| Section | Problems to do | Submit | Target or due date | Comments |
|---|---|---|---|---|
| 1.2 | #1-13,16-20 | None | Wednesday, August 30 | |
| 1.3 | #1-6,9,10,14-21 | None | Thursday, August 31 | Do at least two of #1-6 by hand. |
| 2.1 | #1-9 odd | None | Friday, September 1 | |
| 2.2 | #3,7,9,11,37,39,41,44,45,46 | None | Friday, September 1 | |
| Evolution operators | #1,2,3 from handout | #3 | Monday, September 11 | |
| 2.3 | #3,5,7,11,14,17 | None | Monday, September 11 | |
| 2.4 | #10,11 | #12 | Wednesday, September 13 | |
| 2.6 | #9,11,15,22,27,29,30,31,33,37,39 | None | Thursday, September 14 | |
| 2.5 | #3,5,9,15,17,19 | None | Thursday, September 14 | |
| 2.7 | #7,9,13,22,23,25,27 | None | Monday, September 18 | |
| 2.8 | #1-3,6-8,14,16,20 | None | Wednesday, September 20 | |
| 2.10 | #3,9,12,13 | None | Monday, September 25 | |
| 3.1 | #1,3,5,9,11 | None | Thursday, September 28 | |
| 3.2 | #3,5,9,11,17,21,22 | None | Wednesday, October 4 | |
| 3.11 | #1,3,5 | None | Wednesday, October 4 | |
| 3.3 | #3,9,11,13,19 | None | Wednesday, October 4 | |
| 3.4 | #1,5,9,15,17,18 | None | Thursday, October 5 | |
| 3.5 | #1,2,5,7,11,25,29,33 | None | Monday, October 9 | |
| 3.6 | #3,5,8,9,11,12,13 | None | Wednesday, October 11 | |
| Springs bonus | #1-6 from handout | #1-6 | Friday, October 13 | |
| 3.7 | #11,13,15 | None | Wednesday, October 18 | |
| 3.8 | #3,7,9,37,38,39,41 | None | Wednesday, October 18 | |
| 3.9 | #3,9,13,15 | None | Friday, October 20 | |
| 3.10 | #3,5,7,11 | None | Monday, October 23 | |
| 4.1 | #15,25-28,29-32 | None | Monday, November 6 | |
| 4.2 | #1,13,15,21 | None | Monday, November 6 | |
| 4.3 | #11,14,17,21,25 | None | Monday, November 6 | |
| 4.4 | #19,23,25,27 | None | Wednesday, November 8 | |
| 4.5 | #3,7,9,13,17,19,21 | None | Thursday, November 9 | |
| 4.7 | #7,9,11,16,29 | None | Wednesday, November 8 | |
| 4.6 | #17,21,23,25,29,31,33-36 | None | Thursday, November 16 | |
| 4.8 | #3,7,10,15,35 | None | Friday, November 17 | |
| 6.1 | #5,14,18 | None | Monday, November 20 | You should use the ODE 2D Calculator (or equivalent computing technology) to look at the direction fields and approximate solutions for #18. |
| 6.2 | #7,9,17,24-27,29,35 | None | Monday, November 20 | |
| 6.5 | #1,3 | None | Monday, November 20 |
In Section 6.5, the text gives something roughly equivalent to what I, in class, labeled the linearization theorem. The text's version is phrased in terms of ideas we haven't yet discussed, namely stability and asymptotic stability of equilibrium points and almost linear. We talk about stability and asymptotic stability in class next Monday.
Exam 3 is take-home. It is due Thursday, November 30.
Modeling Project 3 will be due on the last day of classes.
We'll cover the ideas from Chapter 6 in an order that differs from the text. In particular, we've looked at the idea of linearization first. We'll go back and look at stability of equilibrium points (Section 6.3) and conservative systems (Section 6.4) later. For now, you can ignore part (c) for the two problems I've assigned from Section 6.5. I'll assign more problems from this section after we talk about stability.
In dealing with the repeated eigenvalue case, I use notation that differs from the text. I use w0+tw1 where the text uses tv1+w2. I prefer my notation because the subscript matches the power of t. This is particularly convenient when generalizing to higher multiplicities.
There is an alternate way to deal with the repeated eigenvalue case. Rather than first finding eigenvalues, you can go straight to the generalized eigenvector problem and find a basis for the solution set of (A-λI)w0=0. For each vector in the basis, compute a corresponding value of w1 and put the pieces together to get a solution. The number of solutions this generates will equal the dimension of the generalized eigenspace. If this matches the algebraic multiplicity for that eigenvalue, you'll have enough solutions in hand.
Our solution technique for solving constant-coefficient homogeneous linear systems will naturally lead us to the eigenvalue-eigenvector problem for the coefficient matrix. You should find an efficient way to compute eigenvalues and eigenvectors, either by hand or using technology. Note that our text is written for students who have not necessarily had a linear algebra. Section 4.4 has material on the eigenvalue problem that you can read as review. I've assigned problems from Sections 4.4 and 4.5.
In class, we are using the language of linear algebra to describe the structure of solutions. In particular, we have a theorem telling us that the set of solutions to a linear homogeneous n by n system of equations is a subspace of dimension n. Using this, we know that it is enough to find a linearly independent set of n solutions. We can write the general solution as a linear combination of the solutions in this basis.
The text's lanuage is slightly different but the underlying ideas are the same.
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 a solution curve (or an integral curve). A phase portrait is a collection of integral curves for a variety of initial points. The text discusses direction fields, solution curves, and phase portraits in Section 4.5 and Section 6.2
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. (Note: for an alternate location, try this link.) 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
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 4 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 6).
