Use (a) to solve the system x - ( )x. Х. -8 7
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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Question
Transcribed Image Text:Use (a) to solve the system
x = (: :) .
Х.
-8 7
NB: Real solutions are required.
Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of
the two eigenvalues yields two linearly independent solutions. The second will yield solutions
which are identical (up to a constant) to the solutions already found.
a) is as follows
Show from first principles, i.e., by using the definition of linear independence, that if
x+iy, y # 0 is an eigenvalue of a real matrix A with associated eigenvector v = u + iw, then
the two real solutions
Y(t) = et (u cos bt – w sin bt)
and
Z(t) = eat (u sin bt + w cos bt)
are linearly independent solutions of X = AX.
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