Differential Equation

Exercise: Forced Damped Motion

A $4-$lb weight is hung on the lower end of a coil spring suspended from a beam. The weight comes to rest in its equilibrium position, thereby stretching the spring $3$ in $.$ The weight is then pulled down $6$ in $.$ below this position and released at $t=0.$ At this instant an external force given by $F(t)=13sin4t$ is applied to the system. The resistance of the medium in pounds is numerically equal to twice the instantaneous velocity, measured in feet per second.

$($a$)$ Find the displacement of the weight as a function of the time.

$($b$)$ Observe that the displacement is the sum of a transient term and a steady$-$state term, and find the amplitude of the steady$-$state term.