Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas- sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM) . The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe- matical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface Consider a first order differential equation of form x' = f ( t, x). In elemen- tary courses one frequently gets the impression that such equations can usually be "e;solved,"e; i. e. , that explicit formulas for the solutions (in terms of powers, exponentials, trigonometric functions, and the like) can usually be found. Nothing could be further from the truth.

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas- sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM) . The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe- matical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface Consider a first order differential equation of form x' = f ( t, x). In elemen- tary courses one frequently gets the impression that such equations can usually be "e;solved,"e; i. e. , that explicit formulas for the solutions (in terms of powers, exponentials, trigonometric functions, and the like) can usually be found. Nothing could be further from the truth.