Difference Equation

What is Difference Equation?

Difference Equation is an equation that shows the functional relationship between an independent variable and consecutive values or consecutive differences of the dependent variable.

They are mostly reorganized as a recursive formula, so that, a system’s output can be calculated from the input signal and precedent outputs.

The order of difference equation can be stated in terms of consecutive values of y – is the distinction between the peak and lowly subscripts of y, whereas, the degree of a difference equation, free from ∆’s, is the uppermost exponent of the y’s.

Few examples of difference equations are given below.

$\displaystyle {{y}_{{x+2}}}+4{{y}_{{x+1}}}-6{{y}_{x}}=2{{x}^{2}}$
$\displaystyle {{\Delta }^{2}}{{y}_{x}}+5\Delta {{y}_{x}}+3{{y}_{x}}=0$
$\displaystyle {{\Delta }^{3}}{{y}_{x}}+5{{\Delta }^{2}}{{y}_{x}}-3\Delta {{y}_{x}}+5{{y}_{x}}={{x}^{2}}-x+1$

How to solve a Difference Equation?

Generally, a solution of a difference equation of order n is a solution that includes n random constants or n random function which are cyclic with a period equal to the intermission of differencing.

Any function that satisfies the difference equation is also the solution to it. Furthermore, the solution achieved by assigning particular values to the random constant or functions is a particular of a difference equation.

Applications of difference equation

Difference equations can be applied in computer science, numerical solutions of differential equations, Fourier series, signal processing, algebra, time series analysis and queuing theory for the design of the operating system.
Proficiently used for modeling population dynamics or the spread of infectious diseases.
Applications in seismic mitigation, cryptography, coding theory and drill systems.

Linear Difference Equation

When the equation is of the form:

$\displaystyle {{y}_{x}}\,,\,{{y}_{{x+1}}}\,,\,{{y}_{{x+2}}}..................$

is called a linear Difference Equation. The general form of a linear difference equation is,

$\displaystyle {{a}_{0}}{{y}_{{x+n}}}+{{a}_{1}}{{y}_{{x+n-1}}}+................\,{{a}_{{n-1}}}{{y}_{{x+1}}}+{{a}_{n}}{{y}_{x}}=R(x)$

Where, a_i and R(x) are well-known function of x. The above equation is said to be Homogeneous Difference Equation when R(x) is zero, whereas, the aforementioned equation is called a Non-Homogeneous Difference Equation when R(x) is not equal to zero.