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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)=30e−t 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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