Section 1.1 of the text introduces lots of terminology that we will use in class. You will need to learn this language by carefully reading the text and doing the assigned problems. I will not go over this language systematically in class but will begin using it regularly. You will need to ask questions if the language is not making sense to you.
Today we talked about slope fields for first-order differential equations. (The text calls these direction fields but I want to reserve this phrase for something different later in the course.) There are many Java applets available on the web that will draw slope fields. One I've tried and like is the Slope Field Calculator by Marek Rychlik at the University of Arizona. You should try it with the simple examples we did in class and with some of the problems from the Section 1.2 homework assignment. Note that this applet uses x as the independent variable whereas we have been using t in class.
The process for solving many differential equations comes 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.
Over the last two classes, I've tried to give you a way of thinking about the general solution to a nonhomogeneous linear first-order differential equation as given in the text's Equation (6) or the specific solution as given in Equation (8). Each of the two terms has an interpretation.
We have not yet talked about the Existence-Uniqueness Theorem so I've pushed back the dates for assignments. The additional problems from Section 2.3 concern the Existence-Uniqueness Theorem. You should look at Problem 5 from section 2.4 as it deals with an example of continuously compounded interest.
Here's a web site I just learned about: simpsonsmath.com. Check it out if your bored.
Our first exam is scheduled for Tuesday, February 14 from 11:00-12:20. It will cover material from Chapters 1 through 3 of the text.
I'm leaving it to you to read and understand the EUT in Section 4.1.
We will approach the theory of linear differential equations using the language and tools of linear algebra. The text is written for an audience who might not know linear algebra so there will be differences between what we do in class and what is done in the text. The main ideas are the same.
The text defines a fundamental set of solutions for a homogeneous linear differential equation. This is equivalent to saying that the set is a basis for the set of all solutions. The set of all solutions is a subspace of Cn(a,b) and can be thought of as the null spaceof a linear transformation.
The main results from Sections 4.2 and 4.3 are summarized in the language of linear algebra on this handout from today's class.
The text deals with second order linear equations in Chapter 4 and generalizes to nth order linear equations in Chapter 5. I've assigned a few problems from Chapter 5 since we have been doing the general theory in class.
Today, we defined the complex exponetial ea+ib where
a and b are real numbers. Part of the definition is given by
Euler's formula as
eiθ = cosθ + i sinθ.
The other part of the definition is
ea+ib = ea eib.
Putting these together, we have
ea+ib = ea eib
= ea(cos b + i sin b)
= ea cos b + i ea sin
b
Thus, the real part of ea+ib is ea
cos b and the imaginary part of ea+ib is ea sin
b.
I've changed the due date for the problem from Section 4.7 to Monday. Here are some comments that might help with this problem.
You should finish the handout from class as part of your homework. As you do these, you will need to develop some feel for what works as a good "judicious guess." As part of this, think about what happens to specific functions as you run them through the derivatives in the differential equation.
Exam 2 will be on Tuesday, March 22 from 11:00-12:20. It will cover material through Chapter 4 plus the generalizations to nth order equations from Chapter 5.
Modeling Project 2 will be due on Monday, March 27. We've developed enough relevant mathematical tools for you to work on this project now.
Variation of parameters is a method for generating a particular solution to a nonhomogeneous linear problem given the complementary solution for the problem. It has the advantage of working for more general equations than the method of undetermined coefficients and the disadvantage of requiring antiderivatives that may be difficult or impossible to evaluate in terms of elementary functions.
We are 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. The text introduces it later (in Chapter 8).
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 a simple example, namely
As homework, use technology to explore phase portraits of linear systems of the form
for constants a, b, c, and d. As you explore various choices of these parameters, categorize the phase portraits into classes having similar features. Find a simple representative for each of the classes you find. Print out a phase portrait for each representative.
Section 6.1 of the text deals with the calculus of matrix functions. Most of this can be summarized by the statement compute limits, derivatives, and integrals entry by entry. Section 6.2 gives the existence-uniqueness theorem for linear systems. Sections 6.3 and 6.4 give the theory of the structure of solutions for homogeneous systems. The systems theory handout covers the same ideas using the language of linear algebra (so we talk about a basis for the solution space rather than a fundamental set of solutions).
