Answered: Suppose that all the eigenvalues of the…

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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Suppose that all the eigenvalues of the matrix A have negative real part. Then every
solution of the differential equation
x`= Ax satisfies,
|x(t)| ≤ |x(s)|, if t > s.

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$S u p p o s e t h a t a l l t h e e i g e n v a l u e s o f t h e m a t r i x A h a v e n e g a t i v e r e a l p a r t . T h e n e v e r y s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n, x' = A x w e n e e d t o p r o v e t h a t, |x (t)| \leq |x (s)|, i f t > s$

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