| Section | Problems to do | Submit | Target or due date | Comments |
|---|---|---|---|---|
| 1.1 | 3,5,7,8,9,15,17 | 6,16 | Monday, January 26 | |
| 1.2 | 1,3,4,9,11,13,15,17,21,23,29,33,35 | 30,36 | Thursday, January 29 | |
| 1.3 | 1,3,5,9,13,15,17 | 14,24 | Monday, February 2 | You can use any relevant technology to produce slope fields. |
| 1.4 | 7,9,15 | 14 | Thursday, February 5 | |
| 1.5 | 1,5,7,13,15,18 | 12 | Thursday, February 5 | |
| 1.6 | 3,7,11,15,19,31,39 | 28 | Friday, February 6 | |
| 1.7 | 3,9,10,13,21 | None | ||
| 1.8 | 1,3,11,17,19,21,29,33 | None | ||
| 1.9 | 5,11,15,21 | None | ||
| 2.1 | 8-12,15,17 | None | ||
| 2.2 | 3,5,9,11,13,15,17,21,25,27 | None | ||
| 2.1 | 20-23 | None | ||
| 2.2 | 7,19 | 8,12 | Thursday, February 26 | |
| 2.4 | 3,5,11,16 | 6 | Friday, February 27 | |
| 2.3 | 5,7,9,11,19 | None | ||
| 3.1 | 5,9,13,15,17,25,27 | None | ||
| 3.2 | 3,7,9,11,17,19,21 | 20 | Monday, March 9 | |
| 3.3 | 3,5,7,9,13,23,25 | 20 | Monday, March 9 | |
| 3.4 | 1,3,5,7,9,11,13,15,19,21,23 | None | ||
| 3.5 | 3,7,9,11,12,16,19,23 | None | ||
| 3.7 | 1,3,9,11,13 | 10 | Tuesday, March 31 | |
| 3.6 | 1,5,7,9,13,15,17,19,21,23,25,27,31,32,36 | 34 | Tuesday, March 31 | Note that #13-19 pair with #21-27. For #13-19, use the system of first-order equations approach. For #21-27, use the second-order equation approach |
| 3.8 | 5,11,17 | None | ||
| 4.1 | 5,9,13,17,31,36,37,39 | None | ||
| 4.2 | 3,5,9,11,13,15,19 | None | ||
| 4.3 | 1,5,15,17,21 | None | ||
| 5.1 | 1,7,11,15,17,18,19,23,27 | 22,28 | Monday April 20 | |
| 5.2 | 4,5,9,13,15 | None | Note that #5,9,13 here match the system in Section 5.1 #7,11,15. | |
| 5.3 | 1,3,9,11,13,15 | None | ||
| 5.4 | 1,2,3 | None | ||
| 6.1 | 1,3,5,7,9,11,13,15,19,23 | None | ||
| 6.2 | 1,5,7,9,11 | None |
For Monday May 11, I will be busy with exams and other commitments until 2:30. I will be available from 2:30 to 3:30.
For Tuesday May 12, my tentative plan is to be available most of the day between 9:00 am and 4:00 pm. I currently have commitments from 10:30 to 11:00 and noon to 1:30. I will post an update on Tuesday morning.
You are welcome to come find me whenever I'm in my office. If you don't want to risk climbing the stairs to find I'm gone, e-mail or call (3567) to set up a specific time.
For Friday May 8, I will be around from 9:00 to 11:30 in the morning and from 2:30 to 3:30 in the afternoon.
You are welcome to come find me whenever I'm in my office. If you don't want to risk climbing the stairs to find I'm gone, e-mail or call (3567) to set up a specific time.
For Wednesday May 6, I plan to be around my office from 10:00 to 11:30 in the morning and from 2:30 to 4:00 in the afternoon. After 4:30, I'll be at the department's end-of-year picnic in the Harned courtyard (or inside Harned if it's raining). All are welcome.
For Thursday May 7, my tentative plan is to be around from 10:00 to noon in the morning and from 1:00 to at least 3:30 in the afternoon. I'll post an update on Thursday morning.
You are welcome to come find me whenever I'm in my office. If you don't want to risk climbing the stairs to find I'm gone, e-mail or call (3567) to set up a specific time.
Exam #4 is due tomorrow by 5 pm. You can bring it to class or to my office later in the day. If I am not in when you come to drop off your exam, slide it under my door.
At the beginning of class, we looked at this animation for Problem 5.1.22. The animation shows the phase portraits of the system changing as the parameter value changes. You can clear see the bifurcation in which the system goes from having no equilibrium points to having two equilibrium points. There is a second, more subtle, change in which one equilibrium point changes from a spiral source to a source. Some would call this second change a bifurcation while others would not.
