# Math 244 Fall 2005 Sections 1-3: Professor Bumby

• main course page.
• semester course page.
• Recitation Instructor's Course Page
• A Java Applet for Studying Ordinary Differential Equations (written by M. Rychlik of U. of Ariz., maintained at Rutgers by E. Sontag)
• Notes prepared for Math 252 on Euler's method and its application to proving a version of the existence and uniqueness theorems as mentioned in the lecture on September 26. Theory will not play a big role in this course, and computation will only appear in the Maple labs, but it is important to know that initial value problems usually have unique solutions since this assures you that any expression that satisfies the conditions is the only right answer. Correct answers, not methods, are the focus of all exercises and exam problems in this course.
• Notes on Matrix exponentials from Fall 2004 (except that a significant typographical error was corrected on Oct. 15, 2005), that is the basis of a lecture on October 10, 2005. Prior experience shows that this approach gives an easier solution than the method described in the textbook, and that it is also more likely to lead to a correct answer. In particular, this means that exam questions will be written with the expectation that this method will be used. For exam 2, only 2-by-2 matrix examples will appear. The extension to the 3-by-3 triangular case will be revisited in part 3 of the course. Maple Lab 4 includes routines for using this method in Maple, so it will be the next lab done in this section. It should be ready early in the week of Oct. 10, but preparation was actually completed on Oct. 15.
• Notes on Inhomogeneous equations and systems that should have been prepared before exam 3 are now available. They give details underlying the approach taken in lecture. Some exercises are included. An attempt was made to avoid fractions in these exercises. The attempt was mostly successful, but there may be one or two exercises that don't have answers that are as simple as intended.
• Notes on Series methods from Fall 2004 indicating differences between notation in the textbook and in the solutions shown in lecture. The book's emphasis on formulas requiring a fixed notation may make it more difficult to understand the simple principles used in these methods. In particular, the exam will emphasize finding the initial terms in a series rather than the recurrence formula connecting the terms, so the notation for general terms is less significant than finding the coefficient of a given power of the independent variable. Some exercises are included with instructions resembling those that you can expect on the exam. The notes include a brief discussion of nonhomogeneous equations, but the notes were made available so late that this was not included in the exercises and will not be part of the exam.

## Special Announcements

This lecture section has registered with the Maple Adoption Program. This is a special program that Maplesoft offers because the Maple laboratory projects are a required part of the course. One benefit of the program is that students in this section may purchase a copy of the Maple 10 Student Edition for the reduced price of \$75.00 US (download only). You will need a Special Access Code available from the Maple Adoption folder in the Resources section of the Sakai site for this section.

The lecturer's regular office hours are to be found on his home page, but special office hours for this course will be announced here.

