Damped oscillator question

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-$$\beta$$t/2(Acos$$\omega$$'t + Bsin$$\omega$$'t)

= Re e-$$\beta$$t/2$$\hat{x}$$o,hei$$\omega$$'t

where $$\hat{x}$$o,h = |$$\hat{x}$$o,h| ei$$\phi$$.

If the oscillator is at the position xo with velocity $$\vartheta$$o at time t = 0, show that

A = xo

B = \frac{\frac{xo\beta}{2} + $$\vartheta$$o}{$$\omega$$'}

|$$\hat{x}$$o,h| = $$\sqrt{A2 + B2}$$

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

2. Homework Equations

I know that
ei$$\phi$$ = cos $$\phi$$ + isin$$\phi$$
$$\omega$$'2 = $$\omega$$o2 - $$\beta$$2/4

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 - $$\beta$$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...

I couldn't write B exactly. Rigth B is that :

B = [(beta times Xo / 2) + Vo] / w'

I hope it is clear

Is there anyone who can help me?

vela
Staff Emeritus
$$A\cos \omega t + B \sin\omega t = R\cos(\omega t-\phi)$$