We now into the last major topic of the course, namely systems of first-order differential equations. We'll first study linear systems and will be able to find explicit solutions. This is the emphasis in Chapter 6 of the text. After that, we will look at nonlinear systems and generally will not be able to find explicit solutions. Instead, we develop ways of understanding the behaviour or nature of solutions without knowing the solutions explicitly. Much of this will involve a geometric point of view. I will introduce this geometric view while we study linear systems.
Given a system of two first-order differential equations, we can plot a direction field in the plane. On this direction field, we can choose an initial point and then sketch the curve that is tangent to the vectors of the direction field. This is called an integral curve. A phase portrait is a collection of integral curves for a variety of initial points. It is convenient to use technology to draw direction fields and integral curves.
One nice tool for this is the applet ODE 2D Calculator by Marek Rychlik at the University of Arizona. After the applet loads (which make take a few seconds), you'll see a direction field plot and, below it, boxes for making changes. You might want to start with the example we did in class, namely
An online tool you might find handy is the Matrix Calculator at the WIMS site. (Look "Matrix calculator" on the list of online calculators at this site.) You can use this to get eigenvalues and eigenvectors for a matrix you enter. I've tried this with a few small matrices and it seems to be reliable.
Exam #3 is a take-home exam. It is due on Thursday, December 1 at 4 pm.
Modeling Project #3 is due on Wednesday, December 7 at 4 pm.
| Section | Problems to do | Submit | Target or due date | Comments |
|---|---|---|---|---|
| 1.1 | #1-11 odd, 12-16 | None | Wednesday, Aug 31 | |
| 1.2 | #1-6,9,10,14-21 | None | Thursday, Sept 1 | Do at least two of #1-6 by hand. |
| 2.1 | #1-13 odd, 16 | None | Friday, Sept 2 | |
| 2.2 | #3,7,9,14-16 | None | Wednesday, Sept 7 | |
| 2.3 | #3,5,11,15,23,25,28,29,30 | #32 | Friday, Sept 9 | |
| 2.4 | #5,11,12,13,14,15 | #14 | Friday, Sept 16 | |
| 2.5 | #2,4,6,13 | #6 | Wednesday, Sept 21 | |
| 3.1 | #3,7,11 | None | Monday, Sept 19 | |
| 3.2 | #3,7,9,11,13,15,21,25,27,30 | None | Wednesday, Sept 21 | |
| 3.3 | #3,5,7,13 | None | Thursday, Sept 22 | |
| 3.4 | #3,5 | None | Friday, Sept 23 | |
| 3.5 | #3,5,6,7,11 | None | Monday, Sept 26 | |
| 3.5 Bonus | #1,2 from handout | None | Wednesday, Sept 28 | |
| 3.8 | #3,9,11 | None | Friday, Sept 30 | |
| 4.1 | #1,3,8 | None | Monday, Oct 3 | |
| 4.2 | #5,9,11,17 | None | Thursday, Oct 6 | |
| 4.3 | #15,19,26 | None | Thursday, Oct 6 | |
| 5.2 | #5,10 | None | Friday, Oct 7 | |
| Linear theory | #1,2 from handout | None | Friday, Oct 7 | |
| 4.4 | #3,9,11,15,19,21,22 | None | Monday, Oct 10 | |
| 4.5 | #3,5,9,15,19 | None | Wednesday, Oct 12 | |
| 4.6 | #1,2,3,7,9,11,19,21,25 | None | Thursday, Oct 13 | |
| 4.7 | #3,4,5,7 | #10 | Friday, October 21 | |
| 4.8 | #3,5,7,11,15 | None | Friday, October 21 | |
| 4.9 | #1-17 odd, 31 | None | Monday, October 24 | |
| 4.10 | #1,3,7,11,15 | None | Wednesday, October 26 | |
| 6.1 | #13,15,21-24 | None | Friday, November 4 | |
| 6.2 | #1,5,7,9,13,15,19 | None | Monday, November 7 | |
| 6.3 | #5,7,9,13,15 | None | Monday, November 7 | |
| 6.5 | #21,24,25,29,34,36,39 | #40 | Monday, November 14 | |
| 6.6 | #7,11,15,17,23,29 | None | Monday, November 14 | |
| 6.7 | #5,7,12 | None | Thursday, November 17 | |
| 6.8 | #3,5,6,21,25 | None | Friday, November 18 | |
| Matrix exponential | #1-5 from handout | None | Monday, November 20 | |
| 8.1 | #14 | None | Friday, December 2 | |
| 8.2 | #1,5,9,24-27,31,35 | None | Friday, December 2 | |
| 8.5 | #1,5,9 | None | Monday, December 5 |
The Mathematical Atlas describes the many fields and subfields of mathematics. The site has numerous links to other interesting and useful sites about mathematics.
If you are interested in the history of mathematics, a good place to start is the History of Mathematics page at the University of St. Andrews, Scotland.
Check out the Astronomy Picture of the Day.
You can look at exams from last time I taught Math 301. You can use these to get some idea of how I write exams. Don't assume I am going to write our exams by just making small changes to the old exams. You should also note that we were using a different textbook so some of the notation is different. There are also differences in the material covered on each exam. You can use this old exam to prepare for our exam in whatever way you want. I will not provide solutions for the old exam problems.
Today we talked briefly about slope (or direction) fields for first-order differential equations. If you have a TI-86, you can coax it into drawing slope fields. There are some brief instructions below. A better option might be one of the Java applets available on the web. One I've tried and like is the Slope Field Calculator by Marek Rychlik at the University of Arizona.
You should learn how to use some tool that produces slope fields. With whatever tool you try, start with the simple examples we did in class and then try some of the equations you encounter in the text or homework assignments.
Here's some details on slope fields on a TI-86. ( For full details, consult a manual. You can get the manual on-line at the TI education web site.) The first step is to go to the MODE menu and choose the option DifEq. Exit the MODE menu and go to the GRAPH menu. The F1 key should correspond to Q'(t)=. Select this to open the window in which you can enter the differential equation (using t as the independent variable and Q1 as the dependent variable. After you enter the equation, use the F5 key to start the plotting. You will get a slope field and a solution curve for the specified initial condition. You can change the initial condition using the INITC window accessible by the F3 key.
In attacking the problems from Sections 2.2 and 2.3, it is best not to try "plugging in" to the general form of the solution that we derived in class and is in the text. There are things to be learned from having an expression for the general solution in hand, but for solving any specific problem it is best to essentially follow the steps we used in getting the general expression. With a little practice, you'll find the steps to be quite natural.
Many techniques for solving differential equations come down to finding antiderivatives. You will need to recall some basic antidifferentiation techniques such as substitutions and integration by parts. You can also make use of integration tables and machine integration tools such as those available on the TI-89. Another resource is The Integrator web site provided by a company called Wolfram. This software company produces the program Mathematica, a general purpose mathematical program that can do symbolic and numerical calculations and produce graphics of all sorts. I'll demonstrate some of the capabilities in class some time this semester.
As homework, you should finish analyzing the "two tank" mixing model that we set up in class. I've also assigned homework from Section 2.4. This sections covers several modeling applications.
Here's a handout with some notes on writing in mathematics.
Here's the Euler method Excel spreadsheet that I built in class.
I've added an Old exams section below with links to copies of exams from the last time I taught this course.
We are currently looking at the basic theory for homogeneous linear differential equations. The most useful result is that the solution set of an nth order homogeneous linear differential equation is a vector space of dimension n. Thus, we can describe all solutions for a given equation by finding a basis of solutions. We'll soon turn our attention to the issue of finding solutions.