Explanation The general expression for Damped oscillator. − k x − b d x d t = m d 2 x d t 2 (I) Here, k is the force constant, b is the damping constant, m is the mass. Let the solution of the equation (I) as, x = A e − b t / 2 m cos ( ω t + ϕ ) (II) Here, A is the amplitude, ϕ is the phase angle. Differentiate equation (II) with respect to time, d x d t = A e − b t / 2 m ( − b 2 m ) cos ( ω t + ϕ ) − A e − b t / 2 m ω sin ( ω t + ϕ ) (III) Differentiate equation (III) with respect to time d 2 x d t 2 = − b 2 m [ A e − b t / 2 m ( − b 2 m ) cos ( ω t + ϕ ) − A e − b t / 2 m ω sin ( ω t + ϕ ) ] − [ A e − b t / 2 m ( − b 2 m ) ω sin ( ω t + ϕ ) + A e − b t / 2 m ω 2 cos ( ω t + ϕ ) ] (IV) Use equation (II), (III) and (IV) in equation (I), − k A e − b t / 2 m cos ( ω t + ϕ ) + b 2 2 m A e − b t / 2 m cos ( ω<

Principles of Physics: A Calculus-Based Text

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Author: Raymond A. Serway, John W. Jewett

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Chapter 12, Problem 38P

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Show that equation x=Ae−bt/2mcos(ωt+ϕ) is a solution of −kx−bdxdt=md2xdt2.

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Chapter 12, Problem 37P

Chapter 12, Problem 39P

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Show that equation x = A e − b t / 2 m cos ( ω t + ϕ ) is a solution of − k x − b d x d t = m d 2 x d t 2 .

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Show that equation x=Ae−bt/2mcos(ωt+ϕ) is a solution of −kx−bdxdt=md2xdt2.

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