# Simple Harmonic Oscillator

1. Jan 24, 2012

### yakkayakka

1. The problem statement, all variables and given/known data

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$_{o}$
2. Relevant equations

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

3. 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 im 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.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 24, 2012

### Spinnor

You wrote,

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

I think you are missing some ")" somewhere?