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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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- [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_forward5arrow_forward17arrow_forward
- (a) sketch the nullclines, (b) sketch the phase portrait, and () write a brief paragraph describing the possible behaviors of solutions. dx =x(-4x – y + 160) dt dy = y(-x? - y² + 2500) dtarrow_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_forwardProblem 2. Find all of the equilibrium solutions for the following difference equation It+1 = rx(1 – æ), 0arrow_forwardFind the general solution and sketch the phase portrait of z = ( 1 1 Z 4 -2arrow_forwardFind the amplitude, phase angle, and period of the motion governed by the initial value problem x¨+4x = 0, x(0) = 1, x˙(0) =−2arrow_forwardQUESTION 11 Let D (x) =x2– 12x+ 36. 3DX Find the consumer surplus at the equilibrium x=3. E $ 25 $ 12 $ 18 $ 36arrow_forwardPlease help solve differential equation problem attached. How to solve and draw the phase line, and sketch several graphs of solutions in the ty-plane and determine the equilibrium points and state if each is asymptotically stable, semistable, or unstable?arrow_forwardChoose one an unstable equilibrium solution. an asymptotically stable equilibrium solution. no equilibrium solution at all. a semistable equilibrium solution.arrow_forwardFf.208.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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