Answer 6.4
(a) To apply the quadratic formula
to (6.5), write (6.5)
in the form
where
Thus,
or
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
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
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
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.