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x=x²-3xy + 2x y = x + y - 1 x = x² + y² - 13 y = xy - 2x - 2y + 4

x=x²-3xy + 2x y = x + y - 1 x = x² + y² - 13 y = xy - 2x - 2y + 4

Advanced Engineering Mathematics
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

Find the equilibrium point(s) of each nonlinear system given below. Then determine the stability of each 
equilibrium point. Simulate these systems and plot their phase plane portraits and determine if any 
possess periodic orbits. 

2). x= x² 3). 3xy + 2x y = x + y - 1 x = x² + y² - 13 y = xy - 2x - 2y + 4
Transcribed Image Text:2). x= x² 3). 3xy + 2x y = x + y - 1 x = x² + y² - 13 y = xy - 2x - 2y + 4
Expert Solution
Step 1: Solving part 2

Given

x˙=x2-3xy+2y=Fx,yy˙=x+y-1        =Gx,y

Let 0,0 be the critical point

∴J=FxFyGxGy0,0       =2x-3y+2-3y120,0       =2012

Clearly the eigen value of the matrix J is 2, 2

Advanced Math homework question answer, step 1, image 1

Therefore the system is asymptotically unstable

Now

    ∂Fx,y∂x+∂Gx,y∂y=2x-3y+2+1=2x-3y+3

Therefore the point care does not have same sign throughout ℝ2

So the system has closed path and periodic solution in the domain ( By using Bendixons Non Existance Criterion)

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