Math 3D Discussion Sections 30 & 31 (Winter 2006)
Elementary Differential Equations

Free Coffee at Starbuck's!
Posted on March 15, 2006 at 8:59am

Today is National Coffee Break Day at Starbuck's. If you get there between 10am and noon, you get a free coffee. Yum! -- Paul

Office Hours
Posted on March 13, 2006 at 12:50am

I just wanted to update you about office hours this week and next week. First, I will hold my regular office hours this Tuesday from 10:00am to 11:00am. I am also available by appointment on Wednesday and Thursday.

I am leaving for a major research trip at the end of this week. Consequently, my last available time for extra office hours is 11:00am on Thursday, March 16, 2006. Please plan accordingly!!! Due to necessary planning and research, I cannot make exceptions. Thank you for your understanding. -- Paul

Homework 8 Solutions Posted
Posted on March 6, 2006 at 12:03pm

Now that I've received your homework, I've posted the Homework 8 Solutions. Please let me know if you have any questions. - Paul

Self-Quiz 7 Solutions Posted
Posted on March 1, 2006 at 1:03pm

I posted Self-Quiz 7. Enjoy! :-) - Paul

Homework 7 Solutions Posted ; A Note on Repeated Roots
Posted on February 27, 2006 at 11:42am

Now that your homeworks are all turned in, I've posted the solutions to homework 7.

Also, I noticed that many of you made use of the known fact that for the constant coefficients case, if r is a repeated root, then y₂ = t e^rt, and in the Euler's equation case, the second solution is y₂(t) = t^rln(t). I think it's good to know this (this is how things are done "in the field"), but you'll want to be sure to know the method of reduction of order. Also, you'll want to check with Dr. Chung on his expectations for the final exam. He may well want you to work these facts out from scratch on the final exam, rather than use the known results.

I'm checking with him on this detail now, but it is something that you should keep in mind. - Paul

Homework 5 & 6 Solutions, Self-Quiz 6 Solutions Posted
Posted on February 22, 2006 at 11:24am

I just posted the much-delayed solutions to Homework 5 and Homework 6, as well as Self-Quiz 6. If you're in the Tuesday/Thursday section, then I highly recommend that you wait to look at the new self-quiz until after class. This will allow you take the self-quiz in more realistic (test-like) conditions. Thanks - Paul

Self-Quiz 5 Solutions Posted ; More to Follow
Posted on February 22, 2006 at 7:34am

I just posted solutions to Self-Quiz 5 from last week. I'm sorry that it took me so long. I hope to have Homework 5 and Homework 6 solutions posted soon, as well as Self-Quiz 6 solutions. Thanks. - Paul

Correction
Posted on February 9, 2006 at 9:29pm

I had a typo on my approximate grade scale. The "Danger Zone" is for those scoring 15 and below, not 17 and below. Also, I forgot that sucrose is a disaccharide, not a monosaccharide. - Paul

Midterm Grades Posted
Posted on February 8, 2006 at 2:36m

I just posted your Midterm Grades at EEE. Here's the approximate grading scale:

• A: 23 to 30
• B: 19 to 22
• C: 16 to 18
• Danger: 15 and below.

If you scored 15 or below, you'll need to work very hard on your remaining homework and final exam. However, it's still very possible to do well in this class.

I hope to further discuss test-taking strategies that may help on the final. In the meantime, let's keep pace and see what we can do.

Lastly, I'd like to note that this is an approximate guideline to how well you did; your grades aren't final until the corpulent woman vocalizes a melody. (You know, like 5-10 mL of disaccharides makes the administration of pharmaceuticals more palatable.) - Paul

Midterm Solutions Posted
Posted on February 6, 2006 at 9:14pm

I just posted the Midterm Solutions. Please study them carefully. After having graded 40% of the midterm, I'd have to say that the biggest overall problem was basic calculus. Yikes, guys. I'll do my best to give fair partial credit. - Paul

Self-Quiz and Homework 4 Solutions Posted
Posted on February 3, 2006 at 1:49pm

I just posted the solutions to Homework 4 and Self-Quiz 4. If I get a chance, I'll try to post more study materials. Otherwise, have a great weekend! - Paul

Homework 3 Solutions Posted ; Update
Posted on January 31, 2006 at 11:26am

I just posted the Homework 3 solutions. I'm sorry for the delay. Also, please note that Self-Quiz 3 Solutions were posted a few days ago.

