In: Advanced Math
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??