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(c) Let ( ) (In(2) A = In(2) mie). B - i. For a positive real number r compute eA and eB. Explain all steps and show your working. ii. Given square matrices C, D, when is it true that eC+D = eCeD? iii. Comment on the stability of the origin for y = Ay and y' = By with A and B as given at the start of this question. iv. Are there complex matrices A and B such that e = A and eB = B? (You are not asked to compute these matrices, if they exist, but please explain the reason for your answer!)

(c) Let ( ) (In(2) A = In(2) mie). B - i. For a positive real number r compute eA and eB. Explain all steps and show your working. ii. Given square matrices C, D, when is it true that eC+D = eCeD? iii. Comment on the stability of the origin for y = Ay and y' = By with A and B as given at the start of this question. iv. Are there complex matrices A and B such that e = A and eB = B? (You are not asked to compute these matrices, if they exist, but please explain the reason for your answer!)

Advanced Engineering Mathematics
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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part C

2. (a) Consider the initial value problem eva (2)te-Wa (a) ,(z) Ya(0) + cos(x) a. Define dya (x) za(x) = да Find the initial value problem solved by za(x) by using a suitable theorem, without solving the ODE explicitly. (b) Let f : Rx R → R : (x, y) → f(x, y) be continuous in both variables and globally Lipschitz continuous in y with Lipschitz constant L. Consider the initial value problems Y1 (x)' = f(x, y (x), 4(0) = 1, y2(2) = f(r, y2(x)) + 2 cos(r), y2(0) = 1. Show by using a suitable theorem from class that |1 (x) – 2(x)| < 2|r|e4l=l. (c) Let (In(2) B = -2 A = In(2)), i. For a positive real number r compute e4 and eB. Explain all steps and show your working. ii. Given square matrices C, D, when is it true that eC+D = eCeD? iii. Comment on the stability of the origin for y = Ay and y' = By with A and B as given at the start of this question. iv. Are there complex matrices À and B such that ed = A and e = B? (You are not asked to compute these matrices, if they exist, but please explain the reason for your answer!) %3D
Transcribed Image Text:2. (a) Consider the initial value problem eva (2)te-Wa (a) ,(z) Ya(0) + cos(x) a. Define dya (x) za(x) = да Find the initial value problem solved by za(x) by using a suitable theorem, without solving the ODE explicitly. (b) Let f : Rx R → R : (x, y) → f(x, y) be continuous in both variables and globally Lipschitz continuous in y with Lipschitz constant L. Consider the initial value problems Y1 (x)' = f(x, y (x), 4(0) = 1, y2(2) = f(r, y2(x)) + 2 cos(r), y2(0) = 1. Show by using a suitable theorem from class that |1 (x) – 2(x)| < 2|r|e4l=l. (c) Let (In(2) B = -2 A = In(2)), i. For a positive real number r compute e4 and eB. Explain all steps and show your working. ii. Given square matrices C, D, when is it true that eC+D = eCeD? iii. Comment on the stability of the origin for y = Ay and y' = By with A and B as given at the start of this question. iv. Are there complex matrices À and B such that ed = A and e = B? (You are not asked to compute these matrices, if they exist, but please explain the reason for your answer!) %3D
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