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Exercise 2.5:

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.