Repeat Problem 17 using the improved Euler’s method, which has a global truncation error O(h²). See Problem 1. You might need to keep more than four decimal places to see the effect of reducing the order of the error.

Problem 17

Consider the initial-value problem y' = 2x - 3y + 1, y(1) = 5. The analytic solution is

(a) Find a formula involving c and h for the local truncation error in the nth step if Euler’s method is used.

(b) Find a bound for the local truncation error in each step if h = 0.1 is used to approximate y(1.5).

(c) Approximate y(1.5) using h = 0.1 and h = 0.05 with Euler’s method. See Problem 1 in Exercises 2.6.

(d) Calculate the errors in part (c) and verify that the global truncation error of Euler’s method is O(h).

38 -3(x-1)