## Calendar

• Thursday, September 01. First Recitation class. A slope field of a nonlinear equation, with graphs of numerically computed solutions was featured. Other images that may have been shown include: a slope field whose solutions are tangents to a parabola; the slope field of a linear equation, with graphs of exact solutions; equation X together with some graphs of solutions; and a discontinuous slope field whose solutions can easily be seen.
• Wednesday, September 07. First Lecture. Begin segment on chapters 1 (Generalities) and 2 (First order equations). Maple Lab 0 (Practice Lab) assigned. Here are links to the general instructions, Lab0 description and two Maple worksheets: a seed file and a supplementary worksheet. Since Maple is used in the Multivariable Calculus course, it is assumed that you have prior experience with the program. If you have any difficulty with Lab 0, you should visit the lecturer's office for a guided tour of Maple.
• Monday, September 12. A quiz was added to the Sakai site consisting of five True-False questions on definitions from chapter 1. The quiz will be available from 6PM Monday, September 12 to 6PM Friday, September 16. The quiz is required. The grade on the quiz will be counted toward your grade. Moreover, the quiz will also give valuable information about the quiz tool that will be used to allow that tool to aid in the grading of Maple labs.
• Wednesday, September 21. Maple Lab #0 due. Rather than collect printed versions of your work on the project, the "Assignments" tool in Sakai will be used. It will give you a place to upload the file. I will execute the file as part of grading it, so you should use the "Remove Output" tool in the Edit Menu, save the file, and upload that saved file.
• Wednesday, September 28. Exam #1. The exam will emphasize methods for finding exact solutions of differential equations: linear equations (section 2.1); separable equations (section 2.2); exact equations (section 2.6). Another important theme in the course concerns features of solutions that can be found from the equation itself without solving the equation. Many of these features can be seen in the slope field. A question dealing with recognizing a given slope field and recognizing features of the equation visible in the equation is likely (production of the image in a form that can appear on the exam still needs to be tested). However, the key applications - annuities (section 2.3) and population dynamics (section 2.5) - include studies of equilibrium points and the shape of the graphs of solutions that are easier to see in the equation than in formulas for the solution. Problems involving this aspect of applications are also likely to appear. Problems will resemble those done in lecture or assigned for homework. Note that no formula sheets are allowed on exams, but you are expected to have a graphing calculator.
• Tuesday, October 11 through Friday, October 14. Maple Lab #1 due Changed (again) from original date because of delay in grading lab 0. Once again, this will be collected through the Sakai site.
• Monday, October 24. Exam #2. The main parts of chapters 3 and 7. Since higher order equations and systems are two ways of looking at the same thing, there are benefits to treating them together. The emphasis will be on the linear, homogeneous, constant coefficient case. Applications, as well as less special types of equations will be considered in later segments of the course. Linear algebra is useful for organizing work with systems, but there isn't enough time to give a full treatment of that subject. The sections at the beginning of chapter 7 that describe the way in which a standard development of linear algebra applies to the study of systems will be downplayed in favor of special methods for solving systems of differential equations. These methods also apply to finding eigenvalues and eigenvectors wherever they arise, but they do not depend on a thorough understanding of those problems as linear algebra. It is easier to guess solutions of a higher order equation, so this will be used to introduce the subject. However, systems are a better way to approach the solution of the equations.
• Tuesday, October 25 through Monday October 31. Maple lab 4 will be collected through the Sakai site.
• Wednesday, November 16. Exam #3. This will cover chapter 4 and the remainder of chapters 3 and 7. In addition to systems with more than two dependent variables, or single equations of order more than two, this segment will consider inhomogeneous equations (and systems) and applications. The lectures in this segment include an introduction to the phase plane, but this topic will not be tested on the exam. A quick tour of Maple Lab 3 will be useful, since the lab provides an introduction to computations that illustrate both the free motion and forced motion of simple oscillators. This will illustrate some of the applications in sections 3.8 and 3.9.
• Tuesday, November 22 follows a Wednesday schedule, so the lecture will meet, but there will be no recitation class this week.
• Monday, November 28 through Friday November 02. Maple lab 3 will be collected through the Sakai site. The "optional" Note at the end of project description that refers to Section 3 of the supplementary worksheet is a required part of the project in this section.
• Wednesday, December 07. Exam #4. The main topic will be chapter 5 on series solutions. Brief mention will be made of the use of a phase plane to investigate qualitative aspects of autonomous systems with two dependent variables.
• Wednesday, December 21, 12:00PM To 3:00PM. Final Exam

## Lecture details

The textbook exercises done as examples in lecture will be listed here. For many of these, Maple's solution will also be shown and interpreted. These Maple worksheets will be available for download so you can step through them. Although the exams will emphasize pencil-and-paper solutions, you may find Maple's solutions useful in refining your approach to the exercises. In some cases, the Maple worksheet will be shown in lecture, but Maple versions will be posted whenever they reflect the lecture, however the exercise was done in lecture. These worksheets are primarily for working through as a partial record of the lecture, since they are posted in bare form with output removed. You should download it and step through the worksheet in Maple 10. You may also modify this copy to investigate related questions. In some cases, sections were revisited in later lectures. Examples done on later visits will be added to original list (in parentheses).

3.8, 4.2
Date Section Exercises Maple
Sep. 07 2.2 2, 5, (6), (7), 10, (14), 21, (23). yes
Sep. 12 2.3 8, 12.
Sep. 14 2.6 1, 11, 13, 32. yes
Sep. 19 2.4 7, 16. yes
Sep. 21 2.1 14, 16.
2.5 15.
Sep. 26 2.7, 2.8, 8.* .
Sep. 28 Exam 1
Oct. 03 3.1 3, 12, 15.
7.3 17.
Oct. 05 7.5
Oct. 10 3.4; 7.6 Examples from notes
Oct. 12 3.3; 7.7
Oct. 17 3.5 11, 15.
7.8 7.
Oct. 19 review
Oct. 24 Exam 2
Oct. 26
Oct. 31 3.7 3, (15). yes
4.4
7.9 (part) 3, 7.
Nov. 02 3.6, 4.3, 7.9 (part)
Nov. 07 3.9
Nov. 09 9.1, 9.2
Nov. 14 review
Nov. 16 Exam 3

Lectures leading up to Exam 4 are not shown. While they could be linked to sections of the textbook, the plan of the lectures was given in the notes on series solutions.

## Maple

As new versions of the Maple assignments are prepared, they will be linked here as well as on the course page and the semester page.

The course grades will be based on a ranking on a 700 point scale composed of the following items:

• Four class exams, 80 points each, total 320. Expected time for each class exam will be 60 minutes. This allows time before the exam for last minute questions and a preview of the next segment of the course. This buffer will protect the exam from being disrupted by students arriving a little late.
• One three hour final exam, total 200.
• Four graded Maple Labs, 20 points each, total 80. If time permits, there will be five graded labs, and the 100 point total will be scaled to 80 points.
• Recitation grade, typically graded homework and quizzes (details will be announced by recitation instructor), total 100.

