Chapter 6.6, Problem 7E

Interpretation Introduction

Interpretation:

The system $\ddot{\dot{\text{x}}}\text{ + x}\dot{\text{x}}\text{ + x = 0}$ is reversible is to be shown and its phase portrait is to be plotted.

Concept Introduction:

The system $\dot{\text{x}}\text{ = f}\left(\text{x, y}\right)$ and $\dot{\text{y}}\text{ = g}\left(\text{x, y}\right)$ is said to be reversible if $\text{f}$ is even in $\text{x}$ and $\text{g}$ is even in $\text{x}$, that is, if $\text{f}\left(\text{-x, y}\right)\text{ = f}\left(\text{x, y}\right)$ and $\text{g}\left(\text{-x, y}\right)\text{ = -g}\left(\text{x, y}\right)$.

The trajectories of the dynamic system are called phase portrait.

The phase portrait gives the qualitative behavior of the solutions of a differential equation.