Damped Harmonic Oscillator Using Greens Theorem

[dacruick]

Hi there,

I just started an intermediate classical mechanics course at university and was smacked upside the head with this question that I don't know how to even start.

Homework Statement

We are to find the response function of a damped harmonic oscillator given a Forcing function. The forcing function for t>0 is
F(t) = F(0)exp(-λt)*sin(wt)

where λ is a constant, F(0) is the initial force, w is the angular frequency, and t is time.

Homework Equations

The equation of motion of a damped harmonic oscillator which I believe to be
x'' + 2bx' + w_o²x = F(t)

The Attempt at a Solution

I haven't really even attempted this solution. I've been looking online for a direction, and I saw one example that told me to guess a particular solution based on the forcing functions proportionality to e^iwt. It doesn't tell me why I should be doing that, and to be honest, I don't even really understand what Greens theorem is.

Any help would be very greatly appreciated,

Thanks!

 

[vela]

Have you had a course in differential equations? You have a linear equation with constant coefficients. It's straightforward to solve using the method of undetermined coefficients. So you should be able to find the solution even if it's not using the method you're supposed to.

Am I right in assuming when you refer to Green's theorem, you mean that you're supposed to find the solution to the differential equation by finding the Green's function for the linear operator?