Consider the vibrating system described by the IVP: u" + u = 0.5cos t u(0) = 0, u'(0) = 0 Find the solution.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.4: Applications
Problem 8EQ
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Consider the vibrating system described by the IVP:
u" + u = 0.5cos t
u(0) = 0,
u'(0) = 0
Find the solution.
Transcribed Image Text:Consider the vibrating system described by the IVP: u" + u = 0.5cos t u(0) = 0, u'(0) = 0 Find the solution.
Expert Solution
Step 1

Given IVP is 

u''+u=0.5cost , u0=0,u'0=0

The auxiliary equation corresponding to the given equation is 

m2+m=0m=0,-1

Hence, the complementary function is 

C.F: uc=c1+c2e-t

where c1,c2 are constants

Step 2

Let the particular integral be 

up=0.5costD2+D, Dddx=0.5cost-1+D cosatFD2=cosatF-a2=D+1-20.5cost=-14D+1cost=-14cost-sint

Hence, the general solution is 

u=uc+up=c1+c2e-t+14sint-14cost

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