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This is a carefully revised edition of the well-respected ODE text, whose unique treatment provides a smooth transition to critical understanding of proofs of basic theorems. First chapters present a rigorous treatment of background material; middle chapters deal in detail with systems of nonlinear differential equations; final chapters are devoted to the study of second-order linear differential equations. The power of the theory of ODE is illustrated throughout by deriving the properties of important special functions, such as Bessel functions, hypergeometric functions, and the more common orthagonal polynomials, from their defining differential equations and boundary conditions. The text contains several hundred exercises. Prerequisite is a first course in ODE. The previous edition of this book was published in 1978.