Chapter 2.2, Problem 5E

Interpretation Introduction

Interpretation:

The equation $\dot{\text{x}}\text{ = 1+}\frac{\text{1}}{\text{2}}\text{cosx}$ 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 the 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:

For the given equation, agraph is to be plotted with the vector field on the real line.

All the fixed points are to be found. Fixed points are the points where $\dot{x}=0$.

When the vector field plotted on the real lines flows towards the fixed points, they are called stable points or attractors or sink.

When the vector field plotted on the real lines flows away from the fixed points, they are called unstable points.