airy-treatise-pdes / TeX / sections / chapter01.tex

Full commit
\chapter{On Partial Differential Equations}

\section{Preliminary Notice On Integration}

In all that follows, we shall suppose that it is always possible to effect simple integration; inasmuch as any difficulties of integration, connected with the solutions of partial differential equations, do not affect the principle of those solutions. Thus, for instance, we shall not hesitate to represent an unknown function of $x$ by $\chi''(x)$, (the second differential-coefficient of $\chi(x)$), on the assumption that, whatever be the form of $\chi''(x)$, we can in some way find the function $\chi(x)$ of which it is the second differential-coefficient.

\section{Characteristics Of The Solutions Of Simple Differential Equations}

Before entering on the subject of Partial Differential Equations, it may be convenient to consider some of the characteristics of the solutions of Simple Differential Equations.

To begin with Simple Differential Equations of the first order. Suppose, for facility of geometrical illustration, we consider the equation $y\frac{dy}{dx}=a$, or the equation $y\frac{dx}{dy}=b$, of which the former, translated geometrically, indicates that the subnormal of a plane-curve (to be found) is constant, and the latter indicates that the subtangent is constant. The algebraical solutions are easily found: in each, there is a constant, and is not defined by it; a constant of that class described (perhaps improperly) by the term ``arbitrary,'' but which really means ``not yet determined, but enabling us by  proper determination of its value so to fix the value of $x$ corresponding to a given value of $y$ that we can adjust the solution to some specific condition.''

The reader is requested to observe that instead of the term ``arbitrary constant,'' we shall always use the term ``undetermined constant.''