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


m2emx + 2memx - 3emx = 0

Divide by emx (since it cannot equal 0)

m2 + 2m - 3 = 0  <-----auxiliary equation



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.