(a) Show that (0, 0) is an isolated critical point of the plane autonomous system

Chapter 11, Problem 40

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(a) Show that (0, 0) is an isolated critical point of the plane autonomous system Parametric equations for a folium are 3ct 3ct2 x' = x4 - 2xy3 y' = 2x3y - y4 x = 1 + t3' y = 1+t3 [Hint: The differential equation in x and y is homogeneous.] but that linearization gives no useful information about the nature of this critical point. (b) Use the phase-plane method to show thatx3 + y3 = 3cxy. This classic curve is called a folium of Descartes. (c) Use a graphing utility or a numerical solver to obtain solution curves. Based on your phase portrait, would you classify the critical point as stable or unstable? Would you classify the critical point as a node, saddle point, center, or spiral point? Explain.