1550078481-Ordinary_Differential_Equations__Roberts_

110 Ordinary Differential Equations

SOLUTION

Mathematical Analysis

The differential equation of (27) is linear with a(x) = 1/((x-l)(x+2)) and
b(x) = 1/x. The function a(x) is not defined; and, therefore, not continuous
at x = -2 and x = l. The function b( x) is not defined and not continuous at
x = 0. Hence, the functions a(x) and b(x ) are both continuous on the intervals
( -oo, -2), (-2, 0), (0 , 1 ) , and (1, oo). Since the initial point -1 E ( -2, 0) , the

IVP (27) has a unique solution on (-2, 0) and this is the largest interval on

which (27) has a so lution.

Numerical Solution

A graph of the numerical approximation to the solution of the IVP (27)
generated using MAPLE is shown in Figure 2. 15. This output was created
using the following four Maple statements. These statements were obtained
by modifying of the four statements in example 8.

with(DEtools):with(plots):

de:=diff(y(x), x) = y(x) / ( (x-1 )*(x-2) )+ 1/x:

p:=DEplot(de, y(x), x=-2.5 .. 0.5, {[y(-1)=2]}, arrows=NONE, axes=

BOXED):

display(p);

In the second statement, we changed the specification of the differential equ a-

tion from x - y(x) to y(x)/((x - 1) * (x - 2)) + 1/x. In the third statement,

we replaced the range of x values -1..4 with the new range -2.5 .. 0.5. We did

not specify a range for the y values, so the program selected a range based

on the computed so lution. From Figure 2.15 it appears the minimum y value

selected by the software was -5 and the maximum y value selected was 16.

Also, in the third statement we changed the initial condition to y( -1) = 2 and

we indicated we did not want a direction field displayed by replacing LINE

by NONE.

Notice that the graph indicates the presence of vertical asymptotes in the

solution near x = -2 and x = 0.

This example illustrates that it is necessary for you to perform a thor-

ough mathematical analysis for each initial value problem prior to computing

a numerical solution. In this example, the computer generated a numerical

approximation to the solution outside the interval of existence-that is , out-

side of the interval (-2, 0). So you must analyze each initial value problem

separately and determine what the fundamental theorems tell you about the

problem with respect to existence, uniqueness, and continuation of the solu-

tion. When you use your computer software to numerically solve the IVP (27),

what does the graph of the solution look like?