2.8. Сase 8: pn — Хл, дл 3D Ул, Гn — Уп, Sn — Xn In this case, system (1.8) is written as in the form Xn+1 = f' (af (xp-1)+bf (yn-2)), Yn+1 =f'(af (Yn-1)+bf (xp-2)), (2.61) for n E No. Since f is "1– 1", from (2.61) f (Xn+1) = af (xn-1)+bf (yn-2), f(yn+1) = af (yn-1)+bf (xn-2), (2.62) for n E No. By using the change of variables f (xn) = Un, and f (yn) = Vn, n>-2, (2.63) system (2.62) is transformed to the following one Un+1 = aun-1+bvn-2, Vn+1= avn-1+ bun-2, n E No. (2.64) By summing the equations in (2.64) we get Un+1 +Vn+l =a (u,-1+Vn-1) +b (un-2+Vn-2), n E No, (2.65) whereas by subtracting the second one from the first, we have Un+1 - Vn+1 = a (un-1 – Vn-1) – b (un-2 - Vn-2), ne No. (2.66) From (2.5), we can write the solution of equation (2.65) as Un + Vn = (uo+vo) Jn+1+(u-1+v-1) Jn+2+b(u_2+v-2) Jn, (2.67) for n > -2. On the other hand, by taking a = 0, b = a, c = -b in (1.4) and Sn = (-1)" Jn+1, for all n > -2, which is called generalized Padovan sequence, in (1.5), from (2.66), we also have that Un- Va = (-1)" ( (uo – vo) Jn+1 – (u-1 –v-1)Jn+2+b (u-2-v-2)Jn), (2.68) for n > -2. From (2.68) we have (uo – vo) J2n+1 – (u_1 – V–1) J2n+2+b (u_2 – v-2) J2n, (2.69) U2n - V2n= for n>-1 and U2n+1 - V2n+l -(4o – vo)J2n+2+ (u-1 - v-1) J2n+3 –b(u_2 - v-2) J2n+1, (2.70) for n >-1. From (2.67) Uzn+ v2n = (4o+ vo) J2n+1+(u_1 +v=1) J2n+2+b (u_2+v=2) J2n, (2.71) for n> -1. By summing the equations (2.69) and (2.71) u2n = uoJ2n+1 +v-1J21+2+ bu-2J2n, n2 -1. (2.72) By subtracting equation (2.69) from equation (2.71), we have V2n = voJ2n+1+u-1J2n+2+bv-2J2n, n >-1. (2.73) From (2.67) Uzn+1 +V2n+1 = (uo+vo)J2n+2+(u_1 +v-1)J2n+3+b(u_2+v_2)J2n+1; (2.74) for n2-1. By summing the equations (2.70) and (2.74) we get U2n+1 = VOJ21+2+u-1J21+3 +bv_2J21+1, n> -1. (2.75) By subtracting equation (2.70) from equation (2.74), we have v2n+1 = uoJ2n+2+v-1J20+3 + bu-2J21+1, n> –1. (2.76) From (2.63), (2.72), (2.73), (2.75), (2.76) and after some calculation, we obtain X2, = f(f (x0) J2n+1+f (y-1) J2n+2+bf (x-2)J2n), n2 -1, y2n = f(f (vo) J2n+1+f (x-1) J2n+2+bf (y-2) J2n), n2 -1, =f'(f(vo)J2n+2+f (x=1)J2n+3+bf (y-2)J2n+1), n> –1, (2.77) (2.78) (2.79) and Y2n+1 = f(1())2n+2+f (y-1) J2n+3+bf (x-2)J2n+1),n > -1. (2.80) constant coefficients Xn+1 = axn+bxn-1+cxn-2, nE No, (1.4) which has actually the general solution X = xoSn+x-1 (Sn+1 - aSn) +cx_2Sn-1, nE No, (1.5) where (S.) of equation (1L4) satisfving the initial values S 2=S-=0, So= L. The equation axn-IXn-k Xn+1 = nE No, (1.1) bxn-p±cxn-q' where the initial conditions are arbitrary positive real numbers, k, 1, p, q are non- negative integers and a, b, c are positive constants, is one of the difference equations whose solutions are associated with number sequences. Positive solutions of concrete Motivated by this line of investigations, here we show that the systems of differ- ence