Chapter 2.2, Problem 3E

Interpretation Introduction

Interpretation:

The equation ${\dot{\text{x}}\text{ = x - x}}^{\text{3}}$ is to be analyzed graphically; the vector field is to be sketched on the real line; all fixed points are to be determined; their stability should be classified, and graph of $\text{ x(t)}$ is to be sketched for different initial conditions. Also, if possible, the analytic solution for $\text{ x(t)}$ is to be obtained.

Concept Introduction:

If $\dot{\text{x}}\text{ > 0}$, the flow is towards the right, and if $\dot{\text{x}}\text{ < 0}$, it is towards the left. $\dot{\text{x}}\text{ = 0}$ represents no flow.

Fixed points are the points where $\dot{\text{x}}\text{ = 0}$.

Stable points are points at which the local flow is toward them. They represent stable equilibria at which small disturbances damp out in time away from it.

Unstable points are points at which the local flow is away from them. They represent unstable equilibria.