3. Consider the system of DES X = AX, where A is a constant matrix. (a) Show that if v is an eigenvector for X, then vet is a solution. (b) Show that if v is an eigenvector for A and v = = (A - AI)u, then vte + ue is also a solution. (c) Find another solution if v = (A — \I)u = (A − \I)²w. (Hint: for a solution s(t) you know that s(t) = As(t); equivalently ( – XI)s(t) is equal to (A - XI)s(t).)
3. Consider the system of DES X = AX, where A is a constant matrix. (a) Show that if v is an eigenvector for X, then vet is a solution. (b) Show that if v is an eigenvector for A and v = = (A - AI)u, then vte + ue is also a solution. (c) Find another solution if v = (A — \I)u = (A − \I)²w. (Hint: for a solution s(t) you know that s(t) = As(t); equivalently ( – XI)s(t) is equal to (A - XI)s(t).)
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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Transcribed Image Text:3. Consider the system of ODEs x' = Ax, where A is a constant matrix.
(a) Show that if v is an eigenvector for A, then ve is a solution.
(b) Show that if v is an eigenvector for A and v = (A
solution.
XI)u, then vte + ue is also a
(c) Find another solution if v = (A -\I)u = (A − \I)²w. (Hint: for a solution s(t) you know
that s(t) = As(t); equivalently (-AI)s(t) is equal to (A - XI)s(t).)
(d) Now assume the matrix A is the one from the previous question. Find the general solution
to the system of ODES.
Transcribed Image Text:A
00
00
1 1
00
01
01
-1
0
0
OOOOO
TOTOLO
TOHOTO
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