The following are three different methods for solving y'(t) = f(t, y(t)): (A) Yn+1 = Yn + S(tn, Yn) + f(tn + At, yn + Atf(tn, Yn))] , (B) Yn+1 = Yn + Atf (tn + 4, Yn + f(tn, Yn)) , (C) Yn+1 = Yn +...

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Extracted text: The following are three different methods for solving y'(t) = f(t, y(t)): (A) Yn+1 = Yn + S(tn, Yn) + f(tn + At, yn + Atf(tn, Yn))] , (B) Yn+1 = Yn + Atf (tn + 4, Yn + f(tn, Yn)) , (C) Yn+1 = Yn + [f(tn, Yn) + f (tn+1, Yn+1)] . Consider the problem 3' = iXy, y(0) = yo, (1) where i = v-1, A E R, yo E C. (a) Show that you will have the same recursive relationship between y, and yn+1 when you apply either method (A) or method (B) to solve (1). (b) When method (A) or method (B) is applied to solve (1), prove that as long as yo # 0, 1# 0, for any fixed At, \yn| → o as n → o. (c) Prove that if you apply method (C) to solve (1), you will always have |yn| = |y0|-