(c) For to ER and ro € C", prove that the initial value problem (1) ż(t) = Az(t), t E R, z(to) = x0 , %3D has a unique solution. You may assume that r : R + C" given by r(t) exp(A(t – to))zo is a solution of (1). (d) Find the unique solution of the initial value problem (1) when [1 2 0] A = 0 -21 to = 2, xo = 0 2
(c) For to ER and ro € C", prove that the initial value problem (1) ż(t) = Az(t), t E R, z(to) = x0 , %3D has a unique solution. You may assume that r : R + C" given by r(t) exp(A(t – to))zo is a solution of (1). (d) Find the unique solution of the initial value problem (1) when [1 2 0] A = 0 -21 to = 2, xo = 0 2
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
part D
![(a) Given A € C"xn, for n e N, state the definition of the matrix exponential function
exp(-A).
1.
(b) Compute the matrix exponential function exp(-A) where
4
A =
(c) For to ER and ro € C", prove that the initial value problem
(1)
*(t) = Ax(t), teR, 2(to) = o,
has a unique solution. You may assume that x : R + C" given by r(t)
exp(A(t – to))xo is a solution of (1).
(d) Find the unique solution of the initial value problem (1) when
[1 2 01
0 -2 1
A
to = 2, xo = 0
0
(e) For A1 e Cnxn, A3 € Cmxm and A2 € Cnxm, for n, m e N, find an expression for
the matrix exponential of
[A A2
A3
A =
You may use the formula that for appropriately sized matrices X, Y, Z with X and
Z invertible
[X Y]
z]
[x-1 -x-lYZ-1
z-1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F10c15c98-4bb1-4eeb-896c-1799213d2796%2F34d4c079-907b-4f6d-b1d5-fe89dc5e82c8%2Fy2dastc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Given A € C"xn, for n e N, state the definition of the matrix exponential function
exp(-A).
1.
(b) Compute the matrix exponential function exp(-A) where
4
A =
(c) For to ER and ro € C", prove that the initial value problem
(1)
*(t) = Ax(t), teR, 2(to) = o,
has a unique solution. You may assume that x : R + C" given by r(t)
exp(A(t – to))xo is a solution of (1).
(d) Find the unique solution of the initial value problem (1) when
[1 2 01
0 -2 1
A
to = 2, xo = 0
0
(e) For A1 e Cnxn, A3 € Cmxm and A2 € Cnxm, for n, m e N, find an expression for
the matrix exponential of
[A A2
A3
A =
You may use the formula that for appropriately sized matrices X, Y, Z with X and
Z invertible
[X Y]
z]
[x-1 -x-lYZ-1
z-1
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