equations Xn+l =f'(af (Pn-1)+bf(qn-2)), Yntl =f"(af (ra-1)+bf (Sn-2)), (1.8) for n E No, where the sequences Pn, qn, Tn and Sn are some of the sequences x, and Yn, f: Df R is a "1– 1" continuous function on its domain Dr C R, the initial values x-j, y-j, je {0,1,2} are arbitrary real numbers and the parameters and a, b 2.6. Case 6: Pn = Yn, qn = yn, rn = Xn, Sn = Xn In this case, we obtain the system Xn+1 =5"(af (Yn-1)+bf (Yn-2)), Yn+1 = '(af (xn-1)+bf (xn-2)), (2.29) for n e No. Since f is "1– 1", from (2.29) f (Xn+1) = af (Yn-1)+bf (yn-2), f(yn+l) = af (Xn-1) +bf (xn-2), (2.30) for n E No. By using the change of variables f (xn) = = Un, and f(yn) = Vn, n>-2, (2.31) system (2.30) is transformed to the following one Un+l = avn-1+bvn-2, Vn+1 = aun-1+ bun-2, nE No. (2.32) By summing the equations in (2.32) we get Un+l +Vn+l = a (un-1+ Vn-1) +b(un-2+Vn-2), nE No, (2.33) whereas by subtracting the second one from the first, we have Un+l - Vn+l = -a(un-1 - Vn-1) -6(un-2- Vn-2), ne No. (2.34) | In this section, we consider the eight special cases of systems (1.8), where the sequences Pn, qn, ľn, Sn are some of the sequences xn and yn, for n>-2, and initial values x-j, y-j, j e {0,1,2}, are arbitrary real numbers. 2.1 Case 1: Pn =xn, qn = Xn, ľn = yn, Sn == yn In this case, system (1.8) is expressed as Xn+1 =f (af (xn-1)+bf (xn-2)), Yn+1 =f(af (Vn-1)+bf (Yn-2)), (2.1) for n E No. Since f is "1– 1", from (2.1) f (Xn+1) = af (x,-1)+bf (xn-2), f (Yn+1) = af (yn-1)+bf (yn–2), (2.2) for n E No. By using the change of variables f (xn) = Un, and f(yn) = = Vn, n> -2, (2.3) system (2.2) is transformed to the following one Un+1 = au,-1+bun-2, Vn+1 = avn-1+bvn-2, (2.4) for n E No. By taking a = 0, b= a, c = b in (1.4) and S = Jn+1, for all n > -2, which is called generalized Padovan sequence, in (1.5), the solutions to equations in (2.4) are given by Un = ugJn+1+u-1Jn+2+ bu_2Jn, (2.5) Vn = voJn+1+v_1Jn+2+bv_2Jn, (2.6) for n E No. From (2.3), (2.5) and (2.6), it follows that the general solution to system (2.2) is given by Xn = f(f (x0) Jn+1+f (x-1) Jn+2+bf (x-2)Jn), n> -2, Yn =f(f (vo) Jn+1+f(y-1)Ja+2+bf (y-2) J.), n> -2. (2.7) (2.8) 2.2. Case 2: Pn = Xn, qn = Xn, ľn = Xn, Sn Xn In this case, system (1.8) becomes Xp+1 =f"(af (x,-1)+bf (xp-2)), Yn+1=f"(af (xn-1)+bf (xn-2)), (2.9) for n e No. It should be first note that from the equations in (2.9) it immediately follows that x, = yn, for all n E N. From (2.7), the general solution to system (2.9) is %3D X, = Ya = f'(f (xo) Jn+1+f (x=1) Jn+2+bf (x_2)Jn), n E N. (2.10)
