Math 307 A/B, Fall 2006,

Introduction to Differential Equations.

Instructor: Ian Langmore

Email:

Office: Padelford Hall C-8-F (directions).

Office Hours: Tues-Thurs 3:30-5 or by appointment (I am in my office most days, so just email me and ask to meet)

Textbook: Elementary Differential Equations and Boundary Value Problems, 8^th edition.

Click here for the official syllabus (serves as a basic outline)

Weekly Assignments

Grading

Homework:  30%

Midterm 1:  20%

Midterm 2:  20%

Final:         30%

I am teaching two sections, and expect the mean grade to be around 3.0.

Use of computers

You may use any computer program you want to assist you in solving these equations.  If a computer solves the equation for you, you still need to write out all the individual steps in your homework.  If you have any question regarding MATLAB, or a simple Mathematica question, I will be happy to assist.

Comments about the book

For the most part this course will follow the book by Boyce and DiPrima.  This book is a step up in difficulty from Stewart’s Calculus.  It demands more from the reader.  Read actively, i.e. if a statement made in the book is not obviously true, then get out a pencil and paper and check for yourself; do this even if you are simply writing down the same steps that already appear in the book!

Below are some interesting things about differential equations

Why are differential equations fundamental to science?

Science strives for some fundamental understanding of the world around us, and the universe as a whole (see for instance TOE).  In an attempt to understand the motion of the planets, people such as Johannes Kepler described this motion in fairly precise mathematical terms, developing equations describing the path of motion that a planet will take.  This motion was described in terms of ellipses. This sort of description is unsatisfactory since it gives no means of predicting the motion of more than two planets, or cannons, or bowling balls etc…

When Isaac Newton wrote his Principia this changed.  Newton’s insight was that while there are no fundamental laws describing the path of motion (is it a circle, an ellipse, or what?  Imagine trying to come up with a fundamental law describing what the path taken by a tennis ball would be…) there are fundamental laws describing how that motion will change.  Mathematically changes in motion are phrased in terms of derivatives, and therefore we are led to differential equations.

    Since Newton’s time, other fundamental laws have been phrased in terms of differential equations, most notably:

• Maxwell's equations, describing electromagnetics

• The Shrodinger equation, describing the space and time dependence of a quantum mechanical system

• The Navier-Stokes equations describing fluid dynamics

• The Einstein equations describing general relativity

• String Theory, a current attempt at a “theory of everything” uses lots of differential geometry, which as the name implies uses differential equations

• The Black-Scholes equation in mathematical economics

• The Lotka-Volterra equations in Mathematical Biology

Where does math 307 fit in with this?

        Most differential equations are impossible to solve exactly.  Scientists and mathematicians are then often content using tools from analysis, numerical analysis, nonlinear dynamics, and optimization to get information about the equations.  We do however know how to solve some differential equations exactly.  We will study a few of these in math 307.  While this may sound like a cop-out, it is not.  The study of these simple equations is important since it provides intuition that can be used later while analyzing more complex equations.