1550078481-Ordinary_Differential_Equations__Roberts_

The Initial Value Problem y' = f(x, y); y(c) = d 109

y(x)

-1 0 1 2 3 4

x

Figure 2.14 Direction Field and Graph of a Numerical Approximation of

y' = x - y; y(O) = 2

The following example illustrates one type of result we might obtain in the
event we attemp t to produce a numerical approximation to a solution on an
inter val which is too large- that is, on an interval larger than the interval on
which the solution exists.

EXAMPLE 9 A Numerical Approximation Outside of the Interval

of Existence

Use the fundamental existence, uniqueness, and continuation theorems to
analyze mathematically the initial value problem

(27)

y 1

y' = + - ·

(x-l)(x+2) x'

y(-1) = 2

on the interval [-2.5, .5]. Then use MAPLE to calculate and graph a numer-

ical approximation of the solution to the IVP (27).