9. dy/dt = y²(y² - 1), -∞ < Yo <∞
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- Kk.425. Sketch the time dependent solutions based on the on-dimensional phase line of:y and at dy dx = -Y. Problem 5. Consider the dynamical system given by = 3x Draw the phase portrait of this dynamical system. You have to justify how you got to the phase portrait.For the following problems: ii. Identify the equilibrium values Construct a phase line. Identify the signs of y' and y". Sketch several solution curves. a. = (y + 2)(y - 3) dx b. = y²-2y dx
- In 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)Help with parts a through e, having some difficulty with the phase lines.For 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 curves
- a. Identify the equilibrium values. Which are stable and whichare unstable?b. Construct a phase line. Identify the signs of y' and y''.c. Sketch several solution curves. y' = y - √y, y > 0In each question, draw the phase diagram (i.e. the graph with stationary points and arrows), determine the stationary points, and classify each one as stable, unstable or seminstable (by considering the sign of the derivative of the right-hand side). 1. i = -x + 1 2. i = 2x – x² 3. i = -x(1+x)(2 – x) 4. i = x? – x4 5. i = x²(4 – x²)determine the critical (equilibrium) points, and classify each one asymptotically stable, unstable, or semistable (see Problem 5). Draw the phase line, and sketch several graphs of solutions in the ty-plane. dy/dt=y2(1−y)2,−∞<y0<∞
- 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≥ 0[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.)Problems 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,…