Consider the autonomous system dx dt = x, dy dt = 2y + x3. a. Show that the critical

Chapter 9, Problem 17

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Consider the autonomous system dx dt = x, dy dt = 2y + x3. a. Show that the critical point (0, 0) is a saddle point. b. Sketch the trajectories for the corresponding linear system, and show that the trajectory for which x 0, y 0 as t is given by x = 0. c. Determine the trajectories for the nonlinear system for x _= 0 by integrating the equation for dy/dx. Show that the trajectory corresponding to x = 0 for the linear system is unaltered, but that the one corresponding to y = 0 is y = x3/5. Sketch several of the trajectories for the nonlinear system. 1

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