# Green function question

I was reading up on Classical Mechanics and the general method for solving for an undamped harmonic oscillator was given as

$$\frac{d^{2}q}{dt^{2}} + \omega^{2}q = F(t)$$

was solved using the Green function, G, to the equation

$$\frac{d^{2}G}{dt^{2}} + \omega^{2}q = \delta(t-t')$$

and then integrating via the normal procedure.

Then the next case considered was the damped harmonic oscillator which had a damping term proportional to $$\frac{\omega}{Q}$$ times the velocity. The equation has the form

$$\frac{d^{2}q}{dt^{2}} + \frac{\omega}{Q}\frac{dq}{dt} + \omega^{2}q = F(t)$$

Now, the author wants to find the particular solution to this equation and says that his Green function is of the form

G = $$A e^{\frac{{\omega(t-t')}}{2Q} + i\omega(t-t')}$$ plus this term's complex conjugate. This is a solution to STEADY STATE ONLY but does not consider the transient part.

BUT, he also claims that the above Green function is a solution the equation

$$\frac{d^{2}G}{dt^{2}} + \frac{\omega}{Q}\frac{dG}{dt} + \omega^{2}G = \delta(t-t')$$

How can he do this? The Green function which will solve the above equation will also have a transient part apart from the steady state function which he has considered. G will be of the form above PLUS a dying out part. What happens to that?

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