EQUATION with constant coefficients, As a solution, we try un = A mn, where A and m are constants.This choice has been made because un = kn uo was the solution to the equation un = k un-1.After substituting and solving the auxiliary equation, SAY m1, M2, m3 ... mn be the roots, then the general solution will be un = A(m1)n + B(m2)n + ... where A, B ... are constantsCalculation:Given Differential equation is:un+3 - 4n+2 + un+1 + 6UN = 0Substituting:un+3 = A mn+3; un+2 = A mn+2; un+1 = A mn+1; un = A mn; A mn+3 - 4 A mn+2 + A mn+1 + 6 A mn = 0A mn (m3 - 4m2 + m + 6) = 0m3 - 4m2 + m + 6 = 0This is called auxiliary equation.The roots of the auxiliary equation are -1, 2, and 3, i.e.m1 = -1, m2 = 2, m3 = 3;The general solution will be:un = A(-1)n + B(2)n + C(3)n