Section 6.5 looks at solving constant-coefficient homogeneous systems. This leads to the eigenvalue problem for the coefficient matrix. You should be familiar with eigenvalues and eigenvectors from linear algebra. The text gives some important results. You will need to find an efficient way to compute eigenstuff. One option is Mathematica; this Mathematica notebook gives some details on getting eigenvalue/eigenvector pairs using Mathematica.
An online tool you might find handy is the Matrix Calculator at the WIMS site. (Look for "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.
In class, we will develop a geometric view alongside the analytic view. The text takes an exclusively analytic view in Chapter 6. The geometric view is addressed in Chapter 8. Part of Section 8.6 is relevant to what we did in class today. You should refer to Section 8.6 as we look at different combinations of eigenvalues over the next few class sessions.
On Wednesday, I accidently skipped over the complex eigenvalues case and started in on the repeated eigenvalues case. Today we backed up to take care of complex eigenvalues. On Monday, we'll continue with the repeated roots case. This will lead us into thinking about defining the matrix exponential.
I'll put together a handout on the geometry of ellipses and how some linear algebra ideas can be very useful.
The text's approach to finding additional solutions in the case of repeated roots differs slightly from what we did in class. First, there is a minor notation difference: the text uses v1 and v2 where we used v0 and v1. We got the equivalent of the text's Display (8) on page 313. The text approaches this system of equations by solving the top equation for v1 (noting that this is the eigenvalue problem for the matrix A) and then substitutes this solution into the bottom equation to get a nonhomogeneous equation that is solved for v2. In class, we first substituted from the bottom equation for v1 in the top equation to get (A-αI)2v2=0. We solve this for v2 and then use the bottom equation to compute v1. For the case of algebraic multiplicity 2, geometric multiplicity 1, there isn't much difference between the two approaches. For higher algebraic multiplicity, our approach in class can be more efficient. I've assigned some problems in Section 6.7 that deal with higher algebraic multiplicity.
Defining the exponential of a matrix will gives us a unified way to look at solutions for systems of equations. We'll also use a matrix exponential in a version of variation of parameters that generalizes what we did for a single nonhomogeneous linear first-order differential equation. The text discusses the exponential of a matrix in Section 6.11.
The first few problems in Section 6.8 show how to generalize the method of undetermined coefficients (a.k.a judicious guessing) to systems.
We will take a brief digression into some linear algebra to recall tools that can be very useful in a variety of situations. You should review the following ideas: similar matrices, similiarity transformations, diagonalizable matrices, and orthogonal matrices.
We looked at two applications of similarity transformations. In each application, the tranformed thing is easier to work with. In the best case, the relevant matrix is diagonalizable so the transformed thing is particularly nice.
Exam #3 is a take-home exam. It is due on Friday, April 21 at 4 pm.
For Modeling Project #3, you will describe and model a scenario involving two (or more) interacting species. A summary of your scenario and model are due on Monday, April 24. Your full report is due on Wednesday, May 3.
Section 8.1 gives an Existence-Uniqueness Theorem for nonlinear systems. This theorem parallels the EUT of Section 3.1. I've assigned one problem from this section. Section 8.2 deals with direction fields and equilibrium points. I assigned problems from this section when we first looked at systems of equations. We'll skip over Section 8.3 for now and come back to this later. In class, we looked at linearization which the text covers in Section 8.5. I also briefly mentioned the idea of stable and unstable equilibrium points from Section 8.4. We'll discuss these in more detail soon.
The examples we looked at today illustrate that linearization does not always give us an accurate view of what is going on near an equilibrium point. We'll next talk about conditions under which linearization is to be trusted and conditions under which we'll need to be careful.
