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Consider an oscillator satisfying the initial value problem u" +w?u = 0, u(0) = uo, u'(0) = vo- %3D

Consider an oscillator satisfying the initial value problem u" +w?u = 0, u(0) = uo, u'(0) = vo- %3D

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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Consider an oscillator satisfying the initial valueproblem
u''+w2u=0, u(0)=u0, u'(0)=v0 (i)
(a)let x1=u, x2=u', and transformequation (i) into the form:
x'=Ax, x(0)=x0 (ii)
(b)By using the series (23) on page 417 which is (exp(At)=I + Σ∞n=1 (Antn/n!) ), show that
expAt=I cos wt +A (sinwt)/w (iii)
(c)Find the solution of the initial value problem (ii)
1. Consider an oscillator satisfying the initial value problem u" +w?u = 0, u(0) = uo, u'(0) = vo- a. Let x1 = u, x2 = u' , and transform equations into the form X — Ах, х(0) — х°. b. Use the series of expansion to show that sin(wt) exp( At) = I cos(wt) + A c. Find the solution of the initial value problem of ODE system derived in (a). 2. Find the general solution of the system of equations. (3 x' = –1
Transcribed Image Text:1. Consider an oscillator satisfying the initial value problem u" +w?u = 0, u(0) = uo, u'(0) = vo- a. Let x1 = u, x2 = u' , and transform equations into the form X — Ах, х(0) — х°. b. Use the series of expansion to show that sin(wt) exp( At) = I cos(wt) + A c. Find the solution of the initial value problem of ODE system derived in (a). 2. Find the general solution of the system of equations. (3 x' = –1
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