2.8. Сase 8: pn — Хл, дл 3D Ул, Гn — Уп, Sn — Xn In this case, system (1.8) is written as in the form Xn+1 = f' (af (xp-1)+bf (yn-2)), Yn+1 =f'(af (Yn-1)+bf (xp-2)), (2.61) for n E No. Since f is "1– 1", from (2.61) f (Xn+1) = af (xn-1)+bf (yn-2), f(yn+1) = af (yn-1)+bf (xn-2), (2.62) for n E No. By using the change of variables f (xn) = Un, and f (yn) = Vn, n>-2, (2.63) system (2.62) is transformed to the following one Un+1 = aun-1+bvn-2, Vn+1= avn-1+ bun-2, n E No. (2.64) By summing the equations in (2.64) we get Un+1 +Vn+l =a (u,-1+Vn-1) +b (un-2+Vn-2), n E No, (2.65) whereas by subtracting the second one from the first, we have Un+1 - Vn+1 = a (un-1 – Vn-1) – b (un-2 - Vn-2), ne No. (2.66) From (2.5), we can write the solution of equation (2.65) as Un + Vn = (uo+vo) Jn+1+(u-1+v-1) Jn+2+b(u_2+v-2) Jn, (2.67) for n > -2. On the other hand, by taking a = 0, b = a, c = -b in (1.4) and Sn = (-1)" Jn+1, for all n > -2, which is called generalized Padovan sequence, in (1.5), from (2.66), we also have that Un- Va = (-1)" ( (uo – vo) Jn+1 – (u-1 –v-1)Jn+2+b (u-2-v-2)Jn), (2.68) for n > -2. From (2.68) we have (uo – vo) J2n+1 – (u_1 – V–1) J2n+2+b (u_2 – v-2) J2n, (2.69) U2n - V2n= for n>-1 and U2n+1 - V2n+l -(4o – vo)J2n+2+ (u-1 - v-1) J2n+3 –b(u_2 - v-2) J2n+1, (2.70) for n >-1. From (2.67) Uzn+ v2n = (4o+ vo) J2n+1+(u_1 +v=1) J2n+2+b (u_2+v=2) J2n, (2.71) for n> -1. By summing the equations (2.69) and (2.71) u2n = uoJ2n+1 +v-1J21+2+ bu-2J2n, n2 -1. (2.72) By subtracting equation (2.69) from equation (2.71), we have V2n = voJ2n+1+u-1J2n+2+bv-2J2n, n >-1. (2.73) From (2.67) Uzn+1 +V2n+1 = (uo+vo)J2n+2+(u_1 +v-1)J2n+3+b(u_2+v_2)J2n+1; (2.74) for n2-1. By summing the equations (2.70) and (2.74) we get U2n+1 = VOJ21+2+u-1J21+3 +bv_2J21+1, n> -1. (2.75) By subtracting equation (2.70) from equation (2.74), we have v2n+1 = uoJ2n+2+v-1J20+3 + bu-2J21+1, n> –1. (2.76) From (2.63), (2.72), (2.73), (2.75), (2.76) and after some calculation, we obtain X2, = f(f (x0) J2n+1+f (y-1) J2n+2+bf (x-2)J2n), n2 -1, y2n = f(f (vo) J2n+1+f (x-1) J2n+2+bf (y-2) J2n), n2 -1, =f'(f(vo)J2n+2+f (x=1)J2n+3+bf (y-2)J2n+1), n> –1, (2.77) (2.78) (2.79) and Y2n+1 = f(1())2n+2+f (y-1) J2n+3+bf (x-2)J2n+1),n > -1. (2.80) constant coefficients Xn+1 = axn+bxn-1+cxn-2, nE No, (1.4) which has actually the general solution X = xoSn+x-1 (Sn+1 - aSn) +cx_2Sn-1, nE No, (1.5) where (S.) of equation (1L4) satisfving the initial values S 2=S-=0, So= L. The equation axn-IXn-k Xn+1 = nE No, (1.1) bxn-p±cxn-q' where the initial conditions are arbitrary positive real numbers, k, 1, p, q are non- negative integers and a, b, c are positive constants, is one of the difference equations whose solutions are associated with number sequences. Positive solutions of concrete Motivated by this line of investigations, here we show that the systems of differ- ence equations Xn+l =f'(af (Pn-1)+bf(qn-2)), Yntl =f"(af (ra-1)+bf (Sn-2)), (1.8) for n E No, where the sequences Pn, qn, Tn and Sn are some of the sequences x, and Yn, f: Df R is a "1– 1" continuous function on its domain Dr C R, the initial values x-j, y-j, je {0,1,2} are arbitrary real numbers and the parameters and