8. (a) Show that the system i=y+xf(r)/r, j=-x+yf(r)/r (7=x²+y?) (*) has limit cycles corresponding to the zeros of f(r). What is the direction of motion on these curves? (b) Determine all limit cycles of (*) and discuss their stability if f(r)= (r-3)²(r²– 5r+4).

8. (a) Show that the system i=y+xf(r)/r, j=-x+yf(r)/r (7=x²+y?) () has limit cycles corresponding to the zeros of f(r). What is the direction of motion on these curves? (b) Determine all limit cycles of () and discuss their stability if f(r)= (r-3)²(r²– 5r+4).

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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Question

8. (a) Show that the system
i=y+xf (r)/r,
j= - x+yf(r)/r
(P=x²+y?)
(*)
has limit cycles corresponding to the zeros of f(r). What is the direction of
motion on these curves?
(b) Determine all limit cycles of (*) and discuss their stability if f(r)=
(r- 3)°(r² – Sr+4).

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