1550078481-Ordinary_Differential_Equations__Roberts_

Introduction 23

At this point, we need to examine a few initial value problems and bound-

ary value problems to determine whether all initial value problems and all

boundary value problems have a solution or not. That is, we need to examine

the question of existence of solutions to initial value problems and the ques-

tion of existence of solutions to boundary value problems. We also need to

discover if any initial value problem or any boundary value problem has more

than one solution. That is, we need to consider the question of uniqueness of

solutions to initial value problems and the question of uniqueness of solutions

to boundary value problems.

Let us examine the differential equation

(11) xy' -y = 0.

The function y(x) = ex, where e is an arbitrary constant, and its derivative

y'(x) = e are both defined on the interval (-00,00). Substituting y and y'

into the DE (11), we find

xy' - y = xe - ex = 0

for all x and for all choices of e. Thus, y(x) =ex is a one-parameter family of

solutions to the DE (11) on ( -oo, oo). This family of functions is the set of

all lines which pass through the origin except for the line x = 0 (the y-axis).

EXAMPLE 3 An Initial Value Problem with No Solution

Consider the initial value problem

(12) xy' - y = O; y(O) = l.

For x = 0 the differential equation xy' - y = 0 reduces to -y = 0. Conse-

quently, for x = 0 the differential equation xy' - y = 0 is undefined unless

y(O) = 0 also. Hence, the IVP (12) is an example of an initial value problem

which does not have a solution. Furthermore, there is no solution to any

initial value problem of the form xy' - y = O; y(O) = d i:-0. In geometric

terms, there is no solution of the differential equation xy' -y = 0 which passes

through any point on the positive or negative y-axis- that is, any point of the
form (0, d) where d i:-0.