Repeat Problem 13 using the improved Euler’s method. Its global truncation error is O(h²).

Problem 13

Consider the initial-value problem y' = 2y, y(0) = 1. The analytic solution is y = e^2x.

(a) Approximate y(0.1) using one step and Euler’s method.

(b) Find a bound for the local truncation error in y₁.

(c) Compare the error in y₁ with your error bound.

(d) Approximate y(0.1) using two steps and Euler’s method.

(e) Verify that the global truncation error for Euler’s method is O(h) by comparing the errors in parts (a) and (d).