Consider the initial-value problem y'' = x² + y³, y(1) = 1. See Problem 12 in Exercises 9.1.

(a) Compare the results obtained from using the RK4 method over the interval [1, 1.4] with step sizes h = 0.1 and h = 0.05.

(b) Use a numerical solver to graph the solution of the initial-value problem on the interval [1, 1.4].

Problem 12

Although it might not be obvious from the differential equation, its solution could “behave badly” near a point x at which we wish to approximate y(x). Numerical procedures may give widely differing results near this point. Let y(x) be the solution of the initial-value problem y' = x² + y³, y(1) = 1.

(a) Use a numerical solver to graph the solution on the interval [1, 1.4].

(b) Using the step size h = 0.1, compare the results obtained from Euler’s method with the results from the improved Euler’s method in the approximation of y(1.4).