If M and N are the roots applicable to the homogeneous part of the second order non-homogeneous DE y"+ay'+by=p(x), giving the homogeneous component y1 (characteristic equation) of the solution the form Ae^Mx+Be^Nx, where A and B are constants and where y"+ay'+by=0 (characteristic or auxiliary equation (r-M)(r-N)=r^2+ar+b=0, a=-(M+N) and MN=b), then there will be a solution if a particular solution y2 can be found for p(x). The solution is y1+y2, the sum of the two components.