2.2 EXISTENCE AND UNIQUENESS THEOREM FOR A SINGLE FIRST-ORDER NONLINEAR ODE

In considering the solutions of the nonlinear ODE

with IC

the following result can be proven (cf. Ince, 1956, chapter 3; Coddington, 1989, chapter 5).

Theorem

Let f(x, y) be a one-valued, continuous function of x and y in a domain defined as

and shown as the rectangle ABCD in Figure 2.1.

Figure 2.1: Definition of domain for a case where h = b/ M < a.

Let f( x,y) ? M in , and let

If h < a, then is redefined as

and shown as the rectangle A'B'C'D' in Figure 2.1. Furthermore, let f( x, y) satisfy the Lipschitz condition, which implies that if ( x, y) and ( x, Y) are two points in , then

where K is a positive constant.

With these conditions satisfied, the existence theorem ensures that there exists a unique continuous function of x, say y( x), defined for all x in the range x ? x ₀ ? h, which satisfies the ODE

and the IC

The theorem can be proved in more ways than one. The most common proof is by a method of successive approximations, also called Picard iteration. The proof derives from the fact that integrating both sides of eq. (2.2.1) with...