Repeat Problem 19 using the improved Euler’s method, which has global truncation error O(h²). See Problem 5. You might need to keep more than four decimal places to see the effect of reducing the order of error.

Problem 19

Repeat Problem 17 for the initial-value problem y' = e^-y, y(0) = 0. The analytic solution is y(x) = ln(x + 1). Approximate y(0.5).

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)