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 + 4 [f(tn, Yn) + f(tn+1,Yn+1)] . Consider the problem y = idy, 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| → ∞ as n → 0. (c) Prove that if you apply method (C) to solve (1), you will always have |y,| = lyol-
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 + 4 [f(tn, Yn) + f(tn+1,Yn+1)] . Consider the problem y = idy, 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| → ∞ as n → 0. (c) Prove that if you apply method (C) to solve (1), you will always have |y,| = lyol-
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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|-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08b9dd6e-80ad-4d46-b7e9-145d193ffbe7%2F72ae7b84-0c93-4940-a5e0-881ca0681a5f%2Fhzktg4_processed.png&w=3840&q=75)
Transcribed Image 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|-
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