082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

22.2 Basic definitions 683
22.2.2 Thesolutionofadifferenceequation
A solution is obtained when the dependent variable is known for each value of interest
of the independent variable. Thus the solution takes the form of a sequence. There are
frequently many different sequences which satisfy a difference equation; that is, there
aremanysolutions.Thegeneralsolutionembracesalloftheseandallpossiblesolutions
can be obtained from it.
Example22.1 Showx[n] =A2 n , whereAisaconstant, isasolution of
x[n +1] −2x[n] = 0
Solution x[n] =A2 n x[n +1] =A2 n+1 = 2A2 n
Hence,
x[n +1] −2x[n] = 2A2 n −2A2 n = 0
Hencex[n] =A2 n isasolutionofthegivendifferenceequation.Infactx[n]isthegeneral
solution.
If additionally we are given a condition, sayx[0] = 3, the constantAcan be found. If
x[n] =A2 n , thenx[0] =A2 0 =Aand hence
A = 3
The solution is thus x[n] = 3(2 n ). This is the specific solution and satisfies both the
difference equation and the given condition.
22.2.3 Linearandnon-linearequations
Anequationislinearifthedependentvariableoccursonlytothefirstpower.Ifanequation
isnot linear itisnon-linear. For example,
3x[n +1] −x[n] = 10
y[n+1]−2y[n−1]=n 2
kz[k +2] +z[k] =z[k −1]
arealllinearequations.Notethatthepresenceofthetermn 2 doesnotmaketheequation
non-linear, sincenisthe independent variable. However,
(x[n +1]) 2 −x[n] = 10
y[k+1]= √ y[k] +1
areboth non-linear. Also,
z[n +1]z[n] =n 2 +100
sinx[n] =x[n −1]
arenon-linear.Theproducttermz[n +1]z[n]andthetermsinx[n]arethecausesofthe
non-linearity.