This graduate level textbook offers graduate students a rapid introduction to the language of the subject of ordinary differential equations followed by a careful treatment of the central topics of the qualitative theory. In addition, special attention is given to the origins and applications of differential equations in physical science and engineering. Of special interest to mathematics students is a multifaceted approach to existence theory, for solutions and for invariant manifolds, including an integrated new treatment of smoothness of solutions and invariant manifolds based on the fiber contraction principle. Also, supplementary material throughout the text provides connections between the theory of ordinary differential equations and other advanced mathematical topics; for example, differential geometry, Lie group theory, analysis in infinite dimensional spaces, and even abstract algebra. Applications-oriented students are provided with case studies of important physical models, expecially, coupled oscillators. In particular, the pendulum-rotor model is treated in detail. Of special interest is a treatment of the stability of the inverted pendulum, a discussion of the Fermi-Ulam-Pasta model, and an exposition of the theory of capture into resonance. Through its extensive use of examples, exercises and real-world applications, this book provides science and engineering graduates with a thorough grounding to the theory and application of ordinary differential equations.