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Differential Equations: An Introduction to Modern Methods and Applications
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- Please answer part Barrow_forward1) 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_forwardPlease help me solve this problemarrow_forward
- [14] For each of the following equations, find all equilibria; • find general solutions; • solve the initial value problem with initial condition ₁ (0) = 2, x₂(0) = 1; • sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. 10 x1 1 1 - 3-4 963 = (b) Q]=[4][B]+[B] x2 -5 -7 3 (a)arrow_forward[6] An equation dt = f(y) has the following phase portrait. 2 Y (a) Find all equilibrium solutions. (b) Determine whether each of the equilibrium solutions is stable, asymptotically stable or unstable. (c) Graph the solutions y(t) vs t, for the initial values y(1.4) = 0, y(0) = 0.5, y(0) = 1, y (0) = 1.1, y(0) = 1.5, y(-0.5) = 1.5, y(0) = 2, y(0) = 2.5, y(0) = 3, y(0) = 3.5, y(0) = 4, y(0) = 4.5, y(-1) = 4.5. (Without further quantitative information about the equation and the solution formula, it's clearly impossible to draw accurate graphs of y(t) vs t. Here, try to sketch graphs qualitatively to show the correct dynamic properties. The point is that a great deal of info about solution dynamics can be read off from one simple figure of phase portrait.)arrow_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
- Please solve & show steps...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_forward17arrow_forward
- Phase Line Diagrams. Problems 1 through 7 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 or unstable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. 1. dy/dt = y(y - 1)(y-2), yo≥ 0arrow_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_forwardProblems Problems 1 through 4 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 or unstable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. G 1. dy/dt = ay+by2, a> 0, b>0, -∞0 1. The phase line has upward-pointing arrows both below and above y = 1. Thus solutions below the equilibrium solution approach it, and those above it grow farther away. Therefore, o(t) = 1 is semistable. c. Solve equation (19) subject to the initial condition y(0) = yo and confirm the conclusions reached in part b. y k o(t) = k y viszoq #alo odw to nothogong these o(t) = k k t (a) (b) SS FIGURE 2.5.9 In both cases the equilibrium solution (t) = k is semistable. (a) dy/dt ≤0; (b) dy/dt > 0. Problems 6 through 9 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,…arrow_forward
- Discrete Mathematics and Its Applications ( 8th I...MathISBN:9781259676512Author:Kenneth H RosenPublisher:McGraw-Hill EducationMathematics for Elementary Teachers with Activiti...MathISBN:9780134392790Author:Beckmann, SybillaPublisher:PEARSON
- Thinking Mathematically (7th Edition)MathISBN:9780134683713Author:Robert F. BlitzerPublisher:PEARSONDiscrete Mathematics With ApplicationsMathISBN:9781337694193Author:EPP, Susanna S.Publisher:Cengage Learning,Pathways To Math Literacy (looseleaf)MathISBN:9781259985607Author:David Sobecki Professor, Brian A. MercerPublisher:McGraw-Hill Education
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