We now into the last major topic of the course, namely systems of differential equations. We'll first study linear systems and will be able to find explicit solutions. 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.
Chapter 3 of our text is written for students who have not previously seen matrix algebra. For you, most of this will be review. At the very least, you should skim through these parts of the text to review ideas, familiarize yourself with this text's notation, and check for any new ideas.
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
Now try experimenting with other linear systems
If things are working correctly, you should see a plot below this text. There may be a brief delay while the Java applet code is set up. The plot shows some integral curves for the system of Section 3.8 #5. By dragging with your mouse, you can rotate each picture to see things from different views. The curves are colored by a "Red-Green-Blue" function with the red weight given by |c1|, the green weight given by |c2|, and the blue weight given by |c3| where the general solution is
You have Exam #4 in hand. It is due on Wednesday, May 4 at 4 pm. We will not have class on Monday and Wednesday. I'll be available during the regular class time to address questions from reading the text or working on homework problems. I can also clarify questions from the exam but won't give much direct help.
I will grade Exam #4 and have it available for you to pick up by Monday, May 9.
The final exam is scheduled for Wednesday, May 11 from noon to 2 pm. You can bring one page of notes. You can also bring integration aids such as a table of integrals.
| Section | Problems to do | Submit | Due date | |
|---|---|---|---|---|
| 1.1 | #1,3,5,7,13,15,17,24,27 | None | ||
| 1.2 | #3,5,11,13,17,21 | #22,30 | Monday, Jan 24 | |
| 1.3 | #1,2,3,7,9,13,21,23,31 | #22,36 | Thursday, Jan 27 | |
| 1.7 | #1-11 odd | #6 | Monday, Jan 31 | For the problem to be submitted, make the slope field plot by hand on graph paper. |
| 1.4 | #1-6,13 | #14 | Friday, Feb 4 | |
| 1.5 | #1-19 odd | #12,14 | Monday, Feb 7 | |
| 1.9 | #1-15 odd, 19-25 odd | None | ||
| 1.6 | #9-21 odd | #14 | Wednesday, Feb 16 | |
| 2.2 | #1-11 odd, 23 | None | ||
| 2.3 | #3,7,9,17,21,25,32 | None | ||
| 2.4 | #5,13,14 | 15 | Thursday, Feb 24 | |
| 2.5 | #1-17 odd, 27 | 16,26 | Wednesday, March 2 | |
| 2.6 | #1,7,11,13,15,17,18 | None | ||
| 2.7 | #1,5,7,9,11,13 | None | ||
| 2.1 | #2,7 | None | ||
| 2.9 | #4,5,6,8,9,10 | None | ||
| Springs | #1-3 from handout | 3 | Wednesday, March 30 | |
| 2.8 | #1,3,5,13,16 | 12 | Wednesday, March 30 | |
| 6.2 | #1-7 odd, 13,19,23,25 | None | ||
| 6.3 | #1-7 odd,11,13 | 15 | Wednesday, April 6 | |
| 6.5 | #1-13 odd | None | ||
| 6.6 | #1,5,9 | None | ||
| 3.7 | #1,3,7,9 | None | ||
| 3.8 | #3,5,11 | None | ||
| 3.9 | #5,7,11,13,22 | None | ||
| 3.10 | #5,13 | None | ||
| 4.1 | #3,7 | None | ||
| 4.2 | #1-11 odd | None | ||
| 4.3 | #9,13,17,24 | None |
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.
The main goal in Section 1.1 is to give you some sense of how differential equations arise on applications. You should also note the terminology that is introduced. This terminology is summarized in a box at the end of the section. We will begin using this terminology in class without going over it systematically so ask questions if you have any.
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.
There is a typo in Problem 31 of Section 1.3. In part c, the equation for p(t) should have a factor of h(t) multiplying the given integral. This typo appears in the printing of the third edition that I have. It is fixed in later printings of the third edition so your copy may or may not have the typo.
