Problem S5.6. The motion of an harmonically bound Brownian particle moving in one dimension is governed by the Langevin equations, m dv(t) dt -= -yv (1) - mux(t) + (t) and dx(t) = = v(t), dt where v(t) and x(t) are the velocity and displacement of the particle at time t,m is the mass, y is the friction coefficient, wo is the natural frequency of the harmonic oscillator, and (r) is a delta-correlated random force. If the particle at time t = 0 is in equilibrium with the fluid, compute the variance, (((x(t)-xo)))r. [Note that for this case, ((2)(1)) =4ykBT8(12-11), and by the equipartition theorem, (x)=kBT/mw and (v) =kBT/m. Also assume (vo) = (xo) = (xovo) = 0.]