Classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. (Order your answers fromsmallest to largest x, then from smallest to largest y.)x' = xy - 4y - 5y' = y² = x²smaller x-valueX(x, y) =-1.-1T =TraceDeterminantA =XT² - 4A =DiscriminantConclusionunstable spiral point ✓Xlarger x-valueX(x, y) =5,5TraceT =XDeterminantΔ =XDiscriminantT²-4A =Conclusionunstable spiral point

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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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Classify (if possible) each critical point of the given plane autonomous system as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. (Order your answers from smallest to largest x, then from smallest to largest y.) x' = 6x − y2 y' = −y + xy
a.Find all the critical points (equilibrium solutions). b.Use an appropriate graphing device to draw a direction field and phase portrait for the system. c.From the plot(s) in part b, determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. b.Describe the basin of attraction for each asymptotically stable critical point. 9.dx/dt=(2+x)(y−x),dy/dt=y(2+x−x2)
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.
a.Find all the critical points (equilibrium solutions). b.Use an appropriate graphing device to draw a direction field and phase portrait for the system. c.From the plot(s) in part b, determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. b.Describe the basin of attraction for each asymptotically stable critical point.11.dx/dt=(2−x)(y−x),dy/dt=y(2−x−x2)
Sometimes a constant equilibrium solution has the property that solutions lying on one side of the equilibrium solution tend to approach it, whereas solutions lying on the other side depart from it. In this case the equilibrium solution is said to be semistable. Classify the critical (equilibrium) points as asymptotically stable, unstable, or semistable. dy/dt=y^2(7−y^2), −∞<y0<∞
Find the linearization of this system near critical point, by using the Jacobian.Determine the nature of the critical point, and sketch the solution curves.
use the system of equations in obtaining the critical points
Find all the critical points (equilibrium solutions). b.Use an appropriate graphing device to draw a direction field and phase portrait for the system. c.From the plot(s) in part b, determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. b.Describe the basin of attraction for each asymptotically stable critical point. 4.dx/dt=x−xy,dy/dt=y+2xy
Use Cramer's Rule to determine equilibrium solution for two interdependent commodities.
Define a new interpretation of the description “dynamical system” for a collection of interdependent differential equations?
Consider a linear system: x`= ux + y, y`= -x + y, where u is a real constant. When is (0,0) an unstable saddle point? 2 when is (0,0) an unstable spiral point?
Find the general solution in terms of real functions. (b) From the roots of the characteristic equation, determine whether each critical point of the corresponding dynamical system is asymptotically stable, stable, or unstable, and classify it as to type. (c) Use the general solution obtained in part (a) to find a two-parameter family of trajectories x=x1i+x2j=yi+y′j of the corresponding dynamical system. Then sketch by hand, or use a computer, to draw a phase portrait, including any straight-line orbits, from this family of trajectories.

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