Consider the IVP

a) Solve the problem analytically to find .

b) Write a computer program that implements the Taylor
series algorithm of
order **p=2** for this problem. Let your program begin with a fixed step
size and repeatedly halve **h** until a given accuracy, say
, is achieved; i.e., continue to halve until .
Let and start with **n=2**.

c) Repeat (b) using Taylor algorithms of
orders **p=3** and **p=4**
and note
the difference in the final values as **p** increases for fixed **n**. This
example points out that the Taylor algorithms can be quite effective
when the derivatives are easy to evaluate.