How Do You Solve a Damped Oscillator Problem with Initial Conditions?

[burock]

Hi,
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^-\betat/2(Acos\omega't + Bsin\omega't)

= Re e^-\betat/2\hat{x}_o,he^i\omega't

where \hat{x}_o,h = |\hat{x}_o,h| e^i\phi.

If the oscillator is at the position x_o with velocity \vartheta_o at time t = 0, show that


A = x_o

B = \frac{\frac{x_o\beta}{2} + \vartheta_o}{\omega'}

|\hat{x}_o,h| = \sqrt{A² + B²}

tan \phi = -\frac{B}{A}

2. Homework Equations

I know that
e^i\phi = cos \phi + isin\phi
\omega'² = \omega_o² - \beta²/4

3. The Attempt at a Solution

I tried to show the third equation. So I put A² and B² to the square root. And I changed \omega'² to \omega_o² - \beta²/4. 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...

 

[burock]

I couldn't write B exactly. Rigth B is that :B = [(beta times Xo / 2) + Vo] / w'

I hope it is clear

 

[burock]

Is there anyone who can help me?

 

[vela]

Start by finding R and ϕ such that

A\cos \omega t + B \sin\omega t = R\cos(\omega t-\phi)