16(e +g +s) – d(c+ j +|f (d +e + s) – g (b (d +e+g) (b +c+f),
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Show me the steps of determie red
![where po, P1, P2, P3
and
P4 are given by (9).
The characteristic equation associated with Eq.(10) is
15 + pAXt + p3d³ + p2X² + p1d + po = 0,
(11)
Theorem 3 Assume that a < 1 and
16 (e +g +s) – d (c+ f +r)|+|c (d +g +s) – e (b+ f +r)|
+|f (d +e +s) – g (b +c+r)|+ |r (d +e +g) – s (b + c+ f)|
< (d+e+g) (b+ c+ f),
S
(12)
then the positive equilibrium point (7) of Eq. (1) is locally asymptotically sta-
ble.
Proof: It follows by Theorem 1 that Eq.(10) is asymptotically stable if all
roots of Eq.(11) lie in the open disk is |A| < 1 that is if |e4|+ \e3| + |P2l +
lei| + |po| < 1,
d (c + f +r)]
(1— а) [b (е +9 +)
(d +e +g+ s) (b + c +f +r)
|a|+
|(1 – a) [c (d +g +s)
e (b + f +r)]
+
(d +e +g+ s) (b +c + f +r)
|(1 – a) [f (d+e+ s)
(d + e+g+ s) (6 +c+ f +r)
(1 – a) [r (d + e+ g)
(d +e +g+ s) (b + c + f + r)
g (b +c+r)]
s (b +c+ f)]
< 1,
and so
(1 – a) [b (e +g +s) – d (c + f +r)]
(d +e +g+s) (b + c + f +r)
(1– a) [c (d +g +s) – e(b + f +r)]|
+
(d +e +g+s) (b +c+ f +r)
(1 – a) [f (d + e +s)
(d +e +g+ s) (b +c + ƒ + r)
(1 – a) [r (d + e +g) – s (b + c+ f)]|
(d +e +g+s) (b + c + f +r)
а < 1,
g (b +c+r)]
< (1- а),
or
|b(e +g + s) – d (c+ f +r)|+ |c (d +g+s)
+|f (d +e +s) – g (b +c+r)+ |r (d +e +g) – s (b +c+ f)
K (d +e +g) (b+c+f).
- e (b+ f +r)|
Thus, the proof is now completed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25e1145d-beeb-4bcc-a46c-2420650cb54d%2F3ab32344-a3a2-4e7b-880f-cc763bd32e6d%2F6olrfh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:where po, P1, P2, P3
and
P4 are given by (9).
The characteristic equation associated with Eq.(10) is
15 + pAXt + p3d³ + p2X² + p1d + po = 0,
(11)
Theorem 3 Assume that a < 1 and
16 (e +g +s) – d (c+ f +r)|+|c (d +g +s) – e (b+ f +r)|
+|f (d +e +s) – g (b +c+r)|+ |r (d +e +g) – s (b + c+ f)|
< (d+e+g) (b+ c+ f),
S
(12)
then the positive equilibrium point (7) of Eq. (1) is locally asymptotically sta-
ble.
Proof: It follows by Theorem 1 that Eq.(10) is asymptotically stable if all
roots of Eq.(11) lie in the open disk is |A| < 1 that is if |e4|+ \e3| + |P2l +
lei| + |po| < 1,
d (c + f +r)]
(1— а) [b (е +9 +)
(d +e +g+ s) (b + c +f +r)
|a|+
|(1 – a) [c (d +g +s)
e (b + f +r)]
+
(d +e +g+ s) (b +c + f +r)
|(1 – a) [f (d+e+ s)
(d + e+g+ s) (6 +c+ f +r)
(1 – a) [r (d + e+ g)
(d +e +g+ s) (b + c + f + r)
g (b +c+r)]
s (b +c+ f)]
< 1,
and so
(1 – a) [b (e +g +s) – d (c + f +r)]
(d +e +g+s) (b + c + f +r)
(1– a) [c (d +g +s) – e(b + f +r)]|
+
(d +e +g+s) (b +c+ f +r)
(1 – a) [f (d + e +s)
(d +e +g+ s) (b +c + ƒ + r)
(1 – a) [r (d + e +g) – s (b + c+ f)]|
(d +e +g+s) (b + c + f +r)
а < 1,
g (b +c+r)]
< (1- а),
or
|b(e +g + s) – d (c+ f +r)|+ |c (d +g+s)
+|f (d +e +s) – g (b +c+r)+ |r (d +e +g) – s (b +c+ f)
K (d +e +g) (b+c+f).
- e (b+ f +r)|
Thus, the proof is now completed.
