In a

homogeneous,

linear differential equation with

constant coefficients, this is the equation you arrive at when you let

y=e^{mx}. It is also known as the

characteristic equation. For example...

y'' + 2y' - 3y = 0
y=e^{mx}
y'=me^{mx}
y''=m^{2}e^{mx}
m^{2}e^{mx} + 2me^{mx} - 3e^{mx} = 0
Divide by e^{mx} (since it cannot equal 0)
m^{2} + 2m - 3 = 0 <-----auxiliary equation
(m+3)(m-1)=0
m=-3,1
general equation : y=c_{1}e^{-3x}+c_{2}e^{x}

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.