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Suppose a mass of 1 kg is attached to a spring with spring constant k =...

Suppose a mass of 1 kg is attached to a spring with spring constant k = 2, and rests at equilibrium position. Starting at t = 0, an external force of f(t) = e t is applied to the system. Suppose the surrounding medium offers a damping force numerically equal to β times the instantaneous velocity, where β > 0 is some given number.

(a) What is the IVP governing this harmonic motion.

(b) For what value(s) of β will the spring experience over-damped, under-damped, and critically damped motion, respectively.

(c) Using variation of parameters, solve for the general solution of the ODE given in (a). In particular, do not take the initial conditions into consideration; do not solve for c1, c2. (

d) Using (c), describe the behavior of motion as β → ∞. Why does this physically make sense?

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