| Session: Fall 2009 | Instructor: Barry McQuarrie | Office Hours: | ||
| MWF 8:00-9:05am | Office: Science 1380 | MWF 9:15-10:30am | ||
| Location: Sci 1030 | Phone: 589-6302 (I do not use voicemail) | Th 10:30-11:30am | ||
| mcquarrb@morris.umn.edu | drop in (if my door is open we can talk, if it is closed I am not available) | |||
| http://cda.morris.umn.edu/~mcquarrb/ | other times via email appointment |
Course Prerequisites
To succeed in this course you will need to have mastered basic calculus (Calculus I and II). Although the only prerequisite to the course is calculus, we will be using ideas from linear algebra and complex-valued functions extensively. The underlying theory of differential equations has a deep connection with linear algebra, and I will be pointing out these connections as we proceed. We will develop the concepts we need from other areas of mathematics in class or on assignments.
Most students in the class will have some experience with the computer algebra system Mathematica in Calculus I and II. If you have never used Mathematica before, do not despair! I will be using it occasionally in class, and will provide you with resources to help you use Mathematica when you need it on Assignments. Mathematica will never do our thinking for us. It will help us understand concepts and answer questions that would be difficult to answer if we were working the solution out solely by hand. The vast majority of the work in this class will be done using our brains, pencils, and paper!
Goals
A student taking this course should expect to be able to
- classify a given differential equation into the basic types,
- solve first order linear differential equations by various methods,
- solve second order linear differential equations (both constant and variable coefficient) by various methods,
- solve systems of first order linear differential equations using eigensystems,
- determine stability of critical points in linear and locally linear systems.
Beyond the curriculum, you should also expect to
- develop skill at presenting solutions to problems and confidence in your problem solving skills,
- think beyond technique, and understand the problems studied in some depth,
- see the benefit of computers to aid calculation and exposition, but also see the absolute necessity of understanding the theory completely before using a computer.
Textbook
NOTE: TEXTBOOK EDITION IS ACCURATE FOR Fall 2009.
Boyce and DiPrima Elementary Differential Equations and Boundary Value Problems 9th, 8th, or the 7th Ed--the bookstore will have the latest edition, and the course calendar below is based on the 9th Edition. The differences between the editions is minimal, but if you use an earlier edition be aware that some of the sections may be numbered differently, content may be slightly different, and problems listed as practice below may not line up with your older edition.
Another text that presents the concepts with more readability (but lacking the depth of Boyce and DiPrima) is Zill A First Course in Differential Equations with Modeling Applications. If you are looking for a secondary reference this would be an excellent choice. This is NOT the primary text for the course, it is optional.
Time Commitment
University policy says ``one credit is defined as equivalent to an average of three hours of learning effort per week (over a full semester) necessary for an average student to achieve an average grade in the course''. Our course is a four-credit course, meeting approximately three hours per week: 4 credits times 3 hours/week/credit - 3 hours/week in lecture = 9 hours/week outside class. Thus, you are expected to spend 9 hours per week working outside of class, reading the textbook and working problems.
Please make the most of my office hours! The content of the course can be difficult at times and I expect to see you all in my office at some time or other. To get the most out of the course you should
- do homework every day,
- allot time to think about what it is we are doing,
- discuss the techniques we are studying and their implementation with your classmates,
- discuss any difficulties with me during office hours.
Course Components
Practice. Practice questions will not be collected. Solutions to the practice problems are available on the course web page. The purpose of the practice problems is to focus your attention on the important lessons of the day, to serve as a starting point for solving the assignments, to serve as a guide to what I consider a complete solution to problems, and to serve as review for tests. You should do as much extra homework as you deem necessary to enhance your understanding of a topic. I can not stress enough how important it is that you work problems! The practice problems identify the types of problems from the text that should be mastered. Although these practice problems will not be graded, I believe you will have a hard time succeeding in the course unless you spend the time to understand them. Falling behind in this course, as in any university course, can lead to disaster, so it is important that you keep up with the homework.
WeBWorK. (58 questions total) You will be completing some of your assignments using the online homework utility WeBWorK, which you can learn more about here. The WeBWorK problems are provided to give you practice implementing the computational techniques we will be studying, although occasionally a problem will be more theoretical in nature. There are seven WeBWorK problem sets for the course, five due at the midterm and two due before the final exam. I have provided links to the WeBWorK problem sets in the course calender below.
