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Introduction to Ordinary Differential Equations

1972; Elsevier BV; Linguagem: Inglês

10.1016/c2013-0-11341-4

Earl A. Coddington,

Numerical methods for differential equations

This chapter is about the most basic concepts of the theory of differential equations. We will answer some fundamental questions: What is a differential equation? Do differential equations always have solutions? Are solutions of differential equations unique? However, the most important goal of this chapter is to introduce a geometric interpretation for the space of solutions of a differential equation. Using this geometry, we will introduce some of the elements of the subject: rest points, periodic orbits, and invariant manifolds. Finally, we will review the calculus in a Banach space setting and use it to prove the classic theorems on the existence, uniqueness, and extensibility of solutions. References for this chapter include [8], [11], [49], [51], [78], [83], [95], [107], [141], [164], and [179].KeywordsBanach SpacePeriodic SolutionPeriodic OrbitHamiltonian SystemPhase PortraitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.