Differential equations first appeared in the late seventeenth century in the work of Isaac Newton, Gottfried Wilhelm Leibniz, and the Bernoulli brothers, Jakob and Johann. They occurred as a natural consequence of the efforts of these great scientists to apply the new ideas of the calculus to certain problems in mechanics, such as the paths of motion of celestial bodies and the brachistochrone problem, which asks along which path from point P to point Q a frictionless object would descend in the least time. For over 300 years, differential equations have served as an essential tool for describing and analyzing problems in many scientific disciplines. Their importance has motivated generations of mathematicians and other scientists to develop methods of studying properties of their solutions, ranging from the early techniques of finding exact solutions in terms of elementary functions to modern methods of analytic and numerical approximation. Moreover, they have played a central role in the development of mathematics itself since questions about differential equations have spawned new areas of mathematics and advances in analysis, topology, algebra, and geometry have often offered new perspectives for differential equations.

This book provides an introduction to many of the important topics associated with ordinary differential equations. The material in the first six chapters is accessible to readers who are familiar with the basics of calculus, while some undergraduate analysis is needed for the more theoretical subjects covered in the final two chapters. The needed concepts from linear algebra are introduced with examples, as needed. Previous experience with differential equations is helpful but not required. Consequently, this book can be used either for a second course in ordinary differential equations or as an introductory course for well-prepared students.