Answer 6.4

(a) To apply the quadratic formula
 
to (6.5), write (6.5) in the form
 
where
 
 
 
 
Thus,
 
or
(6.7 )

The two explicit expressions for are given by equation (6.7 ).

(b)From the Taylor approximation, we have
 
Now, when
 
So the two approximate expressions for ( ) are
(6.8 )
These approximations are now used to check which of the functions (6.7 ) satisfies the boundary condition. .

(c) Using the plus sign in (6.8 ) gives
 
Then
which says that as

But this contradicts the boundary condition that at
Therefore (6.7+) does not represent a solution of the boundary value problem (BVP).

Using the minus sign in (6.8 ) gives
 
Then
which says that as , in agreement with the boundary condition
Thus it follows that equation (6.7-) gives
 
with p given by (6.6), as the correct solution of the BVP.

Copyright © 1998 Rensselaer Polytechnic Institute. All Rights Reserved.