The Linearization Theorem (aka Hartman-Grobman Theorem) gives us conditions under which we are guaranteed that the linearized system tells us something relevant about the nonlinear system near an equilibrium point. Our text does not refer to this theorem directly but Theorem 8.4 in Section 8.5 is related.
| Section | Problems to do | Submit | Target or due date | Comments |
|---|---|---|---|---|
| 1.1 | #1-11 odd, 12-16 | None | Wednesday, Jan 18 | |
| 1.2 | #1-6,9,10,14-21 | None | Thursday, Jan 19 | Do at least two of #1-6 by hand. |
| 2.2 | #3,7,9,14-16 | None | Monday, Jan 23 | |
| 2.3 | #3,5,11,15,23,25,28,29,30 | None | Tuesday, Jan 24 | |
| 2.1 | #11,13,16,17 | None | Friday, Jan 27 | |
| 2.3 | #31 | #32 | Monday, Jan 30 | |
| 2.4 | #5,11,12,13,15 | #14 | Tuesday, Jan 31 | Note typo in #14: the nonzero output of M1and M2 should be M, not 1. |
| 2.5 | #2,4,8,13 | #6 | Friday, February 3 | |
| 3.1 | #3,7,10 | None | Wednesday, February 1 | |
| 3.2 | #3,7,9,11,13,15,21,25,27,28,30 | None | Friday, February 3 | |
| 3.5 | #3,5,6,11 | 7 | Wednesday, February 8 | |
| 3.3 | #3,5,7,13 | None | Tuesday, February 7 | |
| 3.3 Bonus | #1,2,3 from handout | None | Wednesday, February 8 | |
| 3.4 | #3,5,7 | None | Friday, February 10 | |
| 3.6 | #1,3,4,5,11 | None | Monday, February 13 | |
| 3.8 | #3,9,11 | None | Monday, February 13 | |
| 4.1 | #1,3,8,11 | None | Friday, February 17 | |
| Theory handout | #1,2from handout | None | Tuesday, February 21 | |
| 4.2 | #5,9,11,17 | None | Tuesday, February 21 | |
| 4.3 | #1,9,11,12,15,19,23,26 | None | Wednesday, February 22 | |
| 5.1 | #7 | None | Wednesday, February 22 | |
| 5.2 | #5,10 | #24 | Friday, February 24 | |
| 4.4 | #3,9,11,13,15,19,21,22 | None | Friday, February 24 | |
| 4.5 | #3,5,9,15,19 | None | Monday, February 27 | |
| 4.6 | #1,2,3,7,9,11,19,21,25 | None | Tuesday, February 28 | |
| 4.7 | #3,4,5,7 | #10 | Monday, March 6 | Note typo in #10: In the limits, k/m should be km. |
| 4.8 | #3,5,7,11,15 | None | Monday, March 6 | |
| 4.9 | #1-15 odd, 31 | None | Tuesday, March 7 | |
| 4.10 | #1,3,7,11 | 15 | Friday, March 10 | Note typo in #15: The desired expression is off by a factor of -1. |
| 4.11 | #1,11,14 | None | Friday, March 10 | |
| 8.2 | #1,5,9,11,31,33,35 | None | Wednesday, March 29 | |
| 6.1 | #21-24 | None | Wednesday, March 29 | |
| 6.2 | #1,5,7,9,13,15,19 | None | Wednesday, March 29 | |
| 6.3 | #5,7,9,13,15 | None | Friday, March 31 | |
| 6.4 | #1,13,17 | None | Friday, March 31 | |
| 6.5 | #21,24,25,29,34,36,39 | #40 | Monday, April 3 | |
| 6.6 | #7,11,15,17,23,29 | None | Monday, April 3 | |
| 6.7 | #5,7,12,31,34,35 | None | Wednesday, April 5 | |
| Matrix exponential | #1-5 from handout | None | Monday, April 10 | |
| 6.8 | #3,5,6,21,25 | None | Wednesday, April 12 | |
| 6.10 | #14,15,17,27 | None | Monday, April 17 | |
| 6.11 | #20,21 | None | Tuesday, April 18 | |
| 8.1 | #14 | None | Wednesday, April 19 | |
| 8.5 | #1,5,9 | None | Friday, April 21 | |
| 8.3 | #3,9,12,15 | None | Friday, April 28 |
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.