1550078481-Ordinary_Differential_Equations__Roberts_

26 Ordinary Differential Equations

Figure 1.4 Solutions Curves y = A sin x of y" + y = O; y(O) = 0

EXAMPLE 7 A Boundary Value Problem with No Solution

Consider the boundary value problem

(16) y" + y = O; y(O) = 0, y(7r) = l.

We already know from the previous discussion that if there is a solution to

this boundary value problem, it must be of the form y = Asinx. Imposing

the second boundary condition, y(7r) = 1, results in the following equation

and contradiction

y(7r)=l=Asin7r=O or 1=0.

Hence, there is no solution to the boundary value problem (16).

EXAMPLE 8 A Boundary Value Problem with a Unique Solution

Now consider the boundary value problem

(17) y" + y = O; y(O) = 0, y(7r/2) = l.

Again, if a solution of this boundary value problem exists, it must be of the

form y =A sin x. Imposing the second boundary condition, y(7r /2) = 1, yields

y(7r/2) = 1 = Asin(7r/2) =A.

So A = 1 and the unique solution of the boundary value problem (17) is

y = sinx.