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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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.
For each of the nonlinear systems below, use technology to draw phase portraits (attach to the solution), determine equillibrium solutions and visually estimate whether they are stable or unstable. (a) x' = In(y + 2) x² + y (b) I' y' = y-x² - 8y² X-
Consider the system of equations dx dt dy dt a (1-33 = x 1 - y) - Y Y y (1 - 12/1 - x), 4 taking (x, y) > 0. (a) Write an equation for the (non-zero) vertical (x-)nullcline of this system: (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g., y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria = (Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).) (c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence: If we start at the initial position (2, 3), trajectories ? the point (Enter the point as an (x,y) pair, e.g., (1,2).)
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.
Which of the following is not true about normal distribution O a. Area under the curve equals 1 O b. The curve is symmetrical about the mean OC.The curve is asymptotic to the x axis O d. The curve is U shaped
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
C. Consider the nonlinear system x' = sin(x) cos(y) y = – cos(x) sin(y) () T/2 i) Verify that and are equilibrium solutions. T/2 ii) Linearize the system at the equilibrium points in part (i). Determine whether the equilibrium point is hyperbolic or not. If it is hyperbolic, deter- mine the whether the equilibrium is a source, sink or saddle. iii) Plot the vector field using a vector field plotter.
was 0 > || and 0< (4)tra, the equilibrium point 0 = X for the system X = AX is if stable focus. Error
dx/dt=x-2y dy/dt=4x-x3 Using the method of linearization,determine the nature and stability of the criticalpoint (0, 0) of the system

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