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Theory of Ordinary Differential Equations is designed primarily as a textbook for undergraduate and postgraduate students in various programmes in science. This comprehensive and well-organized book provides various well known mathematical techniques such as the variation of parameters, Clairaut and Sturm-Liouville theory. The book deals with the elementary methods for obtaining explicit solutions of ordinary differential equations and existence as well as uniqueness of solution of differential equations. The topics included in this book would provide students with basic knowledge of ordinary differential equations. Although there are a number of well-established and successful textbooks that treat this subject, they seem either to include a large amount of additional material or to concentrate only on the more advanced topics. We, therefore, collectively materialized this book. Basically, this book explains key concepts and methods in the field of ordinary differential equations. It assumes only minimal mathematical prerequisites but at the same time, introduces the reader to the way ordinary differential equations used in current mathematical research and in scientific modeling. It is designed as a practical guide for students and aspiring researchers in any mathematical science.