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for n E No. Since f is "1-1", from (2.49) S (Xn+1) = af (yn-1)+bf (xp-2), f(yn+1) = af (xp-1)+bf (yn-2), (2.50) for n e No. By using the change of variables f (xn) = un, and f(yn) Vn, n>-2, (2.51) system (2.50) is transformed to the following one Un+l = avn-1 +bu,-2, Vn+1= aun-1+ bvn--2, ne No. (2.52) By summing the equations in (2.52) we get Un+l + Vn+l = a (un-1+ Vn-1) +b(un-2+ Vn-2), ne No, (2.53) whereas by subtracting the second one from the first, we have Un+l - Vn+l = -a(un-1- Vn-1) +b(un-2- Vn-2), ne No. (2.54) From (2.5), we can write the solution of equation (2.53) as un + Vn = (4o+ vo) Jn+1 + (u-1+v-1) Jn+2+b (u-2+v-2) Jn, (2.55) for n>-2. On the other hand, by taking a=0, b=-a, c=bin (1.4) and S,= Jt for all n>-2, which is called generalized Padovan sequence, in (1.5), from (2.54), we also have that Un - Vn = (u0 – vo) Jn+1 +(u-1-v-1) Jn+2+b(u-2 -v-2) J, for n2-2. By summing the equations (2.55) and (2.56) we get Jntl-Jatl vo+ (2.56) Ja+2+J+21 Un = Jn+2-Ja+2, + Jーム u-2+b- v_2, n> -2. (2.57) 2 By subtracting equation (2.56) from equation (2.55), we have Jn+1+Jn+1y Jnt2- Jnt2u uo + Vo+ 2 Vn = 2 2 Jn+2+Jn+2, ニVー」+b 2 Jn+J u-2+b v-2, n>-2. (2.58) From (2.51), (2.57) and (2.58) and after some calculation, we obtain Jn+l - Jt1 げ (xo)+ Ja+2+Jn+2 f(x-1)+ Jn+2-Jn+2 f(y-1) 2 Jn+J +b 2 y-2) n2-2, (2.59) 2 and (In+l - n+l f(xo)+ 2 Ja+l +Jn+lf(yo) Yn = f-1 2

for n E No. Since f is "1-1", from (2.49) S (Xn+1) = af (yn-1)+bf (xp-2), f(yn+1) = af (xp-1)+bf (yn-2), (2.50) for n e No. By using the change of variables f (xn) = un, and f(yn) Vn, n>-2, (2.51) system (2.50) is transformed to the following one Un+l = avn-1 +bu,-2, Vn+1= aun-1+ bvn--2, ne No. (2.52) By summing the equations in (2.52) we get Un+l + Vn+l = a (un-1+ Vn-1) +b(un-2+ Vn-2), ne No, (2.53) whereas by subtracting the second one from the first, we have Un+l - Vn+l = -a(un-1- Vn-1) +b(un-2- Vn-2), ne No. (2.54) From (2.5), we can write the solution of equation (2.53) as un + Vn = (4o+ vo) Jn+1 + (u-1+v-1) Jn+2+b (u-2+v-2) Jn, (2.55) for n>-2. On the other hand, by taking a=0, b=-a, c=bin (1.4) and S,= Jt for all n>-2, which is called generalized Padovan sequence, in (1.5), from (2.54), we also have that Un - Vn = (u0 – vo) Jn+1 +(u-1-v-1) Jn+2+b(u-2 -v-2) J, for n2-2. By summing the equations (2.55) and (2.56) we get Jntl-Jatl vo+ (2.56) Ja+2+J+21 Un = Jn+2-Ja+2, + Jーム u-2+b- v_2, n> -2. (2.57) 2 By subtracting equation (2.56) from equation (2.55), we have Jn+1+Jn+1y Jnt2- Jnt2u uo + Vo+ 2 Vn = 2 2 Jn+2+Jn+2, ニVー」+b 2 Jn+J u-2+b v-2, n>-2. (2.58) From (2.51), (2.57) and (2.58) and after some calculation, we obtain Jn+l - Jt1 げ (xo)+ Ja+2+Jn+2 f(x-1)+ Jn+2-Jn+2 f(y-1) 2 Jn+J +b 2 y-2) n2-2, (2.59) 2 and (In+l - n+l f(xo)+ 2 Ja+l +Jn+lf(yo) Yn = f-1 2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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)
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)
