16.Consider the autonomous system dxdt=y,dydt=x+2x3.dxdt=y,dydt=x+2x3. a.Show that the critical point (0, 0) is a saddle point. b.Sketch the trajectories for the corresponding linear system by integrating the equation for dy/dx. Show from the parametric form of the solution that the only trajectory on which x → 0, y → 0 as t → ∞ is y = −x. c.Determine the trajectories for the nonlinear system by integrating the equation for dy/dx. Sketch the trajectories for the nonlinear system that correspond to y = −x and y = x for the linear system.
16.Consider the autonomous system dxdt=y,dydt=x+2x3.dxdt=y,dydt=x+2x3. a.Show that the critical point (0, 0) is a saddle point. b.Sketch the trajectories for the corresponding linear system by integrating the equation for dy/dx. Show from the parametric form of the solution that the only trajectory on which x → 0, y → 0 as t → ∞ is y = −x. c.Determine the trajectories for the nonlinear system by integrating the equation for dy/dx. Sketch the trajectories for the nonlinear system that correspond to y = −x and y = x for the linear system.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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16.Consider the autonomous system
dxdt=y,dydt=x+2x3.dxdt=y,dydt=x+2x3.
a.Show that the critical point (0, 0) is a saddle point.
b.Sketch the trajectories for the corresponding linear system by integrating the equation for dy/dx. Show from the parametric form of the solution that the only trajectory on which x → 0, y → 0 as t → ∞ is y = −x.
c.Determine the trajectories for the nonlinear system by integrating the equation for dy/dx. Sketch the trajectories for the nonlinear system that correspond to y = −x and y = x for the linear system.
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