Chapter 9.2, Problem 17E

Repeat Problem 16 using the initial-value problem y′ = –2y + x, y(0) = 1. The analytic solution is

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

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

1. (a) Approximate y(0.1) using one step and the RK4 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 the RK4 method.
5. (e) Verify that the global truncation error for the RK4 method is O(h⁴) by comparing the errors in parts (a) and (d).