3.1 Derive the expansion (32) in the text (Hint: Proceed by a succession of one-variable expansions, e.g.,
3.2 Write out the system of equations (36) as in Example 11 for each of the following IVPs.
3.3 Approximate the solution to the following IVPs at
using the
Euler-Cauchy method with step size 
. Do your calculations by hand and
round all results to four decimal places. Compare your numerical results
with the true solutions at 
, and 
3.4 Write a computer program to use the Euler method
(29),
the Euler-Cauchy method (34) with 
,
and the classical Runge-Kutta
method (35) to
approximate the solutions to the IVPs in Exercise 3.3 at 
and . Compare your numerical results with the exact solution at the
indicated 
points.
3.5 Write a computer program to use the classical Runge-Kutta
method (35) to approximate the solution to
the IVP in Example 11 at 
,
, 
, 
. Use a step size . Compare your numerical
results with the exact solution at the indicated points.
3.6 Consider the problem,
on the interval 
. Find the analytical solution 
and show that
is continuous on , but 
is not. Study
the behavior of the error in Euler's method at some fixed points on
as 
.
3.7 When 
depends only on , show that the classical
fourth order
Runge-Kutta formula (35) reduces to Simpson's rule
What is the order of Simpsons's rule; i.e., what is the highest
degree
polynomial 
that the rule integrates exactly? To what quadrature
rule does the Runge-Kutta method of order 2 correspond when 
? When ?