a, b 2.6. Case 6: Pn = Yn, qn = yn, rn = Xn, Sn = Xn In this case, we obtain the system Xn+1 =5"(af (Yn-1)+bf (Yn-2)), Yn+1 = '(af (xn-1)+bf (xn-2)), (2.29) for n e No. Since f is "1– 1", from (2.29) f (Xn+1) = af (Yn-1)+bf (yn-2), f(yn+l) = af (Xn-1) +bf (xn-2), (2.30) for n E No. By using the change of variables f (xn) = = Un, and f(yn) = Vn, n>-2, (2.31) system (2.30) is transformed to the following one Un+l = avn-1+bvn-2, Vn+1 = aun-1+ bun-2, nE No. (2.32) By summing the equations in (2.32) we get Un+l +Vn+l = a (un-1+ Vn-1) +b(un-2+Vn-2), nE No, (2.33) whereas by subtracting the second one from the first, we have Un+l - Vn+l = -a(un-1 - Vn-1) -6(un-2- Vn-2), ne No. (2.34) | In this section, we consider the eight special cases of systems (1.8), where the sequences Pn, qn, ľn, Sn are some of the sequences xn and yn, for n>-2, and initial values x-j, y-j, j e {0,1,2}, are arbitrary real numbers. 2.1 Case 1: Pn =xn, qn = Xn, ľn = yn, Sn == yn In this case, system (1.8) is expressed as Xn+1 =f (af (xn-1)+bf (xn-2)), Yn+1 =f(af (Vn-1)+bf (Yn-2)), (2.1) for n E No. Since f is "1– 1", from (2.1) f (Xn+1) = af (x,-1)+bf (xn-2), f (Yn+1) = af (yn-1)+bf (yn–2), (2.2) for n E No. By using the change of variables f (xn) = Un, and f(yn) = = Vn, n> -2, (2.3) system (2.2) is transformed to the following one Un+1 = au,-1+bun-2, Vn+1 = avn-1+bvn-2, (2.4) for n E No. By taking a = 0, b= a, c = b in (1.4) and S = Jn+1, for all n > -2, which is called generalized Padovan sequence, in (1.5), the solutions to equations in (2.4) are given by Un = ugJn+1+u-1Jn+2+ bu_2Jn, (2.5) Vn = voJn+1+v_1Jn+2+bv_2Jn, (2.6) for n E No. From (2.3), (2.5) and (2.6), it follows that the general solution to system (2.2) is given by Xn = f(f (x0) Jn+1+f (x-1) Jn+2+bf (x-2)Jn), n> -2, Yn =f(f (vo) Jn+1+f(y-1)Ja+2+bf (y-2) J.), n> -2. (2.7) (2.8) 2.2. Case 2: Pn = Xn, qn = Xn, ľn = Xn, Sn Xn In this case, system (1.8) becomes Xp+1 =f"(af (x,-1)+bf (xp-2)), Yn+1=f"(af (xn-1)+bf (xn-2)), (2.9) for n e No. It should be first note that from the equations in (2.9) it immediately follows that x, = yn, for all n E N. From (2.7), the general solution to system (2.9) is %3D X, = Ya = f'(f (xo) Jn+1+f (x=1) Jn+2+bf (x_2)Jn), n E N. (2.10)
Advanced Engineering Mathematics
10th Edition
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Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:2.8. Сase 8: pn — Хл, дл 3D Ул, Гn — Уп, Sn — Xn
In this case, system (1.8) is written as in the form
Xn+1 = f' (af (xp-1)+bf (yn-2)), Yn+1 =f'(af (Yn-1)+bf (xp-2)), (2.61)
for n E No. Since f is "1– 1", from (2.61)
f (Xn+1) = af (xn-1)+bf (yn-2), f(yn+1) = af (yn-1)+bf (xn-2),
(2.62)
for n E No. By using the change of variables
f (xn) = Un, and f (yn) = Vn, n>-2,
(2.63)
system (2.62) is transformed to the following one
Un+1 = aun-1+bvn-2, Vn+1= avn-1+ bun-2, n E No.
