Consider the general initial value problem x_ = f (t, x, y) and y_ = g(t, x, y) with

Chapter 8, Problem 8

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Consider the general initial value problem x_ = f (t, x, y) and y_ = g(t, x, y) with x(t0) = x0 and y(t0) = y0. The Adams- Moulton predictor-corrector method of Section 8.4 generalizes to xn+1 = xn + 1 24 h(55 fn 59 fn1 + 37 fn2 9 fn3), yn+1 = yn + 1 24 h(55gn 59gn1 + 37gn2 9gn3) and xn+1 = xn + 1 24 h(9 fn+1 + 19 fn 5 fn1 + fn2), yn+1 = yn + 1 24 h(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.

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