COMP 205 Homework 6 Mar 30, 1998, Due: Apr 08, 1998
Problems
, 
converges to the exact
solution as the step size h decreases. This is illustrated by the
following example. Consider the initial value problem
.
, show that
, and for a given n choose
. Then 
for any n. Note also that
as 
. Show that substituting for h
in the preceding formula and letting 
gives the desired
result.
is given by
Show that local truncation error in the modified Euler formula is
proportional to 
.
. The
corresponding 3-term Taylor formula is:
To derive other formulas of similar accuracy consider the expression
where 
and 
are arbitrary.
, show that the difference between formulas (1) and
(2) is proportional to 
if a + b = 1, 
,
and 
.
and 
have the
infinity of solutions 
, 
, and 
for any 
.
show that Eq. (2) reduces to the
improved Euler formula (as given in class notes).
, show that Eq. (2) reduces to the
modified Euler formula given in Problem 2 (above).
