1550078481-Ordinary_Differential_Equations__Roberts_

320 Ordinary Differential Equations

EXAMPLE 1 Analyzing an Initial Value Problem

for Existence and Uniqueness

Analyze the initial value problem

(12a)

(12b)

SOLUTION

8

YI

YI(l) = -2, Y2(l) = 4.

In this example, fi(x,yI,Y2) = YIY2/2 and h(x,yI,Y2) = 8/YI· Hence,

8fi/8yI = Y2/2, 8Jif8y2 = yif2, 8h/8yI = -8/yi, and 8h/8y2 = 0.

The functions Ji, h, 8 Ji/ OYI, fJ Ji/ 8y2, 8 h / fJyI, and 8 h / 8y2 are all defined

and continuous in any region of XYIY2-space which does not include the plane

YI = 0- that is, whi ch does not include any point of the form (x, 0, Y2). Since

the initial condition for YI is YI ( 1) = -2 < 0, we let

RI={(x,yI,Y2)I -A<x<B, -C<yI<-E<O, and -D<y2<E}

where A , B , C, D , and E are positive constants which are as large as we choose

and E is a positive constant which is as small as we choose. Since Ji, h,

8Jif 8yI, 8Jif8y2, 8f2/8yI, and 8h/8y2 are all defined and continuous on

RI and since (1, -2, 4) E RI, there exists a unique so lut ion to the initial value

problem (12) on some interval (1 - h, 1 + h).

The fundamental theorem is a "local" theorem because the solution of the
initial value problem is guaranteed to exist and be unique only on a "small"
interval. The following theorem, which we state without proof, is call ed a con-
tinuation theorem and tells us how far we can extend (continue) the unique
solution of an initial value problem, assuming the hypotheses of the funda-
mental theorem are satisfied.