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The solution for x (t) for a driven, underdamped
Chapter 5, Problem 5.33(choose chapter or problem)
The solution for x (t) for a driven, underdamped oscillator is most conveniently found in the form (5.69). Solve that equation and the corresponding expression for \(\dot{x}\), to give the coefficients \(B_{1}\) and \(B_{2}\) in terms of A,\(\delta\), and the initial position and velocity \(x_{0}\) and \(v_{0}\). Verify the expressions given in and (5.70).
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QUESTION:
The solution for x (t) for a driven, underdamped oscillator is most conveniently found in the form (5.69). Solve that equation and the corresponding expression for \(\dot{x}\), to give the coefficients \(B_{1}\) and \(B_{2}\) in terms of A,\(\delta\), and the initial position and velocity \(x_{0}\) and \(v_{0}\). Verify the expressions given in and (5.70).
ANSWER:Step 1 of 3
The general solution of the driven underdamped oscillator is given as,
\(x(t)=A \cos (\omega t-\delta)+e^{-\beta t} \cdot\left(B_{1} \cos \omega_{1} t+B_{2} \sin \omega_{1} t\right)\) .................. (1)
For t = 0.
\(\begin{aligned} x(0) & =A \cos (\omega(0)-\delta)+e^{-p(0)} \cdot\left(B_{1} \cos \omega_{1}(0)+B_{2} \sin \omega_{1}(0)\right) \\ x_{0} & =A \cos \delta+B_{1} \\ B_{1} & =x_{0}-A \cos \delta \end{aligned}\)
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