A Linear differential equation with constant coefficients (ie. one in which all the terms involving y or its derivatives are multiplied only by constants) normally has solutions of the form y=eαx.
A second order homogeneous linear differential equation with constant coefficients has general solution y = Aeαx + Beβx where α and β are different constants, possibly complex.
If we write the differential equation as
y" + A y' + B y = 0 where the 's denote differentiation
then α and β are the two roots of the equation α2 + Aα + B = 0.
If α = β then the solution takes the alternate form y = (A+Bt)eαx.
Euler's equation gives a meaning to a complex power by defining
eiθ = cos(θ) + i sin(θ) where i2 = -1.
Corollary's are
cos(θ) = eiθ + e-iθ and sin(θ) = eiθ - e-iθ
2 2i
These relations let us switch freely between trig forms and complex exponential forms.
Realistic oscillating systems suffer from frictional loss mechanisms that, more or less gradually, remove energy from the system. Most mechanisms produce forces that can be approximated by the form F=-2βv where C is some positive constant and v the instantaneous velocity. In that case the differential equation becomes y" + 2 βy' + ω02 y = 0.
If β < ω0 the system is called under-damped and the solution consists of decaying oscillations with frequency very nearly ω0 and decay constant β.
If β = ω0 the system is called critically-damped and the solution crosses zero only once before coming to rest. This solution reaches rest in the shortest time possible.
If β > ω0 the system is called over-damped and the solution is a decaying exponential that takes longer to reach equilibrium than the critically damped solution.
If we write the equation of motion in the form y" + 2 βy' + ω02 y = 0 then the solutions are:-
If β < ω0 the system is called under-damped and the solution can be written
y = A e-βt cos(ωt + φ) where ω = √(ω02-β2).
If β = ω0 the system is called critically-damped and the solution can be written
y = (A + B t)e-βt.
If β > ω0 the system is called over-damped and the solution can be written
y = A e-(β+γ)t + B e-(β-γ)twhere γ = √(β2-ω02).