^{mx}. It is also known as the characteristic equation. For example...

y'' + 2y' - 3y = 0 y=eThe 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.^{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}