Total Solution

The total solution or the general solution of a non-homogeneous linear difference equation with constant coefficients is the sum of the homogeneous solution and a particular solution. If no initial conditions are given, obtain n linear equations in n unknowns and solve them, if possible to get total solutions.

If y_(h) denotes the homogeneous solution of the recurrence relation and y_(p) indicates the particular solution of the recurrence relation then, the total solution or the general solution y of the recurrence relation is given by
                  y =y_(h)+y_(p).

Example: Solve the difference equation
                 a_r-4a_r-1+4a_r-2=3r+2^r...........equation (i)

Solution: The homogeneous solution of this equation is obtained by putting R.H.S equal to zero i.e.,
                 a_r-4a_r-1+4a_r-2=0

The homogeneous solution is a_r(h)= (C₁+C₂ r).2^r

The equation (i) can be written as (E²-4E+4) a_r=3r+2^r

The particular solution is given as