6.3.3 Example C The equation Yk+1Yk – 2yk = -2 (6.38) can be transformed into the linear equation Xk+2 2xk+1 + 2xk = 0 (6.39) by means of the substitution Yk = *k+1/xk. (6.40) The characteristic equation for equation (6.39) is p2 – 2r + 2 = 0, (6.41) and its two complex conjugate roots are r1,2 = 1+i = V2etin/4 (6.42) Therefore, the general solution of equation (6.39) is æk = 26/2[D1 cos(Tk/4)+ D2 sin(tk/4)], (6.43) and - 15D1 cos[T(k+1)/4] + D2 sin[T(k+1)/4] Dị cos(Tk/4) + D2 sin(rk/4) (6.44)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
icon
Concept explainers
Question

Explain the determine red

6.3.3 Ехample C
The equation
Yk+1Yk – 2yk = -2
(6.38)
can be transformed into the linear equation
Xk+2 – 2xk+1+2xk = 0
(6.39)
by means of the substitution
Yk =
Xk+1/xk.
(6.40)
The characteristic equation for equation (6.39) is
p2 – 2r + 2 = 0,
(6.41)
and its two complex conjugate roots are
r1,2 = 1+i = v2e±i™/4_
(6.42)
Therefore, the general solution of equation (6.39) is
X} = 2k/2[D1 cos(tk/4)+ D2 sin(Tk/4)],
(6.43)
and
D1 cos[T(k+ 1)/4] + D2 sin[r(k +1)/4]
Dị cos(Tk/4) + D2 sin(tk/4)
Yk =
V2-
(6.44)
200
Difference Equations
Now, define the constant a such that in the interval -T/2 < a < T/2,
tan a =
D2/D1.
(6.45)
With this result, equation (6.44) becomes
V2 cos[(T/4)(k + 1) – a]
Yk =
cos(Tk/4 – a)
(6.46)
or
Yk
1- tan(ak/4 – a).
(6.47)
This is the general solution to equation (6.38).
Note that since tan(0+ 7) = tan 0, the solution has period 4; i.e., for given
constant a, equation (6.47) takes on only four values; they are
Yo = ]
tan(-a),
- tan(7/4 – a),
1- tan(7/2 – a),
1– tan(37/4 – a).
Y1 =1-
(6.48)
Y2 =
Y3
An easy calculation shows that yo = Y4.
Transcribed Image Text:6.3.3 Ехample C The equation Yk+1Yk – 2yk = -2 (6.38) can be transformed into the linear equation Xk+2 – 2xk+1+2xk = 0 (6.39) by means of the substitution Yk = Xk+1/xk. (6.40) The characteristic equation for equation (6.39) is p2 – 2r + 2 = 0, (6.41) and its two complex conjugate roots are r1,2 = 1+i = v2e±i™/4_ (6.42) Therefore, the general solution of equation (6.39) is X} = 2k/2[D1 cos(tk/4)+ D2 sin(Tk/4)], (6.43) and D1 cos[T(k+ 1)/4] + D2 sin[r(k +1)/4] Dị cos(Tk/4) + D2 sin(tk/4) Yk = V2- (6.44) 200 Difference Equations Now, define the constant a such that in the interval -T/2 < a < T/2, tan a = D2/D1. (6.45) With this result, equation (6.44) becomes V2 cos[(T/4)(k + 1) – a] Yk = cos(Tk/4 – a) (6.46) or Yk 1- tan(ak/4 – a). (6.47) This is the general solution to equation (6.38). Note that since tan(0+ 7) = tan 0, the solution has period 4; i.e., for given constant a, equation (6.47) takes on only four values; they are Yo = ] tan(-a), - tan(7/4 – a), 1- tan(7/2 – a), 1– tan(37/4 – a). Y1 =1- (6.48) Y2 = Y3 An easy calculation shows that yo = Y4.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Knowledge Booster
Points, Lines and Planes
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 Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,