Note that I have changed the due date on the homework from Section 1.3
In class, we talked a bit about the following question: If f has an antiderivative, can this antiderivative be expressed in terms of elementary functions (i.e., algebraic, exponential, logarthmic, and trigonometric functions)? This is sometimes called the question of integration in finite terms. The article "Integration in Finite Terms: The Liouville Theory" by Toni Kaspar, Mathematics Magazine Vol. 53, No. 4 (Sept. 1980), p. 195-201 gives a nice history of this problem. You can get the full text of this article through JSTOR at the UPS library web site.
Today we talked about slope 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 class, we discussed a way to derive a differential equation that models the chemical concentration in the tank flow problem we've been looking at. If you don't feel comfortable with the reasoning we did, particularly in thinking about the new concentration at time t+Δt, you might try some "warm-up" exercises:
I've assigned some problems from Section 1.4. The first 5 are mixing problems. These focus on the amount of stuff in the water rather than the concentration of stuff and so are probably a little easier to think about. Problem 6 involves a model for temperature change called Newton's law of cooling. I'm leaving it up to you to read about this in the text. Next week, I'll assign more problems from this section and decide which ones to submit.
I won't be here for class tomorrow. You should come to class and work with your group on the tank flow problems. Toward the end of the period, Tony will call time. Each group should present its results. If there are disagreements, try to resolve them.
Exam #1 will be next Thursday, February 10. It will cover the material in Chapter 1. In preparing for the exam, you might want to take note of the Review Problems on pages 84-86. It is good to solve some differential equations that come outside the context of a specific section so that you must choose an appropriate solution technique for yourself.
Someone has pointed out that there are typos in the answers for #23 and #25 in Section 1.9. The derivatives in these should be dρ/dx and not dρ/dt because in these problems you are looking for an integrating factor using the assumption that it is a function only of x and not of t.
We are skipping over Section 2.1 for now. Sections 2.2 through 2.4 deal with the general theory of solutions for nth order linear differential equations. In class, I'll use the language of linear algebra (vector space, subspace, linear independence, basis, dimension) in talking about this material. The main result is that the set of solutions for an nth order linear homogeneous differential equation is a subspace of dimension n. Thus, to completely describe the set of solutions, we need to find a basis. In other words, we need to find a linearly independent set containing n solutions.
Exam #2 will be on Thursday, March 10.
Now that we understand the structure of solutions sets for linear, homogeneous equations, we are turning our attention to the problem of finding solutions for homogeneous linear equations. We'll start with the special case of constant coefficient problems. Dealing with nonconstant coefficient problems is generally much more difficult. In the text, this material comes in Chapter 6. In Chapter 2, after looking at the constant coefficient homogeneous linear case, we'll look at finding solutions for nonhomogeneous linear equations.
Exam #2 will be on Thursday, March 10. It will cover Sections 2.1-2.7 and 2.9. You can think of this material coming in three pieces: first, the general theory of linear differential equations; second, the practical techniques of computing solutons; and third, applications, particularly various models from physics for the motion of an object on a spring.
In class, I demonstrated some capabilities of the program Mathematica. Below is a slightly revised version of the movie we made.
Have a great break.
Below is a movie that might help you understand what is going on in the limit you compute in Problem 3 of the forced spring handout. The green curves are from the solution for the case β≠ω0. The one red curve is from the solution for the case β=ω0. For each case, I set the parameters E0, m, and k equal to 1 (so the natural frequency ω0 is also equal to 1) and used the initial conditions x0=v0=0. You can use the controls to advance the movie one frame at a time.
I have changed the due date for the Section 6.3 homework to Wednesday.
Exam #3 will be a take-home exam covering a few sections in Chapter 2 and most of Chapter 6. I'll distribute it on Friday, April 6 and it will be due on Wednesday, April 13.
Section 6.6 deals with finding solutions centered at a regular singular value of a given differential equation. The idea is to try a solution in the form of a Frobenius series. The unknowns in a Frobenius series are the exponent m and the coefficients ak. Substituting a general Frobenius series into the differential equation and then setting coefficients on powers of t equal to 0 will result in an equation in m called the indicial equation and a recurrence relation involving m and the ak. The indicial equation is quadratic in m so there will be two roots. In the case of real roots, there are several subcases that result: