Underdamped Oscillator Solution: Deriving x(0) and v(0)

[yakkayakka]

Homework Statement

Show that the underdamped oscillator solution can be expressed as x(t)=x_{0}e^{-γt}[cos(Ω't+((v_{o}+γx_{o})/(x_{o}Ω')sinΩ't] and demonstrate by direct calculation that x(0)=x_{o} and \dot{x}(0)=v_{o}

Homework Equations

The underdamped oscillator solution is
x(t)=ae^{-γt}cos(Ω't+\alpha)

The Attempt at a Solution

This problem completely overwhelms me so my solution may be a little lacking...
I took the general form
Acos(ω_{o}t)+Bsin(ω_{o}t)
Where
A=acos(\alpha) and B=-asin(\alpha)
Which according to what I read in the book should yield
x(t)=a[cos(ω_{o}t+\alpha)]
So I am thinking that the equation ae^{-γt}cos(Ω't+\alpha) can be transformed into a more useful form using the same method

and that is sadly as close as I could get

Any input would be appreciated. Thanks.

 

[Spinnor]

You wrote,

x(t)=x0e−γt[cos(Ω't+((vo+γxo)/(xoΩ')sinΩ't]

I think you are missing some ")" somewhere?