Chapter 7.2, Problem 21P

Consider the autonomous system

$\frac{dx}{dt}=y,\frac{dy}{dt}=x+2x^3$.

(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 lines $\text{y=-x}$ and $\text{y=x}$, 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 $\text{y=-x}$ and $\text{y=x}$ for the linear system.