Because your first midterm is next week, it's time to start thinking seriously about studying. I'd highly recommend that you download blank versions of all the self-quizzes and try to work them within the recommended times. For additional practice, you might consider browsing my webpages for previous Math 3D classes. Later today or tomorrow, I also hope to post a supplemental self-quiz to help you study, in addition to this week's self-quiz.

Lastly, on Wednesday and Thursday I'll discuss some integrals and trig identities that I think you'll certainly want to know. Good luck! - Paul

Homework 2 Solutions Posted
Posted on January 21, 2006 at 6:23pm

I just posted the Homework 2 solutions a moment ago. You'll probably want to take a good look at the separation of variables problem, since it shows both the "formal" method and the "normal" method.

Self-Quiz 2 Posted
Posted on January 19, 2006 at 7:57pm

I just posted Self-Quiz 2. Because we didn't have time to work on this during class this weekly, you are strongly encouraged to download and work on this quiz.

If I have time, I'll try to post a few more separation of variables examples. In meantime, please keep working on your homework and the self-quizzes, and have a great weekend! - Paul

Homework 1 Solutions Posted; Links Updated
Posted on January 16, 2006 at 6:29pm

I just posted the Homework 1 solutions a moment ago. I'd like to encourage you to pay particular attention to Problem 18 in Section 1.2, as it has some interesting points to it.

Also, I've updated the links to the class email archives above. If your inbox is full and you're not receiving new mail, then that is the place to look. Well, I hope you've all had a good weekend, and I'll look forward to seeing you all in class. - Paul

Self-Quiz 1 Solutions Posted
Posted on January 12, 2006 at 2:25pm

I posted the solutions to the first self-quiz a minute ago. I hope you find it helpful as a start in preparing for your eventual final. Good work on building some initial momentum; I look forward to a productive quarter! - Paul

Welcome
Posted on January 8, 2006 at 12:16pm

Welcome to Math 3D: Elementary Differential Equations. This topic can be very exciting, and I think you're going to enjoy it. In calculus, you learned about derivatives, a key idea in describing our dynamic world. Our world is always in a state of change, and so it only makes sense that we need derivatives to characterize that.

Now, you'll begin to use the tools of calculus to model these dynamics. Simply stated, a differential equation is an equation that involves a function and one or more of its derivatives. More broadly than that, though, a differential equation is a way to relate how a quantity changes to other quantities. Consider, for example, Newton's second law of motion: The acceleration experienced by an object is proportional to the forces applied to it. (The constant of proportionality is its mass.)

We can express this as a simple differential equation. Let y(t) describe the height of an object as a function of time. Then velocity is y'(t), and acceleration is y''(t). So, if F is the sum total of all the forces and m is the mass, the equation is:

y''(t) = (1/m) F.

Next, we might describe the forces. For short heights, the force of gravity is mass times the gravitational constant g. Air resistance can be modeled as being proportional to the velocity squared, and it acts in the opposite direction. So, we might fill in the forces in this differential equation as

y''(t) = (1/m) ( -mg + (-α y'(t)/|y'(t)|) ( y'(t) )² ).

As another example, let us consider population growth. Let B be the population of gigantic beetles in a stupid SciFi flick. We expect the rate of change in the population to be exponential for smaller populations, and we expect the rate to slow down if it gets close to the carrying capacity B_m of the environment. Furthermore, if the population exceeds B_m, then we expect that beetles will start dying (due to competition for food, lack of oxygen, etc.). A reasonable model for this situation is logistic growth:

B'(t) = α B(t) ( B_m - B(t) ).

Notice that B'(t) > 0 if 0 < B < B_m, and B'(t) < 0 if B > B_m, which matches our expectations. Furthermore, if B = 0 or B = B_m, then B'(t) = 0. In these cases, the population is in equilibrium. This example is good because it illustrates a few key ideas: examining equilibria and checking the sign of the derivative away from these equilibria.

Throughout the quarter, we'll be focusing our attention on solving the differential equations most commonly encountered in mathematics, engineering, physics, chemistry, and biology. When we get bogged down in the details of a technique, it's always good to keep an example in mind of why we're doing this in the first place.

Well, welcome to Math 3D, and I'll look forward to meeting you all this week! - Paul