We are now studying systems of first-order differential equations. You should plot some direction fields using the ODE 2D Calculator by Marek Rychlik at the University of Arizona. To get started, you show explore linear systems. Look at a variety of examples. Try to classify linear systems into groups according to the nature of the solution curves.
In class on Friday, I demonstrated some capabilities of the program Mathematica. Mathematica is available for you to use in many computer labs on campus. For your reference, here is the Mathematica file we generated. I've deleted all of the output Mathematica generated to keep the file size small.
| Section | Problems to do | Submit | Due date |
|---|---|---|---|
| 1.2 | #1-23 odd | #6,12,18 | Thursday, September 2 |
| 1.3 | #1,2,5 | None | |
| 1.4 | #1-13 odd, 14,15-19 odd | #10,18,20 | Monday, September 13 |
| 1.5 | #2,4,6,10,11 | #12 | Friday, September 17 |
| 1.9 | #2,3,9,15 | #4,8 | Wednesday, September 22 |
| 1.10 | #3,5,7,10 | #16 | Monday, September 27 |
| 2.1 | #6,10,15,17 | #12,16 | Thursday, October 7 |
| 2.2.0 (p. 140) | #3,5,8,10,11 | #6,12 | Thursday, October 7 |
| 2.2.1 (p. 144) | #1,3,5,7,10,12,14 | #6,8 | Friday, October 8 |
| 2.2.2 (p. 149) | #1,3,5,7,11,15,19 | #6,8,18 | Monday, October 11 |
| 2.3 | #1,3 | None | |
| 2.4 | #1,3,5 | #2,8 | Thursday, October 14 |
| 2.5 | #1-17 odd | #6,14 | Wednesday, October 20 |
| 2.8.0 (p. 197) | #3,7,9 | #10 | Thursday, October 28 |
| 2.8.1 (p. 203) | #7 | None | |
| 2.8.2 (p. 216) | #1,5,7,17,19,21,23 | #12,24 | Thursday, November 4 |
| 3.8 | #1,5,7,11,13 | #10 | Wednesday, November 17 |
| 3.9 | #1,3,5,7,10 | #4,6 | Wednesday, November 17 |
| 3.10 | #1,3,5,7,9,13,15 | #6,17 | Monday, November 22 |
| 3.11 | #1,3,15,16 | None | |
| 4.3 | #1,5,7,11 | None |
Nothing fun yet. If you're bored, check out the spong monkeys here.
You should read Sections 1.1 and 1.2 of the text and work on the problems from Section 1.2. Some of the problems involve solving a given initial-value problem. There are two common styles of solving an initial-value problem. In the first, which Braun favors, one uses uses definite integrals rather than indefinite integrals. The lower limit of integration is chosen as the initial time t0 and the upper limit is chosen as the variable time t. In the second, one uses indefinite integrals and then applies the initial condition to solve for a specific value of the constant of integration. You should experiment with both styles. You can use either style in your work but you will need to be comfortable reading both styles.
Continue working on the problems from Section 1.2. Be prepared to submit Problems 6, 12, and 18 tomorrow.
Have a first read of Section 1.3. There is a lot going on in this section including a little bit of history, a little bit of chemistry/physics, and a little bit of mathematics. Try to make a list of detailed questions for class tomorrow. You can see Vermeer's work at http://essentialvermeer.20m.com/vermeer_painting_part_one.htm and some of Van Meegeren's work at http://www.tnunn.f2s.com/vm-pics.htm
After class today, I thought of a nicer way to deal with Problem 19 of Section 1.2 than what I had originally done. Here's a handout with a hint.
In class on Friday, someone (no names mentioned) got a little too involved in some tangents. The first concerned the term integrable. Here are three issues to distinguish for a function f with domain an interval [a,b]:
The second tangent involved comparing the sizes of the sets of rationals and irrationals. There are several ways of describing the size of a set containing infinitely many things. Two of these are cardinality and measure. In cardinality, the basic distinction among sets of infinitely many things is between countable and uncountable. A set is countable if there is a one-to-one correspondence between that set and the set of natural numbers. If not, a set is uncountable. The set of rational numbers is countable while the set of irrational numbers is uncountable. A second way of describing the size of a set involves defining something called a measure. The probability ideas we discussed in class involve a specific measure. In this specific measure, the rationals have measure 0. So, the rationals are a small set of infinitely many things in two distinct senses: they are countable and have measure 0. The irrationals are a big set in both of these senses. It is a theorem that any countable set has measure 0. Are there uncountable sets that have measure 0 (i.e., sets that are big in the first sense and small in the second sense)? Yes. One example is the Cantor middle thirds set. You can read a bit more about this at the Mudd Math Fun Facts site.
In class tomorrow, we'll return to the topic of differential equations. I'll start with a few minutes on questions from Section 1.3 and then we'll look at separable first order differential equations.
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. Here's links to two I've tried and liked:
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.
Bring questions on the Section 1.5 problems to class on Wednesday. Also read Section 1.9. I will post problems from Section 1.9 some time on Tuesday.
Exam #1 is on Wednesday. It will cover material from Chapter 1, specifically Sections 1.1-1.5, 1.9, and 1.10. For this exam, a well-prepared student should be able to
This portion of the course centers on very specific techniques for solving specific types of differential equations. In class, we are often motivating a specific technique and showing that the technique will work. Most of the problems from the text focus on using the techniques. You can do these problems by following a recipe. Each time you use a recipe, you should rethink why the recipe works.
I've added a new feature so you can check your scores on assignments and exams. Let me know if my records do not match your records. I will base midterm grades on the scores I currently have.
I've assigned problems from Section 2.8.0. These involve looking for solutions in the form of power series. In some cases, such as the examples we did in class, you can solve the recursion formula for the coefficients by finding a pattern. In other cases, you cannot easily get an explicit formula that gives the nth coefficient in terms of n.
The definition of regular singular value I gave in class on Thursday, October 28 was complete nonsense. You should burn any notes you took on this. I got the definition right in class today.
Here's a few words on Section 2.8 of the text. Section 2.8.0 deals with the general idea of power series solutions. The examples and problems here deal with looking for a power series solution based at a non-singular point of the differential equation. Section 2.8.1 covers Euler equations. The idea here is that Euler equations provide are examples of differential equations with a singular value. We can solve Euler equations without power series solution techniques. Since it is easy to get explicit closed-form solutions, we can explore the behavior of solutions near the singular value. This gives us some idea of what to expect in more general cases and provides motivation for the method of Frobenius. Section 2.8.2 deals with power series solutions based at a regular singular value of the differential equation. The conclusions of this section are summarized in Theorem 8 on page 215. Note the the theorem shows how one can obtain the indicial equation from the power series expansions of tp(t) and t2q(t). This is useful for discussing the theory. In practice, it is fine to get the indicial equation by substituting the assumed form of the solution into the differential equation. Section 2.8.3 gives the proofs of some parts of Theorem 8 that are not proven in Section 2.8.2.