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Phase Line Diagrams. Problems
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Differential Equations: An Introduction to Modern Methods and Applications
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- Question 4: Linearize i + 2i + 2x? - 12x + 10 = 0. Around its equilibrium positionarrow_forwardQuestion 8 :SHOW your work. Consider an autonomous differential equation -f(y) for which the graph of f(y) vs y is shown below. (20) -1 a) List the equilibrium solutions of this dt (4-16). (2,4) (4:0)arrow_forwardProblems 8 through 13 involve equations of the form dy/dt = f(y). In each problem sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable, unstable, or semistable. Draw the phase line, and sketch several graphs of solutions in the ty-plane.arrow_forward
- 1) Find all equilibrium solutions of the equation (1 − x) (x² − 4) - x = and classify each one in terms of stability. Draw a phase space diagram and sketch by hand several typical solution curves. Describe the long term (t → ±∞) behavior of the solutions.arrow_forward1. Graph the phase portrait of the system d Ai where A = -7 12 dt 3.arrow_forwardIn each of the following problems, sketch the graph of f(y) versus y, determine the equilibrium solutions, and classify each one as asymptotically stable, asymptotically unstable, or semi-stable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. Here y0 = y(0)arrow_forward
- plot phase portrait of the following Nonlinear system. x₁ = 29₂2₂ - XG X₂₂ 1x²2₂² = -296²³-96₂arrow_forwardFor the Attached problem, Identify the equilibrium values. Which are stable and which are unstable? Construct a phase line. Identify the signs of and Sketch several solution curvesarrow_forward7) In each of the following problems:a. Sketch the Phase Plot of the ODE.b. Determine the equilibrium solutions.c. Classify the equilibrium solutions.d. Draw the phase line and sketch several graphs of solutions on the ty-plane. (7a) y′ = y(y −1)(y −2) , y0 > 0 (7b) y′ = y (1 −y2) , −∞< y0 < ∞. (7c) y′ = y2(1 −y)2, −∞< y0 < ∞. carrow_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_forwardLove Affairs (Strogatz, Nonlinear Dynamics and Chaos, 1994) Let R(t) = Romeo's love/hate for Juliet at time t J(t) = Juliet's love/hate for Romeo at time t What happens when romantic opposites get together? A model for their romance is R = aJ j=-bR+aJ, a and b positive e. If 4b = a i. Write the general solution. ii. Classify the origin. iii. Summarize what happens in their relationship. (Hint: Think about the eigenvectors. Be sure to consider all the qualitatively different possibilities.)arrow_forward1. Consider the model for population growth below. Use a phase line analysis to sketch solution curves for P(t). Determine if the identified equilibrium is stable or unstable. dP —D P(1 — 2Р) dt 2. Model your own Romeo-Juliet problem. Explain your assumptions and show a plot of the numerical solution. You may add a background story if you want to.arrow_forward
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