(6+ f)Q+ (c+r) P aP + (d + g) Q+ (e+ s) P [a (d + g) – (e + s)] PQ + a (e + s) P² – (d + g) Q² + (b+ f)Q+ (c+r) P | (d+g) Q+ (e+ s) P [a(d+g)-(e+s)][(b+f)-(c+r)][(c+r)(d+g)+a(e+s) (b+f)] [a(e+s)+(d+g)]²[((e+s)-(d+g))(a+1)] + a (e + s) [(b+f)-(c+r)]+8 2[a(e+s)+(d+g)] [(b+f)-(c+r)]-8 (d+g) (2a(e+s)+(d+g)] [(b+f)-(c+r)]+8 2[a(e+s)+(d+g)J. + (e + s) 2 [(b+f)-(c+r)]-8 - (d + g) (2ja(e+s)+(d+g) [(b+f)-(c+r)]-8 + (b+ f) ( a(e+s)+(d+g)]

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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immediately from (13) that the inequality (19) holds. There exist two positive
distinctive real numbers P and Q representing two positive roots of Eq.(18)
such that
[(b + f) – (c+r)] + 8
P =
(20)
2 [a (e + s) + (d + g)]
and
[(b+ f) – (c+r)] – 8
Q
(21)
2 [a (e + s) + (d +g)]
[(b + f) – (c+ r)]²
4[(b+f)-(c+r)][(c+r)(d+g)+a(e+s) (b+f)] Now,
(d+g))(a+1)]
where d =
[((e+s)
two of Eq.(1). To this end, we assume that x_4
P, x-1 = Q, xo = P. Now, we are going to show that x1 =
From Eq.(1) we deduce that
we are going to prove that P and Q are positive solutions of prime period
Q, x-2 =
Q and x2
= P, x-3
Р.
(b+ f) Q + (c+ r) P
aP+
bx –1 + cx_2 + fx_3 + rx-4
X1 = axo-+
(22)
dx-1 + ex-2 + gx_3 + sx –4
(d + g) Q+ (e + s) P'
Substituting (20) and (21) into (22) we deduce that
(b+ f)Q+ (c+ r) P
aP+
(d + g) Q+ (e + s) P
[a (d + g) – (e + s)] PQ + a (e + s) P² – (d + g) Q² + (b+ f)Q+ (c+ r) P
(d + g) Q+ (e + s) P
[a(d+g)-(e+s)][(b+f)-(c+r)[c+r)(d+g)+a(e+s) (b+f)]
[a(e+s)+(d+g)]²[((e+s)-(d+g))(a+1)]
+ a (e + s)
2
[(6+f)-(c+r)]+8
2[a(e+s)+(d+g)]
(d + g)
[(b+f)-(c+r)]-8
2[a(e+s)+(d+g)]
+ (e + s)
[(b+f)-(c+r)]+8
2[a(e+s)+(d+g)]
- (d + g)
2
[(b+f)-(c+r)]-8
2[a(e+s)+(d+g)]
+ (b+ f)
[(b+f)-(c+r)]-8
2[a(e+s)+(d+g)]
(d + g)
[(b+f)-(c+r)]-8
2[a(e+s)+(d+g)]
+ (e + s)
[(b+f)-(c+r)]+8
2[a(e+s)+(d+g)]
(c+r)
[(b+f)-(c+r)]+8
2[a(e+s)+(d+g)]
(d + g) ( 6+f)-(c+r)]-8
[(b+f)-(c+r)]+8
2[a(e+s)+(d+g)]
2[a(e+s)+(d+g)]) + (e+ s)
(23)
Transcribed Image Text:immediately from (13) that the inequality (19) holds. There exist two positive distinctive real numbers P and Q representing two positive roots of Eq.(18) such that [(b + f) – (c+r)] + 8 P = (20) 2 [a (e + s) + (d + g)] and [(b+ f) – (c+r)] – 8 Q (21) 2 [a (e + s) + (d +g)] [(b + f) – (c+ r)]² 4[(b+f)-(c+r)][(c+r)(d+g)+a(e+s) (b+f)] Now, (d+g))(a+1)] where d = [((e+s) two of Eq.(1). To this end, we assume that x_4 P, x-1 = Q, xo = P. Now, we are going to show that x1 = From Eq.(1) we deduce that we are going to prove that P and Q are positive solutions of prime period Q, x-2 = Q and x2 = P, x-3 Р. (b+ f) Q + (c+ r) P aP+ bx –1 + cx_2 + fx_3 + rx-4 X1 = axo-+ (22) dx-1 + ex-2 + gx_3 + sx –4 (d + g) Q+ (e + s) P' Substituting (20) and (21) into (22) we deduce that (b+ f)Q+ (c+ r) P aP+ (d + g) Q+ (e + s) P [a (d + g) – (e + s)] PQ + a (e + s) P² – (d + g) Q² + (b+ f)Q+ (c+ r) P (d + g) Q+ (e + s) P [a(d+g)-(e+s)][(b+f)-(c+r)[c+r)(d+g)+a(e+s) (b+f)] [a(e+s)+(d+g)]²[((e+s)-(d+g))(a+1)] + a (e + s) 2 [(6+f)-(c+r)]+8 2[a(e+s)+(d+g)] (d + g) [(b+f)-(c+r)]-8 2[a(e+s)+(d+g)] + (e + s) [(b+f)-(c+r)]+8 2[a(e+s)+(d+g)] - (d + g) 2 [(b+f)-(c+r)]-8 2[a(e+s)+(d+g)] + (b+ f) [(b+f)-(c+r)]-8 2[a(e+s)+(d+g)] (d + g) [(b+f)-(c+r)]-8 2[a(e+s)+(d+g)] + (e + s) [(b+f)-(c+r)]+8 2[a(e+s)+(d+g)] (c+r) [(b+f)-(c+r)]+8 2[a(e+s)+(d+g)] (d + g) ( 6+f)-(c+r)]-8 [(b+f)-(c+r)]+8 2[a(e+s)+(d+g)] 2[a(e+s)+(d+g)]) + (e+ s) (23)
The objective of this article is to investigate some qualitative behavior of
the solutions of the nonlinear difference equation
бxn-1 + сх,n-2 + fxn-3 + rtn-4
Xn+1 = axn +
n = 0, 1, 2, .. (1)
dxn-1 + en-2 + gxn-3 + sxn-4
where the coefficients a, b, c, d, e, f, g,r, s E (0, 0), while the initial con-
ditions x-4,X –3,X –2, x – 1, xo are arbitrary positive real numbers. Note that
the special cases of Eq.(1) has been studied discussed in [11] when f = g =
0 and Eq. (1) has been studied discussed in [35] in the special case
r = s =
when r = s = 0.
Transcribed Image Text:The objective of this article is to investigate some qualitative behavior of the solutions of the nonlinear difference equation бxn-1 + сх,n-2 + fxn-3 + rtn-4 Xn+1 = axn + n = 0, 1, 2, .. (1) dxn-1 + en-2 + gxn-3 + sxn-4 where the coefficients a, b, c, d, e, f, g,r, s E (0, 0), while the initial con- ditions x-4,X –3,X –2, x – 1, xo are arbitrary positive real numbers. Note that the special cases of Eq.(1) has been studied discussed in [11] when f = g = 0 and Eq. (1) has been studied discussed in [35] in the special case r = s = when r = s = 0.
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