Exam 1 has been graded. The average score was 55.6267 and the median was 56. Individual grades have been entered in the FAS Gradebook. There is also a distribution of scores (but no attempt to assign letter grades) and scaled averages (formed by dividing by the maximum possible score [or base score ] and multiplying by 10) for each problem. Scaling allows easy comparison of the difficulty of problems of different weight, since the maximum score is always 10.

Exam 1
Distribution
Range Count
76 - 80 11
71 - 75 7
66 - 70 3
61 - 65 11
56 - 60 6
51 - 55 7
46 - 50 10
41 - 45 8
36 - 40 4
0 - 35 8
Problems
Prob. # Scaled Avg.
1 9.32
2 8.22
3 6.91
4 5.90
5 5.05

Remember that these are grades out of 80. Grades below 40 indicate a serious weakness.

Exam 2 has been graded. The average score was 62.59 and the median was 65.5. A scatter plot shows the comparison of grades on this exam with the score on exam 1. The line of positive slope in the figure is a regression line (the line that minimizes the sum of the squares of the vertical distances of the points). (Some lines of slope -1 may be added to show the beginnings of clusters in the total scores.) Individual grades have been entered in the FAS Gradebook. There is also a distribution of scores (but no attempt to assign letter grades) and scaled averages (formed by dividing by the maximum possible score [or base score ] and multiplying by 10) for each problem. Scaling allows easy comparison of the difficulty of problems of different weight, since the maximum score is always 10.

Exam 2
Distribution
Range Count
76 - 80 6
71 - 75 15
66 - 70 14
61 - 65 10
56 - 60 9
51 - 55 6
46 - 50 2
41 - 45 2
0 - 40 6
Problems
Prob. # Scaled Avg.
1 8.27
2 6.13
3 8.63
4 7.60
5 8.57

Exam 3 has been graded. The average score was 53.74 (with a median of 57). A scatter plot shows the comparison of grades on this exam with the sum of the scores on exam 1 and exam 2. The line of positive slope in the figure is a regression line (the line that minimizes the sum of the squares of the vertical distances of the points). Individual grades have been entered in the FAS Gradebook. There is also a distribution of scores (but no attempt to assign letter grades) and scaled averages (formed by dividing by the maximum possible score [or base score ] and multiplying by 10) for each problem. Scaling allows easy comparison of the difficulty of problems of different weight, since the maximum score is always 10.

Exam 3
Distribution
Range Count
75 - 80 4
70 3
65 - 69 6
60 - 64 11
55 - 59 12
50 - 54 10
45 - 49 5
40 - 44 6
35 - 99 5
below 35 5
Problems
Prob. # Scaled Avg.
1 6.59
2 7.90
3 8.30
4 1.59
5 8.35
6 5.82

There were difficulties with problem 4 that made assignment of partial credit difficult. The fairest approach seemed to be to assign credit only for complete or nearly complete solutions. The few with credit will have a nice bonus, but other grades will be judged as though this were a 70 point exam.

Exam 4 has been graded. The average score was 65.33 (with a median of 69). A scatter plot shows the comparison of grades on this exam with the sum of the scores on exam 1 and exam 2. The line of positive slope in the figure is a regression line (the line that minimizes the sum of the squares of the vertical distances of the points). Individual grades have been entered in the FAS Gradebook. There is also a distribution of scores (but no attempt to assign letter grades) and scaled averages (formed by dividing by the maximum possible score [or base score ] and multiplying by 10) for each problem. Scaling allows easy comparison of the difficulty of problems of different weight, since the maximum score is always 10.

Exam 4
Distribution
Range Count
76 - 80 15
71 - 75 15
66 - 70 8
61 - 65 9
56 - 60 5
51 - 55 4
46 - 50 3
0 - 45 5
Problems
Prob. # Scaled Avg.
1 9.41
2 8.61
3 8.76
4 5.98
5 7.90

This graph shows the relation between recitation grades and exam scores. A correlation line has been added.

The final exam has been graded and course grade submitted. The scatter plot shows the relation between the total of previous coursework and the exam. As usual. there is a regression line. There are also lines showing totals of 575, 530, 475, 445, 395 marking the divisions between letter grades (more detail is in a table below). Your score on the exam and grade for the course are available in the FAS Gradebook. The average exam score was 151.5 with a median of 156. The tables show the distribution of exam scores and of course grades.

When grades were listed in order, a clear minimal level of competence, corresponding to the lowest passing score, identified itself. The Sakai drop box for each of the students with a lower score contains additional information about grades in various parts of the course.

Summary
Exam
Range Count
190 - 199 7
180 - 189 10
170 - 179 5
160 - 169 8
150 - 159 10
140 - 149 8
130 - 139 2
120 - 129 5
below 120 10