Semistable Equilibrium Solutions. Sometimes a constant

Chapter 2, Problem 7

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Semistable Equilibrium Solutions. Sometimes a constant equilibrium solution has theproperty 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 theequilibrium solution is said to be semistable.(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 thecorresponding equilibrium 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).