differential equations with linear algebra nov 2009

[...]... a differential equation or system of such equations is linear, then we can usually solve it exactly Linear algebra and systems first In most other texts that present the subjects of differential equations and linear algebra, the presentation begins with first-order differential equations, followed by second- and higher order linear differential equations Following these topics, a modest amount of linear. .. is well-suited: a hybrid course in linear algebra and differential equations, or a course in differential equations that requires linear algebra as a prerequisite We address each course separately with some suggestions for instructors Linear algebra and differential equations For a hybrid course in the two subjects, instructors should begin with chapter 1 on linear algebra There, in addition to an introduction... linear algebra and differential equations, and this text is written for precisely such a class At other institutions, linear algebra and differential equations are treated in two separate courses; in settings where linear algebra is a prerequisite to the study of differential equations, this text may also be used for the differential equations course, with its first chapter on linear algebra available... amount of linear algebra is introduced before beginning to consider systems of linear differential equations Here, however, we begin on the very first page of the text with an example that shows the natural way that systems of linear differential equations arise, and use this example to motivate the need to study linear algebra We then embark on a one-chapter introduction to linear algebra that aims... fact that linear algebra and differential equations can be presented as two parts of a mathematical whole that is coherent and interconnected Indeed, we regularly discuss how the structure of solutions to linear differential equations and systems of equations exemplify important ideas in linear algebra, and how linear algebra often answers key questions regarding differential equations • To present an... ideas in the subject, students will encounter a handful of examples on linear differential equations that foreshadow part of the role of linear algebra in the field of differential equations The goal of the chapter on linear algebra is to introduce important ideas such as linear combinations, linear independence and span, matrix algebra, and the eigenvalue problem At the close of chapter 1 Introduction... page intentionally left blank Differential Equations with Linear Algebra This page intentionally left blank 1 Essentials of linear algebra 1.1 Motivating problems The subjects of differential equations and linear algebra are particularly important because each finds a wide range of applications in fundamental physical problems We consider two situations that involve systems of equations to motivate our... time required to introduce the subject of linear algebra to students, the final two chapters of the text (on numerical methods and series solutions) are ones we would normally not expect to be considered in a hybrid course Differential equations with a linear algebra prerequisite For a differential equations course in which students have already taken linear algebra, chapter 1 may be used as a reference... equations 8 Essentials of linear algebra 1.2 Systems of linear equations Linear equations are the simplest of all possible equations and are involved in many applications of mathematics In addition, linear equations play a fundamental role in the study of differential equations As such, the notion of linearity will be a theme throughout this book Formally, a linear equation in variables x1 , , xn is one... systems of linear differential equations or higher order linear differential equations Over the past decade or two, first-order differential equations have become a standard topic that is normally discussed in calculus courses As such, chapter 2 can be treated lightly at the instructor’s discretion In particular, it is reasonable to expect that students are familiar with direction fields, separable differential . alt="" Differential Equations with Linear Algebra This page intentionally left blank Differential Equations with Linear Algebra Matthew R. Boelkins, J. L. Goldberg, and Merle C. Potter 3 2009 3 Oxford. solutions to linear differential equations and systems of equations exemplify important ideas in linear algebra, and how linear algebra often answers key questions regarding differential equations. •. settings where linear algebra is a prerequisite to the study of differential equations, this text may also be used for the differential equations course, with its first chapter on linear algebra available