21-260: Week #6

The reading and homework assignments for Week #6 have been posted. You can follow the link from the Schedule page.

21-260: Week #5

I've posted the reading and homework assignments for Week #5. As I mentioned in class, there is no written assignment this week. There is an online assignment. I said in class that it would be due on Friday, but I've changed the deadline to Sunday evening, because... Well because there was no reason not to.

You can follow the link from the Schedule page.

21-260: Study Break

Engines of our Ingenuity.

21-260: Exam Strategy

I just responded to a student email asking about studying for the exam, and what type of exam you can expect. This is what I wrote:

I would say that working through practice problems is the best way to prepare. You should also review your class notes, and especially your graded homework. When you are working problems, keep track of things you need to look up or ask about. Those are things you should focus on as the exam date gets closer (as it is now!)

On the day of the exam, I would recommend taking a couple minutes to read over the whole exam. Sort out which problems look easy, and which look more challenging. Finish the easiest problems first, leaving you more time to think about the harder problems without distractions.

Regarding the makeup of the exam. You can expect 20-25% of the exam to be pretty straightforward, and another 20-25% to be pretty challenging. The rest somewhere in between. There will be some conceptual material, like on the written homeworks, but most of the exam will be more computational.

Chris Potter's office hours

My office hours this week will be in Wean 6201:

Mon 4-6pm
Tue 2-4pm
Wed (none)
Thu 4-5pm

21-260: Exam Odds and Ends

As announced in class today, our first midterm is this Wednesday from 7:30-8:20 in UC McConomy. There will be no class on Wednesday. I will hold a review session tonight from 6:30-8:00 in PH 100.

I've added "Systems of Differential Equations (Section 7.1)" to the list of topics covered. That is what I've been telling people all along, but I inadvertently left it off the list. I also added 7.1.23 to the practice problems from the text.

21-260: Week #3 Homework Solutions

... are now available in the Solutions section of the Blackboard site.

21-260: Week #4 WileyPLUS Assignment

I've had to remove two problems from the WileyPLUS assignment for this week. Problems 2.7.1a and 2.7.18b were left over from a previous version of the course. We are not covering Euler's method this semester. I thought I removed all those problems before posting the assignment, but these two slipped through. Sorry to anyone who has been working on these problems, and sorry to anyone whose access to the site was interrupted while I edited the assignment.

21-260: Correction to HW#4

Thanks, everyone who pointed out the repeated problem on this week's assignment. I've corrected the assignment page. The correct problems are 2.5.20, 2.5.25 and 7.1.22. I also corrected the quotation marks so they would come out correctly.

21-260: Week #4

The reading and homework assignments for Week #4 have been posted. Follow the link from the Schedule page.

21-260: Homework Solutions

The solutions to HW#1 will be made available in the section of the Blackboard site tomorrow at noon. Solutions to HW#2 will appear on the site Thursday at 5:00. Normally, in weeks unaffected by exam schedules, solutions will appear at 5:00 on the Thursday following the due date.

21-260: Existence and Uniqueness

I recently received an email asking for some clarification about today's lecture. I thought many of you might get something from my reply, so I'm posting it here:

You asked "Why do we need a rectangle for the second theorem?" The short answer is: "because nonlinear equations are not as nice as linear equations." But let me turn the question around: Why doesn't the first theorem require a rectangle?

If we start with the equation y' + p(t)*y = g(t), we rearrange things to get y' = g(t)-p(t)*y. We can apply the second theorem to this equation with f(t,y) = g(t)-p(t)*y.

If p and g are continuous on the interval (a,b), then f is continuous on the rectangle (a,b)x(-infty,+infty).

The partial derivative of f with respect to y is df/dy=-p(t). Since p is continuous on the interval (a,b), when we think of df/dy as a function of t and y, it is continuous on the rectangle (a,b)x(-infty,+infty).

So for linear equations, we can always choose a rectangle that is infinitely tall. There is one other issue, though. The theorem for linear equations says the domain of the solution is all of (a,b). An implication of this is that the solution cannot have a vertical asymptote before t=b. Why is this?

Well, a complete explanation is beyond the scope of this course, but it is basically because the growth of the slope y' only depends linearly on y, it can't grow quickly enough to get to +infty unless p(t) or g(t) has a discontinuity.

Here's a related problem: Consider the differential equation y'=y^p, where p is a constant greater than zero. This is a separable equation, and if you solve it, you will find that if p<=1, then the the solution is defined for all values of t, but if p>1, then the solutions have a vertical asymptote.

21-260: Week #3

The reading and homework assignments for Week #3 have been posted. You can follow the link from the Schedule page.

21-260: Office Hours

I neglected to mention my office hours in class this morning. Like last week, I'll be holding office hours tomorrow from 2:00-4:00. These will be my regular office hours for the semester. If you need to meet with me outside this time, let me know, and we make arrangements to do that.

21-260: Week #2

The reading and homework assignments for Week #2 have been posted. You can follow the link from the Schedule page.