Consider a forced spring-mass equation of the form x′′ + x = cos(ωt) with initial conditions...

Consider a forced spring-mass equation of the form x′′ + x = cos(ωt) with initial conditions x(0) = 1 and x′(0) = 0.
a) Suppose ω doesnt = 1, find the solution to the IVP.

b)If ω = 1, find the solution to the IVP.

c)In which of the two cases does the phenomenon of pure resonance occur? Ex- plain your answer.

d)Verify that with ω = 9/10, x(t) = 100/19 (cos( 9t/10 ) − (81/100) cos t) solves the IVP.

e) Recall the trig identity cos(α) − cos(β) = −2 sin( (α+β)/2 ) sin( (α−β)/2 ). Observe that x(t) ≈ 100/19 (cos( 9t/10 ) − cos t). Use this last form and the trig identity above to approximate x(t) by a product of two sin functions. Make a sketch of the graph, carefully accounting for the periods of the two functions. What would you call the long-term behavior of this solution??

Expert Solution

Colby Messinger answered 7 months ago

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