solution of homogeneous difference equation(HDE): method of characteristic roots

SOLUTION OF HOMOGENEOUS DIFFERENTIATION EQUATION (HDE)

Method of characteristic roots:

Step 1:

           At first set the HDE to its general form as given below:

                C₀ a_n + C₁ a_n-1 + C₂ a_n-2 + C₃ a_n-3 + … + C_k a_n-k

Example:

Step 2:

Let,       a_n = r ^order be the trial solution .

           a_n-1 = r ^order-1

           a_n-2 = r ^order-2

Rules for finding order:

order of an equation = maximum subscript - minimum subscript

Step 3:

compare the equation with the terms of a_{n ,}a_n-1,a_{n-2 ... upto} a_n-k

Step 3:

So, the general solution is ....

          the term of a_n.