(2.64)
By summing the equations in (2.64) we get
Un+1 +Vn+l =a (u,-1+Vn-1) +b (un-2+Vn-2), n E No,
(2.65)
whereas by subtracting the second one from the first, we have
Un+1 - Vn+1 = a (un-1 – Vn-1) – b (un-2 - Vn-2), ne No.
(2.66)
From (2.5), we can write the solution of equation (2.65) as
Un + Vn = (uo+vo) Jn+1+(u-1+v-1) Jn+2+b(u_2+v-2) Jn,
(2.67)
for n > -2. On the other hand, by taking a = 0, b = a, c = -b in (1.4) and
Sn = (-1)" Jn+1, for all n > -2, which is called generalized Padovan sequence, in
(1.5), from (2.66), we also have that
Un- Va = (-1)" ( (uo – vo) Jn+1 – (u-1 –v-1)Jn+2+b (u-2-v-2)Jn),
(2.68)
for n > -2. From (2.68) we have
(uo – vo) J2n+1 – (u_1 – V–1) J2n+2+b (u_2 – v-2) J2n,
(2.69)
U2n - V2n=
for n>-1 and
U2n+1 - V2n+l
-(4o – vo)J2n+2+ (u-1 - v-1) J2n+3 –b(u_2 - v-2) J2n+1, (2.70)
for n >-1. From (2.67)
Uzn+ v2n = (4o+ vo) J2n+1+(u_1 +v=1) J2n+2+b (u_2+v=2) J2n,
(2.71)
for n> -1. By summing the equations (2.69) and (2.71)
u2n = uoJ2n+1 +v-1J21+2+ bu-2J2n, n2 -1.
(2.72)
By subtracting equation (2.69) from equation (2.71), we have
V2n =
voJ2n+1+u-1J2n+2+bv-2J2n, n >-1.
(2.73)
From (2.67)
Uzn+1 +V2n+1 = (uo+vo)J2n+2+(u_1 +v-1)J2n+3+b(u_2+v_2)J2n+1; (2.74)
for n2-1. By summing the equations (2.70) and (2.74) we get
U2n+1 = VOJ21+2+u-1J21+3 +bv_2J21+1, n> -1.
(2.75)
By subtracting equation (2.70) from equation (2.74), we have
v2n+1 = uoJ2n+2+v-1J20+3 + bu-2J21+1, n> –1.
(2.76)
From (2.63), (2.72), (2.73), (2.75), (2.76) and after some calculation, we obtain
X2, = f(f (x0) J2n+1+f (y-1) J2n+2+bf (x-2)J2n), n2 -1,
y2n = f(f (vo) J2n+1+f (x-1) J2n+2+bf (y-2) J2n), n2 -1,
=f'(f(vo)J2n+2+f (x=1)J2n+3+bf (y-2)J2n+1), n> –1,
(2.77)
(2.78)
(2.79)
and
Y2n+1 =
f(1())2n+2+f (y-1) J2n+3+bf (x-2)J2n+1),n > -1.
(2.80)
Transcribed Image Text:constant coefficients
Xn+1 = axn+bxn-1+cxn-2, nE No,
(1.4)
which has actually the general solution
X = xoSn+x-1 (Sn+1 - aSn) +cx_2Sn-1, nE No,
(1.5)
where (S.) of equation (1L4) satisfving the initial values S 2=S-=0, So= L.
