2. (a) Consider the initial value problem Define ya (0) (c) Let = a(*)+-Na(a) + cos(x) 2 = a. A= Find the initial value problem solved by a(z) by using a suitable theorem, without solving the ODE explicitly. Za(x) = (b) Let f: Rx R : R (r,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, y₁(x)), y₁ (0) = 1, y2(x)' = f(x, y₂(x)) + 2 cos(x), 32(0) = 1. Show by using a suitable theorem from class that |31(r) → %2(r)| < 2|r|e&\n\. In(2) Əya (x) да (-28). i. For a positive real number a compute eA and eB. Explain all steps and show your working. ii. Given square matrices C, D, when is it true that eC+D = eeD? 1 In(2)), B= H 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? (You are not asked to compute these matrices, if they exist, but please explain the reason for your answer!)
2. (a) Consider the initial value problem Define ya (0) (c) Let = a(*)+-Na(a) + cos(x) 2 = a. A= Find the initial value problem solved by a(z) by using a suitable theorem, without solving the ODE explicitly. Za(x) = (b) Let f: Rx R : R (r,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, y₁(x)), y₁ (0) = 1, y2(x)' = f(x, y₂(x)) + 2 cos(x), 32(0) = 1. Show by using a suitable theorem from class that |31(r) → %2(r)| < 2|r|e&\n\. In(2) Əya (x) да (-28). i. For a positive real number a compute eA and eB. Explain all steps and show your working. ii. Given square matrices C, D, when is it true that eC+D = eeD? 1 In(2)), B= H 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? (You are not asked to compute these matrices, if they exist, but please explain the reason for your answer!)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
part A
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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