Consider the autonomous system dx dt = y, dy dt = x + 5x

Consider the autonomous system dx dt = y, dy dt = x + 5x3 . (a) Show that the origin is the only critical point. (b) Find the approximating linear system near the origin. Write down the corresponding equation for dy?dx and solve it. Sketch the trajectories of the linear system and observe that the origin is a saddle point. From the parametric form of the solution, show that the trajectories that enter and leave the origin lie on the lines y = ?x and y = x, respectively. (c) Now consider the original nonlinear equation. Write down the corresponding equation for dy?dx and solve it. Sketch the trajectories for the nonlinear system that correspond to y = ?x and y = x for the linear system.