**MATH 404**

CLN , TTh 1:00-2:30, EH 4088
# Intermediate (Applied) Differential Equations

(Prerequisites: MATH 216, 256, 286, or 316)

## Instructor:

Dr. Patrick Nelson

East Hall 3071

tel: 763-3408

fax: 763-0937

pwn@math.lsa.umich.edu

office hours: Tues 3-4:30, Wed 9:30 - 11:00, and by appt.

## Course Description

There are three main objectives to this class. First, we will present and explain mathematical
methods for obtaining approximate analytical solutions to differential equations that
cannot be solved exactly. The material in this section will mostly come from Bender and
Orszag and we will present it in a very introductory manner. Second, we will introduce and
explain the theories of dynamical systems, specifically bifurcation and chaos, phase portraits,
linear stability analysis, and local and global behavior of both linear and non-linear differential
equations. Third, we will use the computer via matlab and maple to visualize what we are learning
and to gain a better understanding of the dynamics. Learning the material will be done through
homework that will require both pen and paper analysis and computation support. I would expect
all students to have had an introductory course in differential equations and
some familiarity with matlab. There will be a special session or two on matlab for students who
do not have familiarity with this computation method.

## Textbooks

- Advanced Mathematical Methods for Scientists and Engineers (required),
Carl M. Bender and Steven A. Orszag,
* McGraw-Hill, 1978*,.
This book presents the methods of asymptotics and perturbation theory for obtaining
approximate analytical solutions to 'real' differential equations that arise in
physics and engineering. These equations are usually not solvable in closed form and
numerical methods may not converge to useful solutions. The aim is to teach the insights
that are most useful in approaching new problems and it avoids the special methods and
tricks that work only for particular problems. We will use this book in an introductory
manner where some of the more advanced topics will not be covered.
__Non-linear Systems (recommended)__, P.G. Drazin, *Cambridge texts in Applied Mathematics*,1994.
__Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields (recommended)__,
J. Guckenheimer and P. Holmes, *Springer*

## Mathematical concepts to be covered

- Ordinary differential equations
- Approximate solutions of linear differential equations
- Approximate solutions of non-linear differential equations
- Stability analysis, bifurcations, and limit cycles
- Phase portraits
- Perturbation theory
- Boundary layer theory
- Bessel, parabolic cylinder, and Airy functions
- Numerical analysis with Matlab

## Learning Objectives and Instructor Expectations

The objective of this course (as worded in Bender and Orszag) is to help young and
established scientists and engineers to build the skill necessary to analyze equations
that they encounter in the real world. Asymptotic and perturbation analysis are some
of the most useful and powerful, as well as beautiful, methods for finding approximate
solutions to equations. Combining these techniques with those of dynamical systems and
computation provide the student with a powerful tool for analyzing most ordinary
differential equations.

## Grading

Homework assignments will count as 40% of grade evaluation. There will also be two midterms
worth 30% of the grade and a final that counts for 25%.
The remaining 5% is student participation.