In class, we had a first look at Laplace transforms. We moved quickly through a lot of ideas so we could see one complete example of using the Laplace transform technique in solving a linear differential equation.
I've assigned problems from Section 6.1. There are a lot of small skills you'll need to practice in order to become proficient with the Laplace transform method. We'll get more practice with these skills in class tomorrow so you can wait until after that to work on these problems. In the meantime, focus on the take-home exam.
In class, we have looked briefly at the Lorenz equations as one example of a 3×3 system. Part of the lesson here is that many of our tools (such as numerical solutions and linearization around equilibrium points) generalize from 2×2 systems to n×n systems. Another part of the lesson is going to higher dimensions opens a richer variety of possible behaviors. If you are interested in further reading or study, you might consider
We ended class looking at an animation of two points "flying" on the Lorenz strange attractor. Here's a version of that animation.
For reference, here's a PDF version of the Mathematica work we did in class for Problem 5.3.3 along with a Mathematica notebook. (The Mathematica notebook will not be readable in your browser. To use it, you'll need to download the file and open it using Mathematica.)
At the end of class, we began looking at the classic paper "Deterministic Nonperiodic Flow" by Edward Lorenz. This paper provided one of the earliest examples of chaos for system of nonlinear differential equations. We'll look at a few more details tomorrow.
In class, we looked at the idea of Lyapunov functions and a few examples. I've assigned a few problems from Section 5.4 that deal with Lyapunov functions. Section 5.4 also has material on gradient systems that we will not cover.
I've decided not to generate a context for your report in Project #3. So your report should be a mathematical analysis of the system given in Lab 5.3 along the lines described in the text's instructions. Your report should focus on assertions that you can make and prove or support. You can phrase your report in terms of generic ``predator'' and ``prey'' or you can choose specific species. Keep in mind that x and y are measures of the population sizes but are not necessarily the numbers in units of 1. These could be numbers in different units (such as hundreds, or thousands, or millions) or different measures such as biomass or density (number per area).
I've assigned a few problems from Section 5.3. I'll assign more after we discuss Hamiltonian systems in more detail during class tomorrow.
For reference, here's a PDF version of the Mathematica work we did in class along with a Mathematica notebook. (The Mathematica notebook will not be readable in your browser. To use it, you'll need to download the file and open it using Mathematica.)
Project #3 will be based on Lab 5.3 from the text. You can start working on the mathematical details based on the lab instructions. I'm writing a handout to give you some additional context for your writing. I'll post that sometime tomorrow and bring paper copies to class on Thursday. The project is due Tuesday, April 21.
Exam #4 will be a take-home exam. I will distribute it on Tuesday, April 28 and it will be due on Tuesday, May 5 (the last day of our class).
Our final exam is scheduled for Monday, May 11 from noon to 2 pm.
I've assigned some problems from Section 5.2. We've talked about the ideas in Section 5.2 without using the specific terminology of nullclines that the text introduces.
Note that I've added #17 to the assigned problems for Section 5.1.
We will not cover material from Sections 4.4 and 4.5. Those of you majoring in physics would benefit from reading Section 4.4. All of you might be interested in the model of the Tacoma Narrows Bridge that is presented and analyzed in Section 4.5.
In class, we analyzed our harmonic oscillator model with assumptions of no damping and sinusiodal external forcing. There are two cases to consider in finding a particular solution: one, the forcing frequency β does not equal the natural frequency ω0 and two, the forcing frequency β does equal the natural frequency ω0. I mentioned the question of whether or not the limit of a specific solution for the first case as β→ω0 is equal to the corresponding specific solution for the second case. This is fun to play with. In class, we determined a specific solution for the first case corresponding to the object starting in the equilibrium position at rest. Take the limit of this as β→ω0. (You might find L'Hopital's rule useful here. Keep in mind that you are taking a limit with β as the variable.) Then, find the specific solution for the second case with the same initital conditions and see if this matches the limit.
Exam #3 will be on Thursday. It will cover material from Sections 3.6, 3.7, 4.1, and 4.2. It will not cover material from Section 3.8 on 3×3 linear systems.
Today (Tuesday), I have office hour from 1:30 to 2:30 and am currently available after that until about 5:00. On Wednesday, I am currently available from 10:15 to 11:45 am and from 2:30-4:00 pm.
Exam #3 will be on Thursday.
In class, we've separated out the mathematics of finding particular solutions for nonhomogeneous linear second-order differential equations from the application of understanding a damped harmonic oscillator with forcing. The text moves back and forth between these topics in Sections 4.1 and 4.2. We'll talk about damped harmonic oscillators with forcing in class on Monday.
Exam #3 will be Thursday, April 9.
