Consider the nonlinear planar systemx'=3y-6y'=3x-x²a. Find all of its stationary solutions.b. Find a nonconstant function H(x,y) such that every trajectory of the systemsatisfies H(x,y) =c for some constant c.c. Classify and give the stability of each stationary solution.d. Sketch the phase plane portrait of the system near each critical point. Marksketched solutions with arrows to indicate direction of motion.

In mathematics and science, a system can be a system within which the change of the output isn't…

Answered: Consider the nonlinear planar system…

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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5. Consider the system <-3-1* -2 x' = x. (a) Find the general solution for the system. (b) Is the equilibrium solution ♂ a source, sink or saddle? Is it unstable or stable? Explain. (c) State the equations of the v- and h-nullclines for the system. (d) Sketch the phase portrait, using nullclines as an aid.
For the system below a. Find the equilibrium points. b. For the equilibrium point (1,1) solve the associated linear system and preform the qualitative analysis dx =x (2 = x- y), dt dy -y(y-x') dt
Consider the system: dx/dt=4x(1-2x)-xy dy/dt=2y-4xy (a) Find all critical (equilibrium) points. (b) Using the Jacobian matrix, classify (if possible) each critical (equilibrium) point as a stable node, a stable spiral point, an unstable node, an unstable spiral point, or a saddle point.
2. For the following coupled system x=y y = x(1-x²)+y find all equilibrium points and determine the nature of the linear system around each of the equilibrium points. Draw the null-clines, directions of the trajectories on each null-clines and hence draw the approximate phase plots the the system.
3. Consider the nonlinear system S a' = y y = 4 – a2 + 2y %3D a) Find all critical points. b) For each critical point find the linearized system and determine its type and stability. Then determine type and stability of each critical point for the nonlinear system. c) Sketch the phase portrait near each critical point. Carefully mark sketched solutions with arrows.
Consider the system dx/dt = -x dy/dt = -4x3+y a. show that the origin is the only equilibrium point. b. find the linearized system at the origin. c. classify the linearized system and sketch its phase portrait.
Consider the following system of equations. dx dt dy =y y=x-x²-y dt (a) Find the equilibrium points of the system. smallest x-value (x, y) = largest x-value (x, y) = (x, y) = 5 (b) Using technology, as directed by your instructor, sketch the direction field and phase portrait of the system. (c) Briefly describe the behavior of typical solutions, As t increases, typical solutions spiral in the Select direction Select one of the outside two solutions depending on the initial condition.
Find and classify the critical point of the linear system. dx/dt = -3x-y-19 dy/dt = 13x+y+69
8. For the system below, find all the critical points, find the linearization at each crit- ical point and classify it. Also indicate if they are stable, asymptotically stable, or unstable. a' = (0.6 – 0.01y)x, y = (0.002r – 0.4)y

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