Consider an oscillator satisfying the initial value problem (50) u′′+ω2u=0,u(0)=u0,u′(0)=v0 a.Let x1 = u, x2 = u′, and transform equations (53) into the form (51) x′=Ax,x(0)=x0. b.Use the series (23) to show that (52) exp(At)=I cos(ωt)+Asin(ωt)ω.
Consider an oscillator satisfying the initial value problem (50) u′′+ω2u=0,u(0)=u0,u′(0)=v0 a.Let x1 = u, x2 = u′, and transform equations (53) into the form (51) x′=Ax,x(0)=x0. b.Use the series (23) to show that (52) exp(At)=I cos(ωt)+Asin(ωt)ω.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Consider an oscillator satisfying the initial value problem
(50)
u′′+ω2u=0,u(0)=u0,u′(0)=v0
a.Let x1 = u, x2 = u′, and transform equations (53) into the form
(51)
x′=Ax,x(0)=x0.
b.Use the series (23) to show that
(52)
exp(At)=I cos(ωt)+Asin(ωt)ω.
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