= a, duo b (e uz+g uz+s u4) – d (c uz+f u3+r us) (d ui+e uz+g uz+su4)² aF(u0,...,us) %3D Ine ƏF(uo,...,us) duz c (d ui+g uz+s u4) - e (b ui+f u3+r u4) (d u1+e uz+g u3+sus)? ƏF(uo,.u4) f (d u1+e uz2+s u4) – g (b ui+c uz+r u4) duz (d ui+e uz+g u3+su4)² ƏF(uo....,u4) dus r (d ui+e uz+g u3) – s (b u1+c uz+f u3) (d ui+e uz+g u3+su4)?
= a, duo b (e uz+g uz+s u4) – d (c uz+f u3+r us) (d ui+e uz+g uz+su4)² aF(u0,...,us) %3D Ine ƏF(uo,...,us) duz c (d ui+g uz+s u4) - e (b ui+f u3+r u4) (d u1+e uz+g u3+sus)? ƏF(uo,.u4) f (d u1+e uz2+s u4) – g (b ui+c uz+r u4) duz (d ui+e uz+g u3+su4)² ƏF(uo....,u4) dus r (d ui+e uz+g u3) – s (b u1+c uz+f u3) (d ui+e uz+g u3+su4)?
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 determine red and it complete
![2
The local stability of the solutions
In this section, we study the local stability of the solutions of Eq.(1). The
equilibrium point a of Eq.(1) is the positive solution of the equation
b +c+f +r
d +e +g+ s'
x = ax +
(6)
If a < 1, then the only positive equilibrium point z of Eq.(1) is given by
b +c+f + r
(7)
(1 – a) ( d+ e + g+s)
Let us now introduce a continuous function F : (0, 00)5 → (0, 00) which
is defined by
buj + cu2 + fu3 + ru4
F(uo,.., u4) = auo +
(8)
dui + eu2 + gu3 + su4
Therefore it follows that
OF (uo,...,u4)
= a,
aF(uo,..,u4)
b (e uz+g uz+8 u4) – d (c u2+f u3+r us)
(d ui+e u2+g u3+su4)2
OF(uo,...,u4)
au2
e (d ui+g uz+s u4) – e (b u1+f u3+r us)
(d u1+e uz+g uz+su4)²
OF (uo,...,u4)
duz
f (d ui+e uz+s u4) – g (b ui+c u2+r us)
(d u1+e u2+g uz+su4)?
aF(uo,...,u4)
r (d ui+e u2+g u3) – s (b u1+c u2+f u3)
(d u1+e u2+g_u3+su4)²
Consequently, we get
OF (ã,...,ã)
Juo
= a = - P4:
aF(7,...,ï)
(1-a) [b (e +g +s) – d (c +f +r)]
(d +e +g+s)(b +c +f+r)
= - P3,
OF (7,..,ã)
(1-a) [e (d +g +s) – e (b +f +r)]
(d +e +g+s)(b +c +f+r)
= - P2,
(9)
(1-a)[f (d +e +s) – g (b +c+r)]
(d +e +g+s)(b +c +f+r)
= - P1,
aF (7,...,ã)
(1-a)[r (d +e +g) - 8 (b +c+f)]
(d +e +g+s)(b +c +f+r)
= - Po.
Thus, the linearized equation of Eq.(1) about a takes the form
Yn+1 + P4Yn + P3Yn-1+ P2Yn-2 + P1Yn-3 + Poyn-4 = 0,
(10)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48dbf4e2-6a0f-454b-99ed-fcbc11bc7c27%2Ff5db6b97-bf25-4059-afa4-a8bfda238b7c%2F9o4lat_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2
The local stability of the solutions
In this section, we study the local stability of the solutions of Eq.(1). The
equilibrium point a of Eq.(1) is the positive solution of the equation
b +c+f +r
d +e +g+ s'
x = ax +
(6)
If a < 1, then the only positive equilibrium point z of Eq.(1) is given by
b +c+f + r
(7)
(1 – a) ( d+ e + g+s)
Let us now introduce a continuous function F : (0, 00)5 → (0, 00) which
is defined by
buj + cu2 + fu3 + ru4
F(uo,.., u4) = auo +
(8)
dui + eu2 + gu3 + su4
Therefore it follows that
OF (uo,...,u4)
= a,
aF(uo,..,u4)
b (e uz+g uz+8 u4) – d (c u2+f u3+r us)
(d ui+e u2+g u3+su4)2
OF(uo,...,u4)
au2
e (d ui+g uz+s u4) – e (b u1+f u3+r us)
(d u1+e uz+g uz+su4)²
OF (uo,...,u4)
duz
f (d ui+e uz+s u4) – g (b ui+c u2+r us)
(d u1+e u2+g uz+su4)?
aF(uo,...,u4)
r (d ui+e u2+g u3) – s (b u1+c u2+f u3)
(d u1+e u2+g_u3+su4)²
Consequently, we get
OF (ã,...,ã)
Juo
= a = - P4:
aF(7,...,ï)
(1-a) [b (e +g +s) – d (c +f +r)]
(d +e +g+s)(b +c +f+r)
= - P3,
OF (7,..,ã)
(1-a) [e (d +g +s) – e (b +f +r)]
(d +e +g+s)(b +c +f+r)
= - P2,
(9)
(1-a)[f (d +e +s) – g (b +c+r)]
(d +e +g+s)(b +c +f+r)
= - P1,
aF (7,...,ã)
(1-a)[r (d +e +g) - 8 (b +c+f)]
(d +e +g+s)(b +c +f+r)
= - Po.
Thus, the linearized equation of Eq.(1) about a takes the form
Yn+1 + P4Yn + P3Yn-1+ P2Yn-2 + P1Yn-3 + Poyn-4 = 0,
(10)
Transcribed Image Text:l stc ksa
12:14 AM
@ 1 60% 4
The objective of this article is to investigate some qualitative behavior of
the solutions of the nonlinear difference equation
bxn-1 + cæn-2+ fxn-3 + ræn-4
Xn+1 = axn +
n = 0, 1, 2, . (1)
dxn-1+ exn-2 + gæn-3 + sxn-4
where the coefficients a, b, c, d, e, f, g, r, s E (0, 00), while the initial con-
ditions a_4,x_3,x_2, x-1, xo are arbitrary positive real numbers. Note that
Cancel
Actual Size (399 KB)
Choose
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,