Assignments. (about 20 questions total) Assignments will be handed out well in advance of the due date. Assignments I have handed out to the class will be available outside my office. The assignments will take some time to complete, and it is important that you begin work on them immediately. Assignments will be handed in during class on the day they are due, unless you have spoken to me beforehand and I have granted an extension. Putting assignments in my mailbox or under my office door while I am teaching another course is severely frowned upon unless we have agreed that you will be doing this. If this is done when I am teaching your class I will not accept the work--believe it or not, people have actually done this!
Assignment solutions should be a self-contained document. They should be written legibly, contain diagrams or tables where appropriate, and should state the problem and clearly explain the solution. If you have used a significant amount of Mathematica in your solution, please print out the Mathematica notebook and include it as part of your solution.
Midterm Exam. The midterm will consist of five problems--possibly some short answer or true/false questions, along with questions dealing with the application of the techniques we have learned. You will not be allowed any outside material on your desks during the midterm exam (calculators will be allowed, but should not be necessary to solve the problems). The exam should not be significantly harder than the assignments; my tests tend to be long, so do not be alarmed if you require the entire class time to complete the test. There will be no Mathematica component to the exam.
Final Exam. The final exam will be similar in format to the midterm exam, except slightly longer, and cover only the material since the midterm exam.
Grading
Here is the University-wide uniform grading policy.
- A: Represents achievement that is outstanding relative to the level necessary to meet course requirements.
- B: Represents achievement that is significantly above the level necessary to meet course requirements.
- C: Represents achievement that meets the course requirements in every respect.
- D: Represents achievement that is worthy of credit even though it fails to fully meet the course requirements.
- F: Represents failure and indicates that the coursework was completed but at a level unworthy of credit, or was not completed and there was no agreement between the instructor and student that the student would be temporarily given an incomplete.
- I: See the catalog.
The grade for the course will be calculated by the following formula:
| WeBWorK | 35% |
| Assignments | 25% |
| Midterm Exam | 20% |
| Final Exam | 20% |
Your numerical grades will be converted to letter grades and finally Grade Points via the following cutoffs (see the UMM Catalog for more on Grades and Grading Policy):
| Numerical | 95% | 90% | 87% | 83% | 80% | 77% | 73% | 70% | 65% | 60% | Below 60% |
| Letter | A | A- | B+ | B | B- | C+ | C | C- | D+ | D | F |
| Grade Point | 4.00 | 3.67 | 3.33 | 3.00 | 2.67 | 2.33 | 2.00 | 1.67 | 1.33 | 1.00 | 0.00 |
Please note that you are not competing against your fellow students. I will adjust the difficulty of the questions and the severity of the grading so that, for example, a B+ score corresponds to what I consider B+ achievement.
Respectful Classroom
- Be in class on time. I nor you fellow classmates enjoy the disruption late arrival causes. I know that situations crop up that will entail late arrival (please come even if you are late!) but try to ensure it is the exception and not the rule.
- If you need to leave class early, let me know before class and slip out as unobtrusively as possible.
- During class, cell phones and music devices should be turned off, and headphones removed from ears.
- As a student you may experience a range of issues that can cause barriers to
learning, such as strained relationships, increased anxiety, alcohol/drug problems,
feeling down, difficulty concentrating, and/or lack of motivation. These mental health
concerns or stressful events may lead to diminished academic performance or reduce a
student`s ability to participate in daily activities.
If you have any special needs or requirements to
help you succeed in the class, come and talk to me as soon as
possible, or visit the appropriate University service yourself.
You can learn more about the range of services available on campus by visiting
the websites:
- The Academic Assistance Center www.morris.umn.edu/services/dsoaac/aac/
- Student Counseling www.morris.umn.edu/services/counseling/
- Disability Services www.morris.umn.edu/services/dsoaac/dso
- Multi-Ethnic Student Program www.morris.umn.edu/services/msp/
- Cooperation is vital to your future success, which ever path you take. I encourage cooperation amongst students where ever possible, but the act of copying or other forms of cheating will not be tolerated. Academic dishonesty in any portion of the academic work for a course is grounds for awarding a grade of F or N for the entire course. Any act of plagiarism that is detected will result in a mark of zero on the entire assignment or test for both parties. If you are in any way unclear about what constitutes academic dishonesty please come and talk to me. UMM's Academic Integrity policy and procedures can be found at www.morris.umn.edu/Scholastic/AcademicIntegrity/.
- Since the assignments are handed out days in advance, only under exceptional circumstances (which can be officially documented) will I accept late work. You will receive a mark of zero if an assignment is submitted late. However, please talk with me asap (do not wait until the next class) if you missed turning something in, even if it is after the deadline.
