A First Course in Ordinary Differential Equations

Abstract

This book presents a modern introduction to mathematical techniques for solv ing initial and boundary value problems in linear ordinary differential equations (ODEs). The focus on analytical and numerical methods makes this book particu larly attractive. ODEs play an important role in modeling complex scientific, technological, and economic processes and phenomena. Therefore, to address students and scientists of various disciplines, we have circumvented the traditional definition-theorem-proof format. Instead, we describe the mathematical background by means of a variety of problems, examples, and exercises ranging from the elementary to the challenging problems. The book is intended as a primary text for courses in theory of ODEs and numerical treatment of ODEs for advanced undergraduate and early graduate students. It is assumed that the reader has a basic knowledge of elementary calculus, in particular methods of integration, and numerical analysis. Physicists, chemists, biologists, computer scientists, and engineers whose work involves the solution of ODEs will also find the book useful both as a reference and as a tool for self-studying. The book has been prepared within the scope of a German-Iranian research project on mathematical methods for ODEs, which was started in early 2012. We now outline the contents of the book. In Chap. 1, an introduction to ODEs and some basic concepts are presented. Chapter 2 deals with scalar first-order ODEs. The existence and uniqueness of solutions is studied on the basis of the well-known theorems of Peano and Picard-Lindelöf. The analytical standard techniques for the determination of the exact solution are given. Moreover, the solution of high-degree first-order ODEs (Clairaut and Lagrange equation), which are often encountered in the applications, is discussed. This chapter ends with the discussion about the family of curves and orthogonal trajectories. Chapter 3 is devoted to analytical methods for the solution of second-order ODEs. Subsequent to the introduction to higher-order ODEs, solution methods for homogeneous and inhomogeneous equations are presented. Chapter 4 focuses on the Laplace transform for scalar ODEs that is also used in Chap. 5 for the solution of systems of first-order ODE