On Tuesday, we looked at these Mathematica results to get a feel for the geometry of an example in space. After we looked at this in class, I added another plot showing more solution curves to give a better feel for the phase portrait. Here's an on-line version you can interact with to rotate and zoom.
Project #2 is due on Monday, April 6.
Exam #3 will be on Thursday, April 9.
I've assigned additional problems from Sections 3.6 and 3.7, including problems to submit next week.
We've covered material in an order that differs from what is in Sections 3.6 and 3.7 of the text. So, we've also skipped over some of the material in these sections that deals with modeling the motion of an object on a spring. We'll begin talking about those ideas in class on Thursday.
I've assigned some problems from Sections 3.7 and 3.6.
The computer lab session today gave you an introduction to Mathematica. You are free to take advantage of Mathematica or similar technology for your work in this course but are not required to do so.
Mathematica has capabilities directly related to differential equations, in particular the function DSolve. Check the built-in help for details on this. More is available in specialized add-on packages for Mathematica. One nice one for differential equations is the DiffEqs package written by Selwyn Hollis (Armstrong Atlantic State University Department of Mathematics).
Exam #2 will be Thursday, March 12 from 11:00 am to 12:20 pm. It will cover material from Sections 2.1-2.4 and 3.1-3.5.
I'll be available today for office hour from 1:30 to 2:30. I should also be available from 2:30 to 3:00 and 4:00 until about 5:00.
On Wednesday, I don't have an official office hour. I am currently available from about 10:00 to 11:45 and 2:15 to 3:30. Let me know if you want to set up an appointment for a specific time. I have commitments from 3:30 to 5:00. I can meet at 5:00 if you let me know in advance.
Exam #2 will be Thursday, March 12 from 11:00 am to 12:20 pm. It will cover material from Sections 2.1-2.4 and 3.1-3.5.
As we saw today, solutions in the xy-plane of a linear system for which the coefficient matrix has complex eigenvalues will generally be elliptical spirals. In each example we've done in class, the spirals were based on fairly nice ellipses with major and minor axes aligned with the x- and y-axes. In general the spirals are based on ellipses for which the major and minor axes do not align with the x-axis and y-axis. In general, determing the specific details of the ellipse geometry is a bit involved. Some tools from linear algebra can be very helpful in this. In this course, we won't be going into detail on how to determine the full details of the ellipse geometry. If you are interested in those details, you can take a look at this handout.
I had originally intended to have problems from Sections 3.2 and 3.3 due tomorrow (Friday). I've changed this to Monday, March 9.
I've assigned problems from Section 3.4 so you can see what's coming. Based on what we did in class, you should be able to do #1 and parts of the other problems. We haven't yet discussed the geometry of solutions wiht complex eigenvalues. We'll do that in class tomorrow.
Exam #2 will be Thursday, March 12 from 11:00 am to 12:20 pm. It will cover material from Sections 2.1-2.4 and 3.1-3.5.
I've assigned problems from Sections 3.2 and 3.3, including problems to be submitted on Friday.
Exam #2 will be Thursday, March 12 from 11:00 am to 12:20 pm.
In class, we looked at a few problems to get experience with sketching phase portraits in the case of real nonzero eigenvalues. In the text, this is discussed in Section 3.3. Tomorrow, we'll review how to compute eigenvalues and eigenvectors. We'll then be able to put together the full story of starting with a linear constant-coefficient system, finding the eigenstuff for the relevant matrix, using the eigenstuff to build the general solution, and finally, using the general solution to sketch the phase portrait.
For now, our focus is on real, nonzero, distinct eigenvalues. In the rest of the week, we'll deal with other cases for eigenvalues (0 as an eigenvalue, repeated eigenvalues, complex eigenvalues).
Exam #2 will be Thursday, March 12 from 11:00 am to 12:20 pm.
We will skip over the ideas in Section 2.5 for now. The ideas are interesting and we will return to them later in the course. You might want to quickly read the section to get a sense of the main ideas.
Chapter 3 of the text is written for an audience that has not necessarily seen linear algebra. (This is a typical approach for an introductory differential equations text since many colleges do not require linear algebra as a prerequisite for differential equations.) Since all of you have had linear algebra, we'll move quickly through some of the background ideas in class. You should read the text carefully and do a self-check on whether or not you understand the relevant ideas. Ask questions (e-mail, in class, in my office,...) when they arise.
Exam #2 will be on Tuesday, March 10 or Thursday, March 12. In either case, we'll use the 80 minute period from 11:00 am-12:20 pm.
I've changed the due date for Section 2.2 problems to submit. It was originally Tuesday and is now Thursday.
