1550078481-Ordinary_Differential_Equations__Roberts_

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

-2 -1 0 1 2

x

Figure 2.16 Graph of a Numerical Approximation of the Solution to

y' = - x/y; y(-1) = 1

EXAMPLE 11 Numerical Approximation of the IVPs:

y' = yfx.Y; y(-1) = d

Mathematically analyze the following two initial value problems.

(29a)

(29b)

y' = py; y(-1) = 0

y' = py; y(-1) = -1/9 ~ -.11111111

113

Then calculate a numerical approximation of the solution to initial value prob-
lems on the interval [-1, 1] using MAPLE and SOLVEIVP.

SOLUTION

Mathematical Analysis

The function f(x, y) =FY is defined and continuous in the first quadrant,

in the third quadrant, on the x-axis (where y = 0), and on the y-axis (where

x = 0). The function of jay = JX1Y is defined and continuous in the first
quadrant, in the third quadrant, and on the y-axis; but it is not defined on
the x-axis. Applying the fundamental theorems to the initial value problem

y' = py; y(c) = d