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Consider the initial value problem mx′′+cx′+kx=F(t), x(0)=0, x′(0)=0 modeling the motion of a damped mass-spring system initially at...
Consider the initial value problem
mx′′+cx′+kx=F(t), x(0)=0, x′(0)=0
modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m=2 kilograms, c=8 kilograms per second, k=80 Newtons per meter, and F(t)=60cos(8t) Newtons.
Solve the initial value problem.
x(t)=
Determine the long-term behavior of the system (steady periodic solution). Is limt→∞x(t)=0? If it 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)≈xsp(t)=
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