2.7. Case 7: Pn= yn, In = Xn, rn = Xp, Sn = yn In this case, system (1.8) is expressed as Xn+1 =f(af (Yn-1)+bf (xn-2)), Yn+1 =f(af (xn-1)+bf (Yn-2)), (2.49) for n € No. Since f is "1– 1", from (2.49) f (Xn+1) = af (yn-1)+bf (xn-2), f(yn+1) = af (xp-1) +bf (yn-2), (2.50) for n e No. By using the change of variables f (xn) = un, and f(yn) =vn, n >-2, (2.51) system (2.50) is transformed to the following one Un+1 = avn-1 +bun-2, Vn+1=aun-1+bv,n-2, n E No. (2.52) By summing the equations in (2.52) we get Un+1+Vn+l = a(un-1+Vn-1)+b(un-2+Vn-2), ne No, (2.53) whereas by subtracting the second one from the first, we have Un+l - Vn+1 = -a(un-1 – Vn-1)+b(un-2 – Vn-2), n E No. (2.54) From (2.5), we can write the solution of equation (2.53) as Un + Vn = (uo+vo) Jn+1+(u-1+v-1) Jn+2+b(u_2+v_2)Jn for n>-2. On the other hand, by taking a =0, b=-a, c=bin (1.4) and S, =Jt for all n > -2, which is called generalized Padovan sequence, in (1.5), from (2.54), we also have that (2.55) Un – Vn = (u0 – o) Jn+1+(u-1 – v-1)Jn+2+b(u-2–- v-2)J,, (2.56) for n> -2. By summing the equations (2.55) and (2.56) we get Jn+l - J+1 tl vo+ 2 Jn+2+Jn+2 u_1 и, — 2 2 Jn+2-J+2, Jn+J v-1+b- u-2+b 2 ーav-2,n2-2. (2.57) 2 By subtracting equation (2.56) from equation (2.55), we have Jn+1 - Ja+1 Vn = Jn+1+Jn+1, uo+ Jn+2- Jn+2u-1 2 2 Jn+2+Jn+2, Jn- J V-1+b 2 Jn+J, 'u-2+b 2 'v_2, n> -2. (2.58) From (2.51), (2.57) and (2.58) and after some calculation, we obtain ri(n+1+Jn+1f (xo)+ Ja+1-Jntl f(yo) Xn = f 2 2 Jat2+ Ja+2 (x-1)+ Jn+2-Jat2 f(y-1) + 2 2 Jn- J Jn+J, +b- 2 n>-2, (2.59) and Yn =f- Jntl-Jn+l f(xo)+ Jn+1 +Jn+1 f (vo) 2 Jn+2 - J+2, Jn+2+J+2 2f(x-1)+ 2 ج. شه. Jn +b 2 Jn+J f(y-2) n>-2. (2.60)
Transcribed Image Text:2.7. Case 7: Pn= yn, In = Xn, rn = Xp, Sn = yn In this case, system (1.8) is expressed as Xn+1 =f(af (Yn-1)+bf (xn-2)), Yn+1 =f(af (xn-1)+bf (Yn-2)), (2.49) for n € No. Since f is "1– 1", from (2.49) f (Xn+1) = af (yn-1)+bf (xn-2), f(yn+1) = af (xp-1) +bf (yn-2), (2.50) for n e No. By using the change of variables f (xn) = un, and f(yn) =vn, n >-2, (2.51) system (2.50) is transformed to the following one Un+1 = avn-1 +bun-2, Vn+1=aun-1+bv,n-2, n E No. (2.52) By summing the equations in (2.52) we get Un+1+Vn+l = a(un-1+Vn-1)+b(un-2+Vn-2), ne No, (2.53) whereas by subtracting the second one from the first, we have Un+l - Vn+1 = -a(un-1 – Vn-1)+b(un-2 – Vn-2), n E No. (2.54) From (2.5), we can write the solution of equation (2.53) as Un + Vn = (uo+vo) Jn+1+(u-1+v-1) Jn+2+b(u_2+v_2)Jn for n>-2. On the other hand, by taking a =0, b=-a, c=bin (1.4) and S, =Jt for all n > -2, which is called generalized Padovan sequence, in (1.5), from (2.54), we also have that (2.55) Un – Vn = (u0 – o) Jn+1+(u-1 – v-1)Jn+2+b(u-2–- v-2)J,, (2.56) for n> -2. By summing the equations (2.55) and (2.56) we get Jn+l - J+1 tl vo+ 2 Jn+2+Jn+2 u_1 и, — 2 2 Jn+2-J+2, Jn+J v-1+b- u-2+b 2 ーav-2,n2-2. (2.57) 2 By subtracting equation (2.56) from equation (2.55), we have Jn+1 - Ja+1 Vn = Jn+1+Jn+1, uo+ Jn+2- Jn+2u-1 2 2 Jn+2+Jn+2, Jn- J V-1+b 2 Jn+J, 'u-2+b 2 'v_2, n> -2. (2.58) From (2.51), (2.57) and (2.58) and after some calculation, we obtain ri(n+1+Jn+1f (xo)+ Ja+1-Jntl f(yo) Xn = f 2 2 Jat2+ Ja+2 (x-1)+ Jn+2-Jat2 f(y-1) + 2 2 Jn- J Jn+J, +b- 2 n>-2, (2.59) and Yn =f- Jntl-Jn+l f(xo)+ Jn+1 +Jn+1 f (vo) 2 Jn+2 - J+2, Jn+2+J+2 2f(x-1)+ 2 ج. شه. Jn +b 2 Jn+J f(y-2) n>-2. (2.60)
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