- If you are going to miss an exam, let me know in advance so we can work out alternate plans. Taking the midterm early can usually be arranged, but not the final exam (unless there are extraordinary circumstances).
Further Course Information and Resources
- I have created a Study Guide for the course.
- The efunda (Engineering Fundamentals) website has some short discussions on the techniques we study.
At UMM the computer algebra system of choice is Mathematica (I typically abbreviate this as MMA), and I use it extensively in most of the courses I teach. FYI, files ending in .nb are Mathematica files.
Mathematica is expensive, and we do not expect our students to purchase it. UMM has a site licence for Mathematica, and it can be found on any computer on campus (PC or MacIntosh). When you need to work with Mathematica outside of class, visit one of the many computer labs on campus.
If you have specific questions about Mathematica while working on homework, bring the file you are working with to office hours on a usb drive and we can look at it together, or email the file to yourself (or me) before you come to office hours. Most questions can be answered in under 10 minutes.
- Mathematica Quick Reference (I will hand this out in class)
- Mathematica Basics (MMA file--a good place to start if you have never used it before)
- Intermediate Mathematica (MMA file--introduction to some of the MMA you see beyond calculus)
Course Calendar
| # | Date | Assignment | Lecture Topic | Practice | Resources |
|---|---|---|---|---|---|
| Aug 24 | |||||
| 1 | Aug 26 | ww: Preliminaries | Course Introduction & Review | Read: How to Succeed in My Courses | MMA (review DE as seen in calculus) |
| 2 | Aug 28 | 1.1 1.2 1.3 Terminology | 1.3: 1-7,14 Read 1.4 | MMA (used in lecture) | |
| Aug 31 | ww: Direction Fields | 2.1 Linear Equations | 2.1: 7,18,28 | 2.1.nb | Homework 2.1.7 has an excellent MMA example, including plotting families of curves and direction fields. | |
| 3 | Sep 2 | 2.2 Separable Equations 2.3 Modeling |
2.2: 1,4,24 | 2.2.nb | MMA (Mixing Problem) | |
| 4 | Sep 4 | ww: Separable | 2.4 Differences Between Linear & Nonlinear Equations | 2.4: 15,20,21 | 2.4.nb | MMA (used in lecture) |
| 5 | Sep 7 | Labour Day--no class | |||
| 6 | Sep 9 | 2.5 Autonomous Equations and Population Dynamics | 2.5: 1,3,8 | Lecture: notes | MMA (autonomous equations) | |
| 7 | Sep 11 | ww: First Order Linear & Exact | 2.6 Exact Equations and Integrating Factors | 2.6: 1,16 | 2.6.nb | |
| 8 | Sep 14 | Taylor Method of Order n: Numerical Solutions to IVP | Read 2.7 | Numerical Solutions | |
| 9 | Sep 16 | 2.8 The Existence & Uniqueness Theorem Concept Map |
2.8: 1,7,14 | 2.8.nb | MMA (Picard's iteration method) | |
| 10 | Sep 18 | 3.1 Homogeneous Equations Constant Coefficients | 3.1: 1,5,7,9 | 3.1.nb 2.9: 36,42 | 2.9.nb |
MMA (used in lecture) | |
| 11 | Sep 21 | 3.2 Solutions of Linear Homogeneous & Wronskian | 3.2: 1,6,9 | 3.2.nb | Handout (includes flowchart of theorems) | |
| 12 | Sep 23 | 3.3 Complex Roots of Characteristic Equation | 3.3: 1,3,7,17 | ||
| 13 | Sep 25 | 3.4 Repeated Roots & Reduction of Order | 3.4: 6,7,11,16,20,23 | 3.4.nb | MMA (repeated, complex, distinct roots) | |
| 14 | Sept 28 | 3.5 Nonhomogeneous: Undetermined Coefficients | 3.5: 3,7,17 | 3.5.nb | ||
| 15 | Sep 30 | Assignment #1 Due | 3.5 Nonhomogeneous: Undetermined Coefficients | lecture | MMA (associated with Handout) | |
