Transfer function's poles and Auxiliary Equation

[unseensoul]

Why does the homogeneous of a second order differential equation/system (i.e. a series RLC circuit) is identical to the transfer function (i.e. H(jw)) denominator in its standard form? Therefore the poles of the transfer function are also the solutions for the auxiliary equation...

I cannot see any link between these two things, but they seem to be interrelated somehow. Is there any proof for this?

 

[John Creighto]

The Laplace transform transforms a differential equation into an algebraic equation. We can work it out in more detail if you wish.