In this chapter, we apply analysis to the study of ordinary differential equations, generally called DEs or ODEs. Ordinary is used to indicate differential equations of a single variable, in contrast to partial differential equations (PDEs), in which several variables, and hence partial derivatives, appear. We will see some PDEs in the chapters on Fourier series, Chapters 13 and 14.
Most introductory courses on differential equations present methods for solving DEs of various special types. We will not be concerned with those techniques here except to give a few pertinent examples. Rather we are concerned with why differential equations have solutions, and why these solutions are or are not unique. This topic, crucial to a full understanding of differential equations, is often omitted from introductary courses because it requires the tools of real analysis. In particular, we will require the Banach Contraction Principle (11.1.6) which was established in the previous chapter.
KeywordsUnique Solution Lipschitz Condition Power Series Expansion Contraction Principle Continuation Theorem
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