The behaviour of an unforced mechanical system is governed by the differential equation 5 2 -1 0 x(t) = 3 6 -9 x(t), x(0) = 1 1 0 (a) Show that the eigenvalues of the system matrix are 6, 3, 3 and that there is only one linearly independent eigenvector corresponding to the eigenvalue 3. Obtain the eigenvectors corresponding to the eigenvalues 6 and 3 and a further generalized eigenvector for the eigenvalue 3. (b) Write down a generalized modal matrix M and confirm that AM = MJ for an appropriate Jordan matrix J. x(t) = MeM-¹x(0) obtain the solution to the given differential equation. (c) Using the result
The behaviour of an unforced mechanical system is governed by the differential equation 5 2 -1 0 x(t) = 3 6 -9 x(t), x(0) = 1 1 0 (a) Show that the eigenvalues of the system matrix are 6, 3, 3 and that there is only one linearly independent eigenvector corresponding to the eigenvalue 3. Obtain the eigenvectors corresponding to the eigenvalues 6 and 3 and a further generalized eigenvector for the eigenvalue 3. (b) Write down a generalized modal matrix M and confirm that AM = MJ for an appropriate Jordan matrix J. x(t) = MeM-¹x(0) obtain the solution to the given differential equation. (c) Using the result
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:The behaviour of an unforced mechanical system
is governed by the differential equation
5 2 -1
0
x(1) = 3
6-9 x(t),
x(0) = 1
1
1
1
0
(a) Show that the eigenvalues of the system
matrix are 6, 3, 3 and that there is only
one linearly independent eigenvector
corresponding to the eigenvalue 3. Obtain the
eigenvectors corresponding to the eigenvalues
6 and 3 and a further generalized eigenvector
for the eigenvalue 3.
(b) Write down a generalized modal matrix M
and confirm that
AM = MJ
for an appropriate Jordan matrix J.
x(t) = Me M-¹x(0)
obtain the solution to the given differential
equation.
(c) Using the result
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