In a
homogeneous,
linear differential equation with
constant coefficients, this is the equation you arrive at when you let
y=emx. It is also known as the
characteristic equation. For example...
y'' + 2y' - 3y = 0
y=emx
y'=memx
y''=m2emx
m2emx + 2memx - 3emx = 0
Divide by emx (since it cannot equal 0)
m2 + 2m - 3 = 0 <-----auxiliary equation
(m+3)(m-1)=0
m=-3,1
general equation : y=c1e-3x+c2ex
The problems don't always end up with such a simple general equation for y, but the auxiliary equation is obtained just the same way. It's then factored to solve for m, which in turn solves the differential equation.