In class, we looked at the standard simple model for the motion of an object acted on by a spring. Two elements go into our model:
I've assigned some additional problems from Sections 2.1 and 2.2 including some problems to submit next Tuesday.
Project #1 is due on Monday.
If you are interested in reading more details, you can access the papers on HIV models that we looked at in class online:
Project #1 is due on Monday.
I've assigned some problems from Section 2.1 that deal with some simple predator-prey models. Section 2.1 also has material that looks at a model for the motion of an object attached to a spring. We'll look at this idea in class later this week.
In class, I distributed a handout giving an overview of expectations for projects and the assignment for the first project. You first project report is due Monday February 23.
I have not yet assigned problems from Section 2.1 and there is more to discuss and to give you time to start work on your first project.
Exam #1 will be on Thursday from 11:00 am to 12:20 pm. It will cover material from Chapter 1. For Exam #1, a well-prepared student should be able to
Note: I have deliberately skipped over the examples and problems that involve modeling RC circuits. The differential equation model is based on physical laws that not all of you have seen in a physics or electronics course. If you have seen some circuits in a physics or electronics course, you might have interest in those examples and problems.
I will be available this afternoon (Tuesday) for office hour from 1:30 to 3:00 and for appointments from 4:00 to 5:00. On Wednesday, I'm available for appointments from 10:00 to 11:30 and after 2:00 in the afternoon. E-mail or call (3567) if you want to set up a time to meet.
Sections 1.8 and 1.9 give two different approaches to analyzing a linear first-order differential equation.
In class, the two most examples we examined concern a first-order differential equation that involves a parameter (in addition to the independent variable and the dependent variable). One question we ask is "How does the structure of the phase line depend on the value of a parameter?" Structure of the phase line includes the number and nature (sink, source, or node) of equilibrium points.
Exam #1 will be next week, either Tuesday or Thursday. I would like to see if we can use the 80-minute period from 11:00-12:20 for the exam. Note that class normally ends at 11:50. Please check you schedule to see if you are available for the extra 30 minutes on Tuesday or Thursday next week.
Note that I've assigned problems from Sections 1.4 and 1.5 with two to be submitted later this week. We'll address any questions from the other problems in class on Monday and Tuesday.
There are lots of computing technology options available for producing slope fields. At the end of class, I gave you a quick look at the JOde applet. Another option is the tool HPGSolver that is included in the software on the CD that comes with the text. You can use any tool with roughly equivalent functionality that you find convenient.
I've assigned quite a few problems from Section 1.2 so that you get the practice necessary to master separation of variables. In doing separation of variables, you'll need to recall various integration rules and results. You should make use of all the resources you have available, including tables of integrals and technology. There are many integration tools available on-line. One example is the Wolfram Online Integator. (Wolfram is the company that produces Mathematica.)
A few of the problems I've assigned from Section 1.2 involve what are known as mixing problems. There is an extended example of a mixing problem in Section 1.2. I'll leave to you to read this and then bring questions about the reading or assigned problems to class or to me outside of class.
In class, we looked at the model described in the paper "Mathematical Analysis of HIV-I: Dynamics in Vivo", Alan S. Perelson and Patrick W. SIAM Review, Vol. 41, No. 1 (Mar., 1999). You are not responsible for understanding this in any detail but I am including a link here in case you want to read more carefully than our quick look in class allowed. (Note: The link I've used here is for the JSTOR database that our library subscribes to. You may need to be on the campus network in order to access this. If you are having problems with the link and want to access the article, let me know.)
The wording of Problems 15 and 16 in Section 1.1 is a bit muddled. Writing "harvested each year" seems to imply that the harvest happens all at once. This would be hard to model using a continuous time variable; a discrete time variable model would be more appropriate. Since we are using differential equations (and thus a continuous time variable), you should think of the harvesting as an ongoing process for these problems. You can interpret 15(a) as saying that fish are continually harvested at a rate of 100 fish per year. You can interpret 15(b) as saying that fish are continually harvested at a rate proportional to the fish population with a proportionality constant of 1/3. This is not the same as saying that we will pick a particular date and then harvest 1/3 of the fish on that date each year.
I've assigned a few problems from Section 1.2. I'll assign more next week once we've done more on separation of variables in class.
I've assigned a few more problems from Section 1.1 including two problems to be submitted on Monday.
I've assigned a few problems from Section 1.1. The context for these problems is explained in the paragraph that begins at the bottom of page 15. Note that these problems do not require that you solve the differential equation models.
I'll assign more problems from Section 1.1 after class on Thursday, including one or two to submit.
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. Be aware that the text I previously used differed substantially in organization and emphasis from the text we are using this semester. You can use these old exams 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 can use these old exams to prepare for our exam in whatever way you want. I will not provide solutions for the old exam problems.