| 16 | Oct 2 | 3.6 Variation of Parameters | 3.6: 1,4,13 | 3.6.nb | ||
| 17 | Oct 5 | 3.7 Application: Vibrations | 3.7: 1,6,11 | 3.7.nb | MMA (Three Types of Damping) | |
| 18 | Oct 7 | 3.8 Application: Forced Vibrations Meet in Imholte 11 Computer Lab |
MMA (Numerical solutions) | ||
| 19 | Oct 9 | ww: Higher Order Linear | 4.1 General Theory 4.2 Homogeneous Equations Constant Coefficients |
4.1: 4,6,7,11 4.2: 1,3,4,8,11,13,18 | 4.2.nb |
MMA (repeated roots) Roots of Unity |
| 20 | Oct 12 | 4.3 Undetermined Coefficients 4.4 Variation of Parameters |
4.3: 1,4 4.4: 1,3 | 4.3.nb | Lecture: notes | MMA | |
| 21 | Oct 14 | Chapter 1, 2, 3 & 4 Review | Review Notes | Previous Test Questions | Concept Map | ||
| 22 | Oct 16 | Midterm Exam on Chapters 1, 2, 3 & 4 | |||
| Oct 19 | Fall Break--no class | ||||
| 23 | Oct 21 | Assignment #2 Due | 5.1 Power Series Review | 5.1: 1,8,15,16,21,25 | 5.1.nb | MMA (series review) |
| 24 | Oct 23 | 5.2 & 5.3 Series Solution about Ordinary Point Concept Map |
5.2: 1,2,21 | 5.2.nb | MMA (related to lecture) The homework has an in-depth discussion of Hermite Polynomials (5.2.21) |
|
| 25 | Oct 26 | 5.2 & 5.3 Series Solution about Ordinary Point | 5.3: 5,7,11 | 5.3.nb | MMA (further explorations of the problem solved by hand in class) | |
| 26 | Oct 28 | 5.4 Euler Equations & Regular Singular Points | 5.4: 1,2,9,17,20,21,41 | MMA (series solution: series products, what's special about singular points) | |
| 27 | Oct 30 | 5.5 & 5.6 Series Solutions about Regular Singular Point | 5.5: 1, 3 5.6: 1, 11 |
||
| 28 | Nov 2 | Series Solutions in General (not looking for patterns) | Lecture: notes | MMA | Notes from 1986 | ||
| 29 | Nov 4 | 5.7 Bessel's Equation | Lecture: notes | MMA & Physics Today Article: Special Functions |
||
| 30 | Nov 6 | ww: Laplace Transforms | 6.1 Definition of Laplace Transform 6.2 Solution of Initial Value Problems |
6.1: 2, 5, 6, 17 | 6.1.nb 6.2: 5, 10, 27 | 6.2.nb |
MMA (basics for Laplace transform) |
| 31 | Nov 9 | 6.3 Step Functions | 6.3: 4, 7, 18 | 6.3.nb | MMA (basics for step functions) | |
| 32 | Nov 11 | 6.4 DEs with Discontinuous Forcing Functions | 6.4: 3, 11, 19 | 6.4.nb | ||
| 33 | Nov 13 | Assignment #3 Due | 6.5 Impulse Functions | 6.5: 5, 14 | 6.5.nb | MMA (basics for impulse functions) FYI: Generalized Functions |
| 34 | Nov 16 | 7.1 Introduction | 7.1: 1,4,6,8 | 7.1.nb Read 7.2 |
||
| 35 | Nov 18 | ww: Systems | 7.3 Systems of Algebraic Equations | 7.3: 15,18,21 | 7.3.nb | MMA (eigensystems) |
| Nov 20 | 7.3 Systems of Algebraic Equations 7.4 Basic Theory |
7.4: 4 | Pronunciation of the Greek Alphabet | ||
| 36 | Nov 23 | 7.5 Homogeneous Linear Systems Constant Coefficients | 7.5: 16 | MMA (eigensystems in Mathematica) | |
| 37 | Nov 25 | 7.6 Complex Eigenvalues 7.8 Repeated Eigenvalues |
7.6: 9 7.8: 9,13 |
||
| 38 | Nov 27 | Thanksgiving Holiday--No Class | |||
| 39 | Nov 30 | 7.9 Nonhomogeneous Systems | 7.9: 1 | ||
| 40 | Dec 2 | 9.1 The Phase Plane: Linear Systems | 9.1: 1 | MMA (phase plane and stability) | |
| 41 | Dec 4 | 9.2 Autonomous Systems and Stability | Numerical Solutions | ||
| 42 | Dec 7 | 9.3 Locally Linear Systems | |||
| 43 | Dec 9 | 9.4 Competing Species | |||
| 44 | Dec 11 | Assignment #4 Due | Chapter 5, 6, 7 & 9 Review | Review Notes | Practice Test Questions | |
| Tues Dec 15 8:30-10:30am | Final Exam on Chapter 5,6,7, and bits of 9 | ||||