Chapter 9.1, Problem 15E

Repeat Problem 13 using the initial-value problem y′ = x – 2y, y(0) = 1. The analytic solution is

$y(x)=\frac{1}{2}x-\frac{1}{4}+\frac{5}{4}{e}^{-2x}.$

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

1. (a) Approximate y(0.1) using one step and Euler’s method.
2. (b) Find a bound for the local truncation error in y₁.
3. (c) Compare the error in y₁ with your error bound.
4. (d) Approximate y(0.1) using two steps and Euler’s method.
5. (e) Verify that the global truncation error for Euler’s method is O(h) by comparing the errors in parts (a) and (d).