Chapter 2.2, Problem 4E

Interpretation Introduction

Interpretation:

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

Concept Introduction:

The graph of the given equation could be plotted with the vector field on the real line.

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

If $\dot{\text{x}}\text{ > 0}$, the flow is toward the right, and if $\dot{\text{x}}\text{ < 0}$, the flow is toward the left.

The points where the flow is toward them are called stable points, and the points where the flow is away from them are called unstable points.