Physics 325 Spring 2011 Problem Session 7 - x t = x p t B...

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Physics 325 Spring 2011 Discussion 7 March 14, 2011 Consider an overdamped driven harmonic oscillator with driving force proportional to e - σt , for some constant σ > 0 . The equation of motion is therefore ¨ x + 2 β ˙ x + ω 2 0 x = F e - σt . (a) Find a particular solution of this differential equation. (b) Let e - λ 1 t and e - λ 2 t be the homogeneous solutions. Determine λ 1 and λ 2 in terms of β and ω 0 . (c) Suppose the initial conditions are such that the full solution is
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Unformatted text preview: x ( t ) = x p ( t ) + B e-λ 1 t + C e-λ 2 t , where B and C are not zero. Here, x p ( t ) is the particular solution from part (a). If σ < λ 1 and σ < λ 2 , what is the behavior of x ( t ) as t → ∞ ? (d) Assume F is positive. For what range of σ will x p ( t ) be negative?...
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