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UNL MATH 221H - MATH 221: DIFFERENTIAL EQUATIONS

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MATH 221: DIFFERENTIAL EQUATIONSGlenn Ledder, University of Nebraska–LincolnMAY, 2000This portfolio is based on the way I taught the course in Spring 1999.Math 221, Differential Equations, is the introductory course in ordinary differential equations atthe University of Nebraska–Lincoln. Students in the course have completed the three-semester calculussequence and have generally not taken a linear algebra course. Most of the students are engineeringmajors, although there are some math and science majors and an occasional student with a major notrelated to the course content. Nearly all students in the course are taking it as a required course.Math 221 is taught in classes of fewer than 40 students. The enrollment in the course then dictatesthat there be several sections of the course each semester. Because the course plays a key servicerole and is also a prerequisite for other courses, the Department of Mathematics and Statistics hasworked to make the course essentially the same across all sections in the same semester, with onlyminor changes from one semester to the next. This standardization inevitably requires that individualfaculty members agree to compromises that they may not fully endorse. In particular, there is anofficial statement of course goals. I was the principal author of the current version of this statement.In order to try to achieve the goal of uniformity among different sections of the course, all instructorsare expected to use the official textbook chosen by a committee appointed from among faculty whohave an interest in the course. In Spring 1999, the official textbook was Elementary DifferentialEquations, sixth edition, by Boyce and DiPrima. I was constrained to use this text although I did notsupport its adoption.Instructors generally follow a syllabus prepared by a faculty member appointed by the Departmentto be the convener of the course. They devise their own policies for the course and may alter thesyllabus. There is frequent informal communication among the instructors, and this tends to result insimilar practices for the different instructors, especially in grading systems, assignments, and exams.It is not possible to offer common exams because each section has its own exam schedule; however, theadoption of common goals, use of a common textbook, and frequent communication generally result inmidterm and final exams that differ little across the sections. In practice, variations between sectionsare generally minor and deal more with administrative policies and course delivery than with topicscovered or methods of assessment.Within the constraints imposed by the Department, I made my course design decisions individually.I was the convener in Spring 1999, so I prepared the official syllabus. I did not use the official syllabusin my own section. I used the textbook only as a secondary reference and a source of problems,prefering instead to organize the course around my own lecture notes. I chose my own methods ofassessment, which differed in significant ways from those of my colleagues.11 OBJECTIVES AND TOPICSMath 221 concerns the study of ordinary differential equations. Motivation for the importance ofordinary differential equations comes from their use in mathematical modeling. Physical laws generallytake the form of differential equations. In particular, physical laws describing quantities that vary intime, but not space, and physical laws describing quantities that vary in one spatial dimension, andare independent of time, are generally ordinary differential equations. Typical applications of ordinarydifferential equations include decay processes, one-dimensional motion under the influence of forces,changes in homogeneous populations, and modes of vibration in a string or beam.The subject matter of differential equations includes a body of theory of differential equations andtheir solutions, a collection of techniques for analyzing various categories of equations, and mathemat-ical modeling. Since explicit solution formulas can be obtained for only a limited class of differentialequations, some of the important techniques are graphical (yielding qualitative information about solu-tions) or numerical (yielding a set of approximate solution values). Nevertheless, symbolic techniquesremain a large part of the course, as many mathematical models in common use include equationsthat can be solved exactly by symbolic methods.MATH 221 AND ITS PLACE IN THE CURRICULUMMost of the students who take Math 221 are majors in engineering or science who are expected to takethe course as part of the requirements for their degree. The design of the course must acknowledgethe service role that the course takes for the bulk of its students. The course is also required formajors in the Department; thus, it is also important for the course to be designed so as to preparestudents for courses in advanced ordinary differential equations, partial differential equations, andapplied mathematics.COURSE GOALS AND RATIONALEThe primary goals of Math 221 can be divided into those that deal with modeling, those that dealwith concepts, and those that deal with techniques.ModelingMath 221 is a mathematics course rather than a course in science or engineering, but the developmentof differential equation models to represent physical situations is an appropriate mathematical activity.The emphasis in Math 221 should be on the modeling process rather than the specific applications ofmodels. In their subsequent work, students will learn about mathematical models in their own field ofinterest. No specific mathematical model is common to all fields of science and engineering, but theskills of building, analyzing, and improving mathematical models is.1. Math 221 students should learn to formulate differential equation models for phys-ical processes and use them to explore questions of scientific interest.ConceptsMath 221 is not primarily a course about the theory of differential equations; however, a certain amountof theoretical knowledge is needed in order to be able to solve differential equations and interpret theresults. In particular, results obtained by a computer algebra system are not always correct, and it isconceptual knowledge that allows a student to tell when answers provided by technology are wrong.2. Math 221 students should understand the difference between a solution and a so-lution formula and know what conditions are necessary to guarantee


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