Chapter 5.1, Problem 2E

Interpretation Introduction

Interpretation:

To show that for the system $\dot{\text{x}}\text{ = ax, }\dot{\text{y}}\text{ = - y}$, where $\text{a < -1}$, all trajectories become parallel to the $\text{y-}$ direction as $\text{t}\to \infty$, and parallel to the $\text{x-}$ direction as $\text{t}\to \text{- }\infty$

Concept Introduction:

A two dimensional linear system is given as

$\begin{array}{l}\dot{\text{x}}\text{ = ax + by}\\ \dot{\text{y}}\text{ = cx + dy}\end{array}$

If the slope $\frac{\text{dy}}{\text{dx}}\text{ = }0$, all trajectories become parallel to the $\text{x-}$ direction.

If the slope $\frac{\text{dy}}{\text{dx}}\text{ = }\infty$, all trajectories become parallel to the $\text{y-}$ direction.