How Does the Driven Oscillator ODE Describe Long-Term Motion?

[Eric_meyers]

Homework Statement

"The equation mx'' + kx = F0 * Sin (wt) governs the motion of an undamped harmonic oscillator driven by a sinusoidal force of angular frequency w. Show that the steady-state solution is

x = F0 * Sin (wt) /(m * (w0^2 - w^2))

Homework Equations

x(t) = xta(t) + xtr(t) where xta = long term behavior and xtr = transient piece of solution

xta(t) x0 cos (wt - (phi)

where x0 = w0^2 X0/[(w0^2 - w^2)^2 + v^2*w^2]^1/2

and phi = tan^-1(vw/(w0^2 - w^2)

The Attempt at a Solution

Honestly I'm not sure if the above equations are necessary. All my lecture notes tell me is that the long term behavior follows x0 * cos(wt - (phi)) so I'm not sure how I'm suppose to take that and turn it into F0 * Sin (wt) /(m * (w0^2 - w^2))

I mean obviously I have a sin force driving my harmonic oscillator, but how do I use that to determine my long term solution?

I really can't go much further than this at this point, I've been looking at this problem for a long time now and I can't scrap together anything useful.

 

[Redbelly98]

Homework Statement

"The equation mx'' + kx = F0 * Sin (wt) governs the motion of an undamped harmonic oscillator driven by a sinusoidal force of angular frequency w. Show that the steady-state solution is

x = F0 * Sin (wt) /(m * (w0^2 - w^2))

How about substituting this expression for x, and also the expression for x'', into the original differential equation?