions.
  - Linear independence and dimension.
  - Linear transformations and matrices.
  - Homogeneous and inhomogeneous linear equations.
- Linear differential operators.
  - Existence and uniqueness for appropriate initial conditions.
  - n independent solutions without initial conditions.
- Solving homogeneous constant coefficient ODE.
  - How complex numbers can give real answers.
- Solving inhomogeneous ODE via undetermined coefficients.
- Application: Vibrations. Damping, forcing, and resonance.
MIDTERM.
Systems of linear ODE. (E&P, parts of Chapters 4 and 5)
- Generalities.
  - Reduction of higher order equations to first order systems.
  - Existence and uniqueness theorems.
- More linear algebra.
  - Review of determinants.
  - Eigenvalues and eigenvectors.
- Solving constant coefficient linear systems.
Introduction to dynamics of nonlinear systems. (In my opinion, this is the coolest part of the course, and much of the above material comes together here. For a preview, look at the pictures in Chapter 6 of E&P.)
- Critical points of two dimensional systems.
- Application: ecological models.
Laplace transforms. (E&P, some of Chapter 7)
- Basic properties of LT's.
- Using LT's to solve initial-value problems.
REVIEW.
FINAL.

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Last updated: Jan. 4, 1999.