Kaplan Wilfred. Ordinary Differential Equations

Addison-Wesley, 1958. — 560 p.This book is intended to serve as an introduction to the theory of ordinary differential equations. A knowledge of calculus is assumed. Enough material is provided for a year's course in the subject.
Throughout it has been the author's aim to treat the subject from the point of view of "functional analysis." A differential equation is equivalent to a specification of certain functions; we try to obtain properties of the functions from the differential equation which determines them.
Certain differential equations describe a relationship between two functions; if one (the input) is known and the differential equation is stable, the other (the output) is known approximately, the error being a transient. A study of the dependence of output function on input function provides much insight into differential equations.
The branch of engineering known as systems analysis or instrumentation is concerned to a great extent with the point of view described. It has now become clear that among! the differential equations occurring in the physical sciences, those for which all ciuestions are answered by a simple formula are very rare, and that a deeper understanding of the functional relationships is reciuired. The concepts of input and output, which arose in engineering, are an important means to this end. These concepts are beginning to find wider applications: for example, in biology and economics.