Differential equation that governs the motion of the particle?

[Nusc]

A particle of mass m undergoes one-dimensional damped harmonic oscillations with a damping constant gamma and a natural frequency omega nought. In addition the particle is subject to a time dependent external force given by:

Fext = f1t + f2t^2
a) What is the differential equation that governs the motion of the particle?

I found Xp(t) but I don't know what the homogeneous solution is because it doesn't specify if it's underdamped, overdamped, or critcally damped.

How do I know?

b) Determine what the "steady-state" solution will be at late times after all the transient motions have damped out.

So the particular solution will disappear because at t approaches infinity those terms with t will vanish. But I can't complete the question if I don't know the homogenous solution.

 

[Dr.Brain]

A simple harmonic motion has differential equation:

$\frac {d^2x}{dt^2} + kx/m = 0$

Now besides a 'kx; restoring forces , there are more forces which take part in the harmoic motion. Just make those forces part of this differential equation and solve for steady state.

BJ

(Try to convert damping force to complex form..)

 

[Nusc]

$\frac {d^2x}{dt^2} + 2gamma \frac {dx}{dt} + omeganought^2{x}= \frac {f_1t}{m} + \frac {f_2t^2}{m}$

That is the differential equation, i don't understand what you're trying to say.

How do you know that gamma is zero?
Why do you assue a homogeneous equation? It's non-homo.

 

[Dr.Brain]

I am telling u about the part (a) which asks for a Simple Differential equation which governs the motion of the Harmonic motion.

$\frac {d^2x}{dt^2} + kx/m = 0$

The above diff. equation is for SHM which is undamned , in the case of your question certain other forces add to the above diff. eqn , thus giving you the answer for part (a) . Anyways other forces that add to the above diff. eqn are:

f1t + f2t^2 ( acts as the driving force)

and a damping force for which damping constant is $\gamma$
, What will be the damping force in terms of the constant $\gamma$ ?

 

[Nusc]

I still don't understand where your getting to.

All I asked is for Xh(t) from the general soln

Xgen(t) = Xh(t) + Xp(t)

I have already said that I found Xp(t), I want Xh(t). But I can't obtain it since I don't know whether it is over, under or crtically damped. It's damped, so I'm not going to assume an undamped solution which is what you wrote.