7.1 Introduction

The general form of a first-order differential equation is

(7.1a)

where y ? =dy/dx and ( x, y) is a given function. The solution of this equation contains an arbitrary constant (the constant of integration). To find this constant, we must know a point on the solution curve; that is, y must be specified at some value of x, say at x=a. We write this auxiliary condition as

(7.1b)

An ordinary differential equation of order n

(7.2)

can always be transformed into n first-order equations. Using the notation

(7.3)

the equivalent first- order equations are

(7.4a)

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