Differential equations can bring mathematics to life, describing phenomena originating in physics, chemistry, biology, economics, and more. Used by scientists and engineers alike, differential equations are also the starting point of much purely mathematical activity. They also play a role in the formulation and resolution of problems in harmonic analysis, differential geometry, and probability calculus. A large part of functional analysis has therefore been motivated by the need to solve questions in the analysis of differential systems, as with numerical analysis.Differential equations are doubly relevant, then: as significant in many areas of mathematics, and as important machinery for applying mathematics to real-world problems. This book therefore aims to provide a rigorous introduction to the theoretical study of differential equations, and to demonstrate their utility with applications in many fields.Ordinary Differential Equations and Applications originates from several courses given by the author for decades at the University of Seville. It aims to bring together rigorous mathematical theory and the rich variety of applications for differential equations. The book examines many aspects of differential equations: their existence, uniqueness, and regularity, alongside their continuous dependence on data and parameters. Delving into permanent interpretation of the laws of differential equations, we also look at the role of data and how their solutions behave. Each chapter finishes with a collection of exercises, many of which also contain useful hints.Key Features: oThe book can be very useful as a support for basic courses of Ordinary Differential Equations. It contains the main theoretical results and their proofs and a lot of connections to applicationsoThe text "invites" the reader to revisit fundamental results of mathematical and functional analysis (inverse and implicit function theorems, the chain rule for differentiation, fixed-point theorems, etc.). On the other hand, the text shows the way differential equations can be used to model and understand many real-life phenomenaoSome collateral aspects of differential equations have also been included. In particular, the method of characteristics for the solution of the Cauchy problem for a partial differential equation of first order has been presented. There is also one chapter devoted to exhibit basic results from control theoryoIn a final chapter, a considerable amount of additional comments and results have been included. This may be viewed as a list of possible continuations of the theory, useful to those students with interest in the subject

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