The theory of continuous groups is employed to solve ordinary differential equations (ODEs). The presentation is self-contained and presumes for background only a rudimentary exposure to ordinary differential equations. A prior knowledge of group theory, algebraic concepts, or the method of characteristics is not required: when needed, these topics are developed in the text. The book is thus intended for upper division or graduate students in applied mathematics, engineering, or the sciences. As a book on applied mathematics, it may be useful as a monograph. The author is an engineer, not a mathematician, and is unaware of any university course that treats the subject matter of this book. The principal audience is probably readers who are self-instructing themselves. If you are in this category, be sure to examine the worked examples at the end of the chapters in Part II.