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  3. consider the initial value problem mx??+cx?+kx=f(

Consider the initial value problem mx??+cx?+kx=F(t),   x(0)=0,   x?(0)=0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected...

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Advance Math

Consider the initial value problem

mx??+cx?+kx=F(t),   x(0)=0,   x?(0)=0
modeling the motion of a spring-mass-dashpot system initially atrest and subjected to an applied force F(t), where the unit offorce is the Newton (N). Assume that m=2 kilograms, c=8 kilogramsper second, k=80 Newtons per meter, and F(t)=80cos(8t)Newtons.

Solve the initial value problem.

x(t)=

Determine the long-term behavior of the system. Is limt??x(t)=0? Ifit is, enter zero. If not, enter a function that approximates x(t)for very large positive values of t.

For very large positive values of t, x(t)?

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