Tomorrow's exam will cover material from Section 3.1 to 3.11. It might be useful to categorize the material into
We will have Exam #2 next Thursday, October 26 from 11:00-12:20. It will cover material from Chapter 3.
Have a great Fall Break.
The University of New South Wales has a nice web site on Foucault pendulums. It's kind of amusing to note that they think about the South Pole as an extreme location on the Earth. (Think about where this university is located to understand why the South Pole would be a natural choice there.)
The text defines the complex exponential ez, where z is complex, in terms of a power series. This is an elegant approach but requires looking at the issue of convergence for the power series. The text glosses over the convergence issue. (If you take complex analysis, you would deal with this convergence issue in more detail.) In class, I've defined ez using two steps:
News flash: Our move has been delayed so I will be in my current office (Thompson 321D) through next Monday, October 9.
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.
In class, we are working to develop the theory of nth order linear differential equations. The text deals with this theory for 2nd order equations in Section 3.2 and with the general nth order case in Section 3.11. I'll assign problems from Sections 3.2 and 3.11 tomorrow.
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 space of a linear transformation. We'll talk about this in more detail tomorrow. Before then, you should review the ideas of linear transformation and null space.
You should be thinking about the modeling project. Come talk with me if you have questions or need direction.
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.
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.
We are skipping over the material in Section 2.9. If you are a physics major, you might want to read this section. I'm happy to address any questions you have from it.
For your reference, here's the Excel spreadsheet implementing Euler's method that we put together in class.
For Modeling Project #1, you should begin thinking about how you want to approach this modeling situation. Think about what happens when you take some medication like a decongestant. What's involved with the medication getting into your bloodstream and leaving your bloodstream? You might try generating a list of questions that you would want to ask to clarify the situation. You might also start thinking about relevant quantities that you would want to have measured.
For Exam #1, it might be helpful to to categorize material we have covered as theory, analysis tools, and modeling. Theory includes existence-uniqueness theorems and the general solution to linear first-order problems. Analysis tools include slope fields and specific solution techniques. Modeling includes setting up a mathematical model, analyzing that model, and interpreting results to get insight on some real-world phenomenon.
After class, Alex pointed out something from Problem 20 in Section 2.8. We showed that A–B is constant and named this D so A–B=D. We then showed that A satisfies the logistic equation and I sketched a quick slope field with some representative solutions. Note that I implicitly assumed D>0 which is equivalent to assuming A>B. For an initial value of A between 0 and D, the solution is increasing. This seems inconsistent with the interpretation of A as the amount of the first reactant. The quantity A should be decreasing. Alex pointed out that the only relevant part of the slope field is the region above the line A=D. In this region, all solution curves are decreasing as we expect. To see that only this region is relevant, note that A=D+B and B must be greater than or equal to 0. Thus, A must be greater than or equal to D (under the assumption that D>0).
Exam #1 will be on Thursday, September 21 from 11:00 am to 12:20 pm. It will cover material from Chapters 1 and 2 of the text.
Please check your schedule to see if you are available for an exam next Thursday (September 21) for the 80-minute period 11:00 am to 12:20 pm.
Here's a few more thoughts on the evolution operator handout: We are working with a general nonhomogeneous linear first-order problem which we can write as y'=-p(t)y+g(t). Think of this as telling us about the rate of change in the quantity y. Each term on the right-hand side tells us about a process that contributes to change in y. The term -p(t)y describes some internal process that "knows" about the quantity y. The term g(t) describes some external process. For example, in our investment account example, the process of continuously compounding interest is internal because it uses the amount itself while the income stream is external because it can be set without knowing the amount itself. The interesting thing to consider is that the external process described by g(t) has the effect of changing the quantity y. Any change in y related to the external process then become part of the internal process. Consequences of the external process are thus mingled with consequences of the internal process. Think of y as measuring how much "stuff" is in a big pool. Some (magical?) process cause the stuff in the pool to change at a rate that is proportional to how much is in the pool. The proportionality "constant" can change in time. This is the process described by -p(t)y. In addition, we can dump more stuff in the pool at any rate we choose. We can change the rate at which we dump stuff in. This is the process described by the term g(t). In an (infinitesimal) interval of length dt, the amount we dump in is g(t)dt. Of course, anything we dump in is then part of the total and starts changing at the rate that is proportional to how much is in the pool. Both processes are happening continuously and simultaneously.
You should also be careful with thinking that e-P(t) describes exponential growth. The function P(t) can have many forms, depending on p(t). Only with p(t)=-k= with k a constant do we get P(t)=kt and thus exponential growth ekt.
One last thing: The negative sign in the exponent of e-P(t) is just a consequence of the sign choice in the differential equation y'+p(t)y=g(t). If we start with y'-p(t)y=g(t), we could then write y'=p(t)y+g(t) and the relevant evolution operator would be eP(t).
We analyzed the problem of computing the future value of an income stream in two ways
For many problems, the second approach is more efficient. It's often easier to understand how different processes contribute to the rate of change in some quantity. This allows you to write down a differential equation. Note that our models so far are for very simple situations so that the resulting differential equation is one we can solve.
One of the ideas I hoped to convey today is that we can get to exponential growth by looking at continuous compounded interest in two ways:
The first approach gives you a fundamental construction of exponential growth. The second approach amounts to writing down a differential equation and then solving it. It's indirect but efficient. The differential equations approach generalizes quickly to a nonconstant interest rate.
Section 2.1 includes material on something called the Existence-Uniqueness Theorem for first-order linear problems. We haven't discussed this yet in class and none of the assigned problems deal with it. We'll come back to this soon.
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.
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 1.3 homework assignment. Note that this applet uses x as the independent variable whereas we have been using t in class.
Section 1.2 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.
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.