This book provides a comprehensive introduction to the theory of ordinary differential equations with a focus on mechanics and dynamical systems as important applications of the theory. The text is written to be used in the traditional way (emphasis on the theory with the computer component as optional) or in a more applied way (emphasis on the applications and the computer material). The accompanying CD contains Maple worksheets to use in working the exercises and extending the examples. The disk also contains special Maple code for performing various tasks. In addition to its use in a traditional one- or two- (there is enough material for two) semester graduate course in mathematics, the book is organized to be used for interdisciplinary courses in applied mathematics, physics, and engineering. Researchers and professionals may also find the supplementary material on the disk on discrete dynamical systems, theory of iterated maps, and code for performing specific tasks on the disks particularly useful.