This book provides a comprehensive introduction to the theory of ordinary differential equations at the graduate level and includes applications to Newtonian and Hamiltonian mechanics. It not only has a large number of examples and computer graphics to illustrate the theory, but also has a complete collection of proofs for the major theorems ranging from the usual existence and uniqueness results to the Hartman-Grobman theorem and the Jordan canonical form theorem. The book can bc used almost exclusively in the traditional way for graduate math courses, or it can bc used in an applied way for interdisciplinary courses involving physics, engineering, and other science majors. For this reason, an extensive computer component using Maple(r) is provided on the accompanying CD-ROM and is cross-referenced in the text. The material on the CD is an in-depth supplement and complement to the material in the text and contains (1) an introduction to discrete dynamical systems and iterated maps ; (2) special-purpose Maple(r) code for animating phase portraits, stair diagrams, N-body motions, rigid-body worksheets pertaining to all aspects of using Maple(r) to study the topics in the text.