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I have a question about damped oscillator. Actually, although I have read courses about oscillator, I couldn't solve this. I think this is very easy question :(

1. Homework Statement

Consider the solution for the damped ( but not driven ) oscillator,

x = e

^{-[tex]\beta[/tex]t/2}(Acos[tex]\omega[/tex]'t + Bsin[tex]\omega[/tex]'t)

= Re e

^{-[tex]\beta[/tex]t/2}[tex]\hat{x}[/tex]

_{o,h}e

^{i[tex]\omega[/tex]'t}

where [tex]\hat{x}[/tex]

_{o,h}= |[tex]\hat{x}[/tex]

_{o,h}| e

^{i[tex]\phi[/tex]}.

If the oscillator is at the position x

_{o}with velocity [tex]\vartheta[/tex]

_{o}at time t = 0, show that

A = x

_{o}

B = \frac{\frac{x

_{o}\beta}{2} + [tex]\vartheta[/tex]

_{o}}{[tex]\omega[/tex]'}

|[tex]\hat{x}[/tex]

_{o,h}| = [tex]\sqrt{A

^{2}+ B

^{2}}[/tex]

tan [tex]\phi[/tex] = -[tex]\frac{B}{A}[/tex]

2. Homework Equations

I know that

e

^{i[tex]\phi[/tex]}= cos [tex]\phi[/tex] + isin[tex]\phi[/tex]

[tex]\omega[/tex]'

^{2}= [tex]\omega[/tex]

_{o}

^{2}- [tex]\beta[/tex]

^{2}/4

3. The Attempt at a Solution

I tried to show the third equation. So I put A

^{2}and B

^{2}to the square root. And I changed [tex]\omega[/tex]'

^{2}to [tex]\omega[/tex]

_{o}

^{2}- [tex]\beta[/tex]

^{2}/4. But I couldn't reach the solution. Also I couldn't find A or B.

This is the first time I am using Latex. I hope I did no mistake.

Thanks for helping...