Linear Difference Equations

Linear Difference Equations

A linear difference equation with constant coefficients is of the form

a0 yn+r + a1 yn+r -1 + a2 yn+r -2 + . . . . +aryn = f(n).

i.e., (a0Er + a1Er-1 + a2 Er-2 + . . . . + ar)yn = f(n)   ------(1)

where a0,a1, a2, . . . . . ar  are constants and f(n) are known functions of n.

The equation (1) can be expressed in symbolic form as

f(E) yn = f(n) ----------(2)

If  f(n) is zero, then equation (2) reduces to

f (E) yn = 0 ----------(3)

which is known as the homogeneous difference equation corresponding to (2).The solution of (2) consists of two parts, namely, the complementary function and the particular integral.

The solution of equation (3) which involves as many arbitrary constants as the order of the equation is called the complementary function. The particular integral is a particular solution of equation(1) and it is a function of „n‟ without any arbitrary   constants.

Thus the complete solution of (1) is given by  yn = C.F + P.I.

Example 1

Form the difference equation for the Fibonacci sequence .

The integers 0,1,1,2,3,5,8,13,21, . . . are said to form a Fibonacci sequence.

If yn be the nth term of this sequence, then

yn = yn-1 + yn-2 for n > 2

or  yn+2 - yn+1 - yn = 0 for n > 0