Consider the equationdy/dt = a y2. (ii)(a) Find all of the

Chapter 2, Problem 25

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Consider the equationdy/dt = a y2. (ii)(a) Find all of the critical points for Eq. (ii). Observe that there are no critical points ifa < 0, one critical point if a = 0, and two critical points if a > 0.(b) Draw the phase line in each case and determine whether each critical point isasymptotically stable, semistable, or unstable.(c) In each case sketch several solutions of Eq. (ii) in the ty-plane.(d) If we plot the location of the critical points as a function of a in the ay-plane, we obtainFigure 2.5.10. This is called the bifurcation diagram for Eq. (ii). The bifurcation at a = 0is called a saddlenode bifurcation. This name is more natural in the context of secondorder systems, which are discussed in Chapter 9.