![In this section, we study the local stability of the solutions of Eq.(1). The
equilibrium point ã of Eq.(1) is the positive solution of the equation
b +c+f+r
d +e +g+s
* = ax +
(6)
If a < 1, then the only positive equilibrium point ã of Eq.(1) is given by
b +c+f +r
(1 – a) ( d + e + g+ s) *
(7)
F : (0, 00)5 → (0, 00) which
Let us now introduce a continuous function
is defined by
bui + cu2 + fu3 + ru4
F(uo, ..., u4) = auo +
(8)
dui + eu2 + gu3 + su4
Therefore it follows that
aF(u0,...,u4)
duo
= a,
aF(u0,...,us)
dui
6 (e u2+g uz+s u4) – d (c u2+f u3+r u4)
(d ui+e u2+g u3+su4)2
%3D
aF(u0,...,u4)
du2
c (d u1+g uz+s u4) – e (b u1+f u3+r u4)
(d u1+e u2+g u3+su4)²
ƏF(u0,...,u4)
duz
f (d u1+e u2+s u4)
- g (b u1+c u2+r u4)
%3D
(d u1+e u2+g u3+su4)²
ƏF(u0,...,u4)
r (d u1+e uz+g u3) – s (b u1+c u2+f u3)
(d u1+e uz+g u3+su4)²
Consequently, we get
ƏF(ĩ,...,T)
duo
= a = -
P4,
ƏF(ĩ,...,7)
(1–a)[b (e +g +s) – d (c +f +r)]
(d +e +g+s)(b +c +f+r)
= -
P3,
%3D
OF(ĩ,...,î)
du2
(1-a)[c (d +g +s) – e (b +f +r)]
(d +e +g+s)(b +c +f+r)
(9)
= -
P2,
%3D
ƏF(ĩ,...,¤)
du3
(1-a)[f (d +e +s) – g (b +c+r)]
(d +e +g+s)(b +c +f+r)
= -
P1,
aF(ĩ,…..,Ã)
du4
(1-a)[r (d +e +g) – s (b +c+f)]
(d +e +g+s)(b +c +f+r)
= -
Po.
%3D
Thus, the linearized equation of Eq.(1) about ã takes the form
Yn+1 + P4Yn + P3Yn-1+ P2Yn-2 + P1Yn-3 + Poyn-4
0,
(10)
%3D
The objective of this article is to investigate some qualitative behavior of
the solutions of the nonlinear difference equation
bxn-1+ cxn-2 + fxn-3+ran-4
dxn-1+ exn–2+ gxn-3 + sxn-4
Xn+1 = axn +
n = 0,1, 2, .. (1)
where the coefficients o b c de f arsf (0∞) while the initial con-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25e1145d-beeb-4bcc-a46c-2420650cb54d%2F3ab32344-a3a2-4e7b-880f-cc763bd32e6d%2Ffwfw5kr_processed.png&w=3840&q=75)
Transcribed Image Text:In this section, we study the local stability of the solutions of Eq.(1). The
equilibrium point ã of Eq.(1) is the positive solution of the equation
b +c+f+r
d +e +g+s
* = ax +
(6)
If a < 1, then the only positive equilibrium point ã of Eq.(1) is given by
b +c+f +r
(1 – a) ( d + e + g+ s) *
(7)
F : (0, 00)5 → (0, 00) which
Let us now introduce a continuous function
is defined by
bui + cu2 + fu3 + ru4
F(uo, ..., u4) = auo +
(8)
dui + eu2 + gu3 + su4
Therefore it follows that
aF(u0,...,u4)
duo
= a,
aF(u0,...,us)
dui
6 (e u2+g uz+s u4) – d (c u2+f u3+r u4)
(d ui+e u2+g u3+su4)2
%3D
aF(u0,...,u4)
du2
c (d u1+g uz+s u4) – e (b u1+f u3+r u4)
(d u1+e u2+g u3+su4)²
ƏF(u0,...,u4)
duz
f (d u1+e u2+s u4)
- g (b u1+c u2+r u4)
%3D
(d u1+e u2+g u3+su4)²
ƏF(u0,...,u4)
r (d u1+e uz+g u3) – s (b u1+c u2+f u3)
(d u1+e uz+g u3+su4)²
Consequently, we get
ƏF(ĩ,...,T)
duo
= a = -
P4,
ƏF(ĩ,...,7)
(1–a)[b (e +g +s) – d (c +f +r)]
(d +e +g+s)(b +c +f+r)
= -
P3,
%3D
OF(ĩ,...,î)
du2
(1-a)[c (d +g +s) – e (b +f +r)]
(d +e +g+s)(b +c +f+r)
(9)
= -
P2,
%3D
ƏF(ĩ,...,¤)
du3
(1-a)[f (d +e +s) – g (b +c+r)]
(d +e +g+s)(b +c +f+r)
= -
P1,
aF(ĩ,…..,Ã)
du4
(1-a)[r (d +e +g) – s (b +c+f)]
(d +e +g+s)(b +c +f+r)
= -
Po.
%3D
Thus, the linearized equation of Eq.(1) about ã takes the form
Yn+1 + P4Yn + P3Yn-1+ P2Yn-2 + P1Yn-3 + Poyn-4
0,
(10)
%3D
The objective of this article is to investigate some qualitative behavior of
the solutions of the nonlinear difference equation
bxn-1+ cxn-2 + fxn-3+ran-4
dxn-1+ exn–2+ gxn-3 + sxn-4
Xn+1 = axn +
n = 0,1, 2, .. (1)
where the coefficients o b c de f arsf (0∞) while the initial con-
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
