In Differential Equations with Linear Algebra, we endeavor to introduce students to two interesting and important areas of mathematics that enjoy powerful interconnections and applications. Assuming that students have completed a semester of multivariable calculus, the text presents an introduction to critical themes and ideas in linear algebra, and then, in its remaining seven chapters, investigates differential equations while highlighting the role that linearity plays in their study. Throughout the text, we strive to reach the following goals:
– To motivate the study of linear algebra and differential equations through interesting applications in order that students may see how theoretical results can answer fundamental questions that arise in physical situations.
– To demonstrate the 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 exposition that is intended to be read and understood by students. While certainly every textbook is written with students in mind, often the rigor and formality of standard mathematical presentation takes over, and books become difficult to read. We employ an examples-first philosophy that uses an intuitive approach as a lead-in to more general, theoretical results.
– To develop in students a deep understanding of what may be their first exposure to post-calculus mathematics. In particular, linear algebra is a fundamental subject that plays a key role in the study of much higher level mathematics; through its study, as well as our investigations of differential equations, we aim to provide a foundation for further study in mathematics for students who are so interested.
Whether designed for mathematics or engineering majors, many universities offer a hybrid course in 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 as a review of previously studied material. More details on the ways the book can be implemented in these courses follows shortly in the section How to Use this Text. An overriding theme of the book is that if a differential equation or system of such equations is linear, then we can usually solve it exactly.