solve the 2.8.8

2.8.7 (Error estimate for Euler method) In this question you'll use Taylor series expansions to estimate the error in taking one step by the Euler method. The exact solution and the Euler approximation both start at x x0 when t = to. We want to compare the exact value x(t.)ER(to+) with the Euler approximation a) Expand x(t.), x(, + 1) as a Taylor series in , through terms of 0(2) Express your answer solely in terms of xo. At, and f and its derivatives at xo . b) Show that the local error lx(t)-x,-cal)2 and give an explicit expression for the constant C. (Generally one is more interested in the global error in- curred after integrating over a time interval of fixed length T nAr . Since each step produces an 0(1), error, and we take n=T7at=0(') steps, the global error x)-xs O(Ar), as claimed in the text.) 2.8.8) (Error estimate for the improved Euler method) Use the Taylor series ar- guments of Exercise 2.8.7 to show that the local error for the improved Euler method is O(Ar)