Chapter 7.2, Problem 22P

Consider the autonomous system

$\frac{dx}{dt}=x,\frac{dy}{dt}=-5y+x^3$

(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 $\text{dy/dx}$ 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 $\text{y}$-axis and the $x$-axis, respectively.

(c) Now consider the original nonlinear equation. Write down the corresponding equation for $\text{dy/dx}$ and solve it. Sketch the trajectories for the nonlinear system that correspond to $\text{x=0}$ and $\text{y=0}$ for the linear system.