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10
2.2.Secondequation
ElsayedM.Elsayed
Inthissectionwegiveaspecificformofthesolutionsofthedifference
equation
(4)
xn+1=
xn11(xn−1)
xn
j
n=0j1j...
wheretheinitialconditionsx11,xoarearbitraryrealnumberswithx11jxo/
∈
{0j1},x11+xo/=xox11.
Theorem4.Let{xn}∞
nl11beasolutionofEq(4).Thenequation(4)
haveallsolutionsandthesolutionsare
x5n11=kj
x5n=hj
x5n+2=
(k+h−hk)
1
j
wherex11=k,x1o=h.
x5n+1=
k(h−1)
h
j
x5n+3=
h(k−1)
k
j
Proof.Forn=0theresultholds.Nowsupposethatn>0andthat
ourassumptionholdsforn−1.Weshallshowthattheresultholdsforn.
Fromourassumptionforn−1,wehavethefollowing:
x5n16=kj
x5n15=hj
x5n13=
(k+h−hk)
1
j
Now,itfollowsfromEq(4)that
x5n14=
k(h−1)
h
j
x5n12=
h(k−1)
k
j
x5n11=
x5n13(x5n12−1)
x5n12
=
(k−h(k−1))
k(k+h−hk)
=k.
x5n=
x5n12(x5n11−1)
x5n11
=
hk(k−1)
k(k−1)
=h.
x5n+1=
x5n11(x5n−1)
x5n
=
k(h−1)
h
.
x5n+2=
x5n(x5n+1−1)
x5n+1
=
(h+k−hk)
1
.
x5n+3=
x5n+1(x5n+2−1)
x5n+2
=
h(k−1)
k
.
Thus,theproofiscompleted.
.
