The equation of motion for a damped oscillator is m [(d2x)/(dt2)] = -kx - b [(dx

Question

The equation of motion for a damped oscillator is
m [(d2x)/(dt2)] = -kx - b [(dx)/(dt)]
For a critically damped oscillator, b2 = 4mk; then the general solution of the differential equation is
x(t) = ( c1 + c2 t ) e-bt/2m

(A) Derive the solution x(t) with initial values x(0)=0 and v(0)=v0 (at time t=0).
(B) Determine the maximum displacement, in terms of m, k, v0.
(C) Determine the time of maximum displacement.
(D) Suppose the initial velocity were twice as large. How would the maximum displacement and the time of maximum displacement change?

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