Solved: Semistable Equilibrium Solutions. Sometimes a constant equilibrium solution has
Chapter 2, Problem 7(choose chapter or problem)
Semistable Equilibrium Solutions. Sometimes a constant equilibrium solution has the property that solutions lying on one side of the equilibrium solution tend to approach it, whereas solutions lying on the other side depart from it (see Figure 2.5.9). In this case the equilibrium solution is said to be semistable.FIGURE 2.5.9 In both cases the equilibrium solution (t) = k is semistable.(a) dy/dt 0; (b) dy/dt 0.(a) Consider the equationdy/dt = k(1 y)2, (i)where k is a positive constant. Show that y = 1 is the only critical point, with the correspondingequilibrium solution (t) = 1.(b) Sketch f(y) versus y. Show that y is increasing as a function of t for y < 1 and alsofor y > 1. The phase line has upward-pointing arrows both below and above y = 1. Thussolutions below the equilibrium solution approach it, and those above it grow farther away.Therefore (t) = 1 is semistable.(c) Solve Eq. (i) subject to the initial condition y(0) = y0 and confirm the conclusionsreached in part (b).
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