# Damped driven harmonic oscillator in a steady state

In the limit as t goes to infinity the solution approaches

x(t)=Ksin[Ï‰(t-t0)]

where K and t0 depend on Ï‰ (note: t0 = t subscript 0)

I am attempting to show that K(Ï‰) = 1/(Ï‰^4-Ï‰^2+1)^1/2 and failing

My thoughts:

I believe that at t=0. The only important part of the general solution to the original equation is the exponential part (because e^0=1while as t increases (and eventually to the limit) only the particular solution is of importance

hence i have tried equating Ksin[Ï‰(t-t0)] with the particular solution and failing.

https://brainmass.com/math/basic-algebra/damped-driven-harmonic-oscillator-steady-state-563122

## SOLUTION This solution is **FREE** courtesy of BrainMass!

The solution is attached below in two files. The files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore, you can choose the format that is most suitable to you.

The derivation required starts on page 6 (I deleted the variation of parameters part).

Anyway - if you want to try it yourself -

You want to look at the solution when t goes to infinity, so the exponent disappears and you are left with the particular solution.

Now, recall that Asin(a+b) = Asin(a)cos(b) + Asin(b)cos(a)

If in your case "a" is wt and "b" is a constant, you can equate the coefficients of this expansion with the particular solution.

Solve for A and B and you will get the required results.

https://brainmass.com/math/basic-algebra/damped-driven-harmonic-oscillator-steady-state-563122