1550078481-Ordinary_Differential_Equations__Roberts_

24 Ordinary Differential Equations

EXAMPLE 4 An Initial Value Problem with an Infinite Number

of Solutions

The differential equation of the initial value problem

(13) xy' - y = O; y(O) = 0

is the same as the differential equation in the previous example. As we noted
earlier, y = ex is a one-parameter family of solutions on (-00,00). Sub-
stituting the initial condition y(O) = 0 into the solution y(x) = ex results
in the equation y(O) = 0 = e(O) which is satisfied by any constant e. Thus,

y(x) =ex is a solution of the IVP (13) on the interval (-oo, oo) for any choice

of the constant c. Consequently, the IVP (13) is an example of an initial value

problem with an infinite number of solutions.

EXAMPLE 5 An Initial Value Problem with a Unique Solution

Now consider the initial value problem

(14) xy' - y = O; y(l) = 4.

Again, the function y(x) = ex, where c is an arbitrary constant, is a one-

parameter family of so lutions of the differential equation xy' - y = 0 on the

interval (-oo, oo). Imposing the initial condition y(l) = 4 = c(l), we find

c = 4. So the IVP (14) has the unique solution y(x) = 4x on (-oo, oo ).

Let us examine initial value problems of the form xy' - y = O; y(a) = b

where a-=/:-0. The solution of the differential equation is y(x) =ex. Imposing

the initial condition y(a) = b, we find c must satisfy the equation y(a) =

b = ca. Since we have assumed a -=/:-0, the unique solution of this equation

is c = b/ a; and, therefore, the unique solution of the initial value problem
xy' - y = O; y(a) = b where a-=/:- 0 is y(x) =bx/a. Interpreted geometrically,
this means that for any point (a, b) where a-=/:-0- that is, for any point which
is not on the y-axis- there is a unique solution of the differential equation
xy' -y = 0 which passes through (a, b).