Consider the initial value problem x = f(t, x, y) and y = g(t, x, y) with x(t0) = x0 and

Chapter 8, Problem 9

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Consider the initial value problem x = f(t, x, y) and y = g(t, x, y) with x(t0) = x0 and y(t0) = y0. The generalization of theAdamsMoulton predictorcorrector method of Section 8.4 is xn+1 = xn + 1 24h(55fn 59fn1 + 37fn2 9fn3), yn+1 = yn + 1 24h(55gn 59gn1 + 37gn2 9gn3) and xn+1 = xn + 1 24h(9fn+1 + 19fn 5fn1 + fn2), yn+1 = yn + 1 24h(9gn+1 + 19gn 5gn1 + gn2). Determine an approximate value of the solution at t = 0.4 for the example initial value problem x = x 4y, y = x + y with x(0) = 1, y(0) = 0. Take h = 0.1. Correct the predicted value once. For the values of x1, ... , y3 use the values of the exact solution rounded to six digits: x1 = 1.12735, x2 = 1.32042, x3 = 1.60021, y1 = 0.111255, y2 = 0.250847, and y3 = 0.429696.