Calculus I,II. That is, a thorough knowledge and understanding of differential and integral calculus in one variable. Some acquaintance with the basic notions of physical science, for the purposes of motivating and applying differential equations. Calculus III (multi-variable) and Linear Algebra are helpful, but not required.
Differential equations (DEs) form the backbone of mathematical modeling throughout modern science. They are important and interesting because they are the mathematical expression of physical, biological, economic or other laws, whenever those laws concern the rates of change of continuous variables. For instance, DEs describe mechanical oscillations, chemically reactions, electrical circuits, ecological population dynamics, and much more. Their solutions constitute the quantitative predictions that follow from the scientific laws.
In this course we study only ordinary differential equations, which are DEs whose unknowns depend on one variable. Most often the independent variable represents time. The dependent variable, or variables, may represent all sorts of different quantities. When there is just one dependent variable we refer to a differential equation, and when there are several dependent variables (or equivalently a dependent vector) we refer to a system of differential equations.
Our goal in learning about differential equations and systems is two-fold. First, we must understand how these mathematical models arise from problems in science and engineering. Then, we must find ways to solve the equations, in order to extract the predictions of the models. This interaction between modeling and analysis is a key feature of the subject, and we will constantly go back and forth between formulating equations and calculating their solutions.
The Kreyzig textbook takes a standard approach, mixing modeling and theory with applications and solution techniques. The book is very clear and concise. In each chapter it covers the topics in almost the order that the lectures will proceed. However, the order of the chapters is slightly altered, in that Chapter 4 will be discussed after Chapter 6 (and Chapters 3 and 5 is not included in this course).
Students should also note that the Kreyzig textbook is a comprehensive introduction to the advanced mathematical methods used in science and engineering, and consequently that the ODE portion of the book is only Part A. Parts B through E treat other courses (linear algebra, partial differential equations, complex analysis, probability and statistics, and numerics and optimization). Keep this book handy for future reference when you take courses on these other subjects, or when you need some piece of information about any of these topics.
It is essential to do a variety of problems when learning mathematics.
Textbook problems will be recommended for each section. Students should do these problems promptly, and they should raise questions about them in class if necessary. Not all of these problems will be handed in for grading. A selection of most important problems will be assigned for grading, at intervals between one and two weeks, as topics are covered. Sufficient time will be given before the deadline for handing-in, and late homeworks cannot be accepted unless there is an acceptable excuse (note from doctor, mandated sports activity, etc.).Always keep up to date with your reading and re-reading of the textbook. It should be clear which section we are going to discuss each day of class, and you are strongly encouraged to read that section before the class. That way you will get the most out of our presentation and discussion. Then shortly after class you should re-read the section in the textbook, connect its presentation to the lecture, and tackle the assigned homework problems. In advance of any examination, you should consolidate your knowledge and try a few extra problems to test it.