Rayleigh's equation is y” +4 (½(1¹)² − 1) y′ + y = 0, where µ is a constant. Show that differenti- ation of this equation and setting y' = z reduces Rayleigh's equation to the Van der Pol equation. y" +μ(y²-1)y + y = 0.

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
Question

PLEASE HELP WITH THIS QUESTION 

4.
(a) Rayleigh's equation is y" + 4 ((1′)² − 1) y′ + y = 0, where µ is a constant. Show that differenti-
ation of this equation and setting y' = z reduces Rayleigh's equation to the Van der Pol equation,
y" +μ(y² - 1)y + y = 0.
(b) The Van der Pol equation is equivalent to the system u' = v and v' = u(1-²)v-u. Show that
(0,0) is the only critical point of the system. Determine the nature and stability of the critical point
when μ< 2, μ = 2 and μ > 2.
Transcribed Image Text:4. (a) Rayleigh's equation is y" + 4 ((1′)² − 1) y′ + y = 0, where µ is a constant. Show that differenti- ation of this equation and setting y' = z reduces Rayleigh's equation to the Van der Pol equation, y" +μ(y² - 1)y + y = 0. (b) The Van der Pol equation is equivalent to the system u' = v and v' = u(1-²)v-u. Show that (0,0) is the only critical point of the system. Determine the nature and stability of the critical point when μ< 2, μ = 2 and μ > 2.
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
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,