Differential Equations
4th Edition
ISBN: 9780495561989
Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall
Publisher: Cengage Learning
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Chapter 5.1, Problem 7E
a.
To determine
Find the equilibrium points for the given system.
b.
To determine
Find the behavior of the system.at the given point.
c.
To determine
Draw a actualphase portrait for the given system.
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Draw the phase diagram of the system; list all the equilibrium points; determine the stability of the equilibrium points; and describe the outcome of the system from various initial points. You should consider all four quadrants of the xy-plane.
All the following must be included, correct and clearly annotated in your phase diagram: The coordinate axes; all the isoclines; all the equilibrium points; the allowed directions of motion (both vertical and horizontal) in all the regions into which the isoclines divide the xy plane; direction of motion along isoclines, where applicable; examples of allowed trajectories in all regions and examples of trajectories crossing from a region to another, whenever such a crossing is possible.
Draw the phase diagram of the system; list all the equilibrium points; determine the stability of the equilibrium points; and describe the outcome of the system from various initial points. You should consider all four quadrants of the xy-plane.
All the following must be included, correct and clearly annotated in your phase diagram: The coordinate axes; all the isoclines; all the equilibrium points; the allowed directions of motion (both vertical and horizontal) in all the regions into which the isoclines divide the xy plane; direction of motion along isoclines, where applicable; examples of allowed trajectories in all regions and examples of trajectories crossing from a region to another, whenever such a crossing is possible.
Chapter 5 Solutions
Differential Equations
Ch. 5.1 - Consider the three systems (i) dxdt=2x+y (ii)...Ch. 5.1 - Consider the system dx dt=2x+y dy dt=y+x2 (a) Find...Ch. 5.1 - Prob. 7ECh. 5.1 - Prob. 9ECh. 5.1 - Prob. 11ECh. 5.1 - Prob. 13ECh. 5.1 - Prob. 15ECh. 5.1 - If a nonlinear system depends on a parameter, then...Ch. 5.3 - For the system dxdt=Ydydt=x3x (a)show that the...
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- Draw the phase diagram of the system; list all the equilibrium points; determine the stability of the equilibrium points; and describe the outcome of the system from various initial points. You should consider all four quadrants of the xy-plane. All the following must be included, correct and clearly annotated in your phase diagram: The coordinate axes; all the isoclines; all the equilibrium points; the allowed directions of motion (both vertical and horizontal) in all the regions into which the isoclines divide the xy plane; direction of motion along isoclines, where applicable; examples of allowed trajectories in all regions and examples of trajectories crossing from a region to another, whenever such a crossing is possible.arrow_forwardDraw the phase diagram of the system; list all the equilibrium points; determine the stability of the equilibrium points; and describe the outcome of the system from various initial points. You should consider all four quadrants of the xy-plane. Include the coordinate axes; all the isoclines; all the equilibrium points; the allowed directions of motion (both vertical and horizontal) in all the regions into which the isoclines divide the xx plane; direction of motion along isoclines, where applicable. dx dt || 7-y₁ dy dt =x-7.arrow_forwardFor the following two-population system, first describe the type of x- and y-populations involved (exponential or logistic) and the nature of their interaction-competition, cooperation, or predation. Then find and characterize the system's critical points (as to type and stability). Determine what nonzero x- and y-populations can coexist. Finally, construct a phase plane portrait that enables you to describe the long-term behavior of the two populations in terms of their initial populations x(0) and y(0). dx dt dy dt=xy-4y = 5xy-10x CICCES Describe the type of x- and y-populations involved. Select the correct choice below. OA. The populations involved are naturally declining populations in competition. OB. The populations involved are naturally growing populations in cooperation. OC. The populations involved are naturally declining populations in cooperation. OD. The populations involved are naturally growing populations in competition.arrow_forward
- plot phase portrait of the following Nonlinear system. x₁ = 29₂2₂ - XG X₂₂ 1x²2₂² = -296²³-96₂arrow_forwardDraw the phase portraits of the following linear systems and justify the choice of the direction of trajectories. (4.1) * = ( 13 1³ ) *. X, -3 (4.2) = *-(34)x X.arrow_forwardDraw the phase (a) (b) (c) portraits of the following systems, using isoclines +0+0.50=0 +0+0.50=1 +8² +0.50=0arrow_forward
- Interaction of two species of squirrels fiercely competing for the same ecological niche on an island is described by Lotka-Volterra-Gause equations dN1 N1(2 – N1 – 2N2) = f(N1, N2), dt (1) dN2 N2(3 – N2 – 3N1) = g(N1, N2), dt where N1 = N1(t) and N2 = N2(t) are the population densities of the competing species.arrow_forwardFor each of the phase portraits shown below, give a specific example of the possible general solution for the corresponding 2 x 2linear system, and classify the origin as a type of equilibrium point. Explain your process and answer. (Note: There isn't just one correct answer for each phase portrait. Answers will vary, so make sure you explain your choices.) (a) (b) 0- 大 元 (c)arrow_forwardDon't give handwritten answer Thanksarrow_forward
- (3.3) Find the fixed points of the following dynamical system: -+v +v, v= 0+v? +1, and examine their stability.arrow_forwardThe state-space equation of an autonomous linear system is given below. a)Find the equilibrium points of this system and determine the type of equilibrium point. b)Obtain the line equations where the slopes of the orbits are -1,0,1 and ∞. c)Give the correct equations of linear trajectories. d)Draw the phase diagram of the equilibrium points by showing the direction of movement of the orbits with respect to time.arrow_forwardI consider the following system 71=X₂-1 x₂ = x₁²³-X₁-X₂+1 a) find the equilibrium points of the nonlinear system b) plot phase portrait of the system based on Linearizationarrow_forward
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