To derive the S-Domain equation for a lowpass filter we are going to derive the equation below. The image of the lowpass filter is below.
Vo - Voltage output
Vi - Voltage input
PI - pie = 3.14156927
F - Frequency
OMEGA = OM - 2*PI*F
SQRT - Square Root
^ - exponent ex. 2^2 = 4
j = imaginary part of a number. ex. 2 + j3
L - Inductance
R - Resistance
C - Capacitance

Xc = 1/(j*OM*C) - Capacitive Reactants
Xl = j*OM*L - Inductive Reactants
OMc = 1 / SQRT(L*C) - OMEGA cutoff
d = SQRT(L / (R^2*C)) - damping factor

Vo/Vi = Xl || R / (Xc + Xl || R)

= (j*OM*L*(R) / (j*OM*L + R)) / ((1 / j*OM*C) + (j*OM*L*(R) / (j*OM*L + R)))

(1 / j*OM*C) + (j*OM*L*R / (j*OM*L + R))
= ((j*OM*L+R)+(j*OM*L*R)(j*OM*C)) / J*OM*C(j*OM*L + R)
= ((j*OM*L*R) / (j*OM*L+R)) * (J*OM*C(j*OM*L + R) / ((j*OM*L+R)+(j*OM*L*R)(j*OM*C)))
(j*OM*L+R) is cancelled on both sides, and j^%2 = -1
so the equation becomes:

= (-OM^2*L*R*C) / (j*OM*L + R - OM^2*L*R*C)
d*OMc = SQRT(L / (R^2*C)) * SQRT(1 / (L*C)) = SQRT(1 / (R^2*C^2)) = 1/(R*C)
we then take L*R*C out and we use d*OMc we get.

= (-OM^2) / (-OM^2 + (j*OM / R*C) + (1/(L*C)))
= (-OM^2) / (-OM^2 + (j*d*OMc*OM) + OMc^2)