7.1 Introduction

The general form of a first-order differential equation is

(7.1a)

where y ?=dy/dx and f(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 transformed into n first- order equations. Using the notation

(7.3)

the equivalent first-order equations are

(7.4a)

The solution now requires the knowledge n auxiliary conditions. If these conditions are specified at the same value of x, the problem is said to be an initial value problem. Then the auxiliary conditions, called initial conditions, have the form

(7.4b)

If y _i are specified at different values of x, the problem is called a boundary value problem.

For example,

is an initial value problem since both auxiliary conditions imposed on the solution are given at x=0. On the other hand,

is a boundary value problem because the two conditions are specified at different values of x.

In this chapter we consider only initial value problems. The more difficult boundary value problems are discussed in the next chapter. We also make extensive use of...