(Hard) work done by damped, driven oscillator as function of time

[NucEngMajor]

Homework Statement

Force F = const is applied to H.O. initially at rest with mass m, freq w0, damping T. Find x(t). Find work as function of time.

Homework Equations

mx'' + Tx' + kx = F for F= Constant

The Attempt at a Solution

First obtain complimentary solution for free H.O. which I get after some work is x(t) = x0e^-(Tt)coswt + ((v0 + T *x0) / w )*e^(-Tt)sinwt. This agrees with textbook, but NOTE: w here is not equal to w0 for initial frequency and v0 can be taken to be zero. Now...if I try to apply variation of constant and use Wronskian I get a mess for the integrals. So where do I go from here to get my particular solution and then if I obtain it how to I obtain work as function of time?

Thanks[/B]

 

[NucEngMajor]

For work could I just plug x(t) = x(particular) + x(complimentary) into my initial ode and integrate w.r.t. x?

 

[NucEngMajor]

However, x free should 0 as the oscillator is at rest so I just need the forced solution for F=const.