Consider the improved Euler method for solving the illus

Consider the improved Euler method for solving the illustrative initial value problem y? = 1 ? t + 4y, y(0) = 1. Using the result of Problem 17(c) and the exact solution of the initial value problem, determine en+1 and a bound for the error at any step on 0 ? t ? 2. Compare this error with the one obtained in Eq. (25) of Section 8.2 using the Euler method. Also obtain a bound for e1 for h = 0.05, and compare it with Eq. (26) of Section 8.2. In each of Problems 19 and 20, use the actual solution ?(t) to determine en+1 and a bound for en+1 at any step on 0 ? t ? 1 for the improved Euler method for the given initial value problem. Also obtain a bound for e1 for h = 0.1, and compare it with the similar estimate for the Euler method and with the actual error using the improved Euler method