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(a) Using the fact that the solution to the linear state equ * = Ax; x (to) = xo is x(t) = e(t-to)xo, show that -1 eA(t+T) eAteAT and (et) = e = e - At

(a) Using the fact that the solution to the linear state equ * = Ax; x (to) = xo is x(t) = e(t-to)xo, show that -1 eA(t+T) eAteAT and (et) = e = e - At

Advanced Engineering Mathematics
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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2. [The Matrix Exponential, Laplace Transforms and Solution to Linear Systems] In this question, we use the matrix exponential function defined by e At = I +At+ A²t² A³ +³ + 2! 3! The Laplace Transform defined by F(s) = L{f(t)} = f(t)e-stdt. S (a) Using the fact that the solution to the linear state equations x = xo, show that Ax; x (to) = xo is x(t) = eª(t—to), eA(t+T) = eAteAT and (eat) -1 - At =
Transcribed Image Text:2. [The Matrix Exponential, Laplace Transforms and Solution to Linear Systems] In this question, we use the matrix exponential function defined by e At = I +At+ A²t² A³ +³ + 2! 3! The Laplace Transform defined by F(s) = L{f(t)} = f(t)e-stdt. S (a) Using the fact that the solution to the linear state equations x = xo, show that Ax; x (to) = xo is x(t) = eª(t—to), eA(t+T) = eAteAT and (eat) -1 - At =
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