Chapter 9.1, Problem 20E

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.

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). See Problem 5 in Exercises 2.6.

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

$y(x)=\frac{1}{9}+\frac{2}{3}x+\frac{38}{9}{e}^{-3(x-1)}.$

1. (a) Find a formula involving c and h for the local truncation error in the nth step if Euler’s method is used.
2. (b) Find a bound for the local truncation error in each step if h = 0.1 is used to approximate y(1.5).
3. (c) Approximate y(1.5) using h = 0.1 and h = 0.05 with Euler’s method. See Problem 1 in Exercises 2.6.
4. (d) Calculate the errors in part (c) and verify that the global truncation error of Euler’s method is O(h).