The equation
axn-IXn-k
Xn+1 =
nE No,
(1.1)
bxn-p±cxn-q'
where the initial conditions are arbitrary positive real numbers, k, 1, p, q are non-
negative integers and a, b, c are positive constants, is one of the difference equations
whose solutions are associated with number sequences. Positive solutions of concrete
Motivated by this line of investigations, here we show that the systems of differ-
ence equations
Xn+l =f'(af (Pn-1)+bf(qn-2)), Yntl =f"(af (ra-1)+bf (Sn-2)), (1.8)
for n E No, where the sequences Pn, qn, Tn and Sn are some of the sequences x, and
Yn, f: Df R is a "1– 1" continuous function on its domain Dr C R, the initial
values x-j, y-j, je {0,1,2} are arbitrary real numbers and the parameters and a, b
2.6. Case 6: Pn = Yn, qn = yn, rn = Xn, Sn = Xn
In this case, we obtain the system
Xn+1 =5"(af (Yn-1)+bf (Yn-2)), Yn+1 = '(af (xn-1)+bf (xn-2)), (2.29)
for n e No. Since f is "1– 1", from (2.29)
f (Xn+1) = af (Yn-1)+bf (yn-2), f(yn+l) = af (Xn-1) +bf (xn-2),
(2.30)
for n E No. By using the change of variables
f (xn) =
= Un,
and f(yn) = Vn, n>-2,
(2.31)
system (2.30) is transformed to the following one
Un+l = avn-1+bvn-2, Vn+1 = aun-1+ bun-2, nE No.
(2.32)
By summing the equations in (2.32) we get
Un+l +Vn+l = a (un-1+ Vn-1) +b(un-2+Vn-2), nE No,
(2.33)
whereas by subtracting the second one from the first, we have
Un+l - Vn+l = -a(un-1 - Vn-1) -6(un-2- Vn-2), ne No.
(2.34)
|
In this section, we consider the eight special cases of systems (1.8), where the
sequences Pn, qn, ľn, Sn are some of the sequences xn and yn, for n>-2, and initial
values x-j, y-j, j e {0,1,2}, are arbitrary real numbers.
2.1
Case 1: Pn =xn, qn = Xn, ľn = yn, Sn == yn
In this case, system (1.8) is expressed as
Xn+1 =f (af (xn-1)+bf (xn-2)), Yn+1 =f(af (Vn-1)+bf (Yn-2)), (2.1)
for n E No. Since f is "1– 1", from (2.1)
f (Xn+1) = af (x,-1)+bf (xn-2), f (Yn+1) = af (yn-1)+bf (yn–2),
(2.2)
for n E No. By using the change of variables
f (xn)
= Un, and f(yn) =
= Vn, n> -2,
(2.3)
system (2.2) is transformed to the following one
Un+1 = au,-1+bun-2, Vn+1 = avn-1+bvn-2,
(2.4)
for n E No. By taking a = 0, b= a, c = b in (1.4) and S = Jn+1, for all n > -2,
which is called generalized Padovan sequence, in (1.5), the solutions to equations in
(2.4) are given by
Un = ugJn+1+u-1Jn+2+ bu_2Jn,
(2.5)
Vn = voJn+1+v_1Jn+2+bv_2Jn,
(2.6)
for n E No. From (2.3), (2.5) and (2.6), it follows that the general solution to system
(2.2) is given by
Xn = f(f (x0) Jn+1+f (x-1) Jn+2+bf (x-2)Jn), n> -2,
Yn =f(f (vo) Jn+1+f(y-1)Ja+2+bf (y-2) J.), n> -2.
(2.7)
(2.8)
2.2. Case 2: Pn = Xn, qn = Xn, ľn = Xn, Sn Xn
In this case, system (1.8) becomes
Xp+1 =f"(af (x,-1)+bf (xp-2)), Yn+1=f"(af (xn-1)+bf (xn-2)), (2.9)
for n e No. It should be first note that from the equations in (2.9) it immediately
follows that x, = yn, for all n E N. From (2.7), the general solution to system (2.9) is
%3D
X, = Ya = f'(f (xo) Jn+1+f (x=1) Jn+2+bf (x_2)Jn), n E N.
(2.10)
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