Chapter 6.4, Problem 8E

Interpretation Introduction

Interpretation:

To show that if the initial conditions satisfy ${\text{x}}_0{\text{+ y}}_0\text{ = 1}$, then $\text{x}\left(\text{t}\right)\text{ + y}\left(\text{t}\right)\text{ = 1}$ for all $\text{t}$, for the system ${\dot{\text{x}}\text{ = ax}}^{\text{c}}\text{- }\varphi {\text{x, }\dot{\text{y}}\text{ = by}}^{\text{c}}\text{- }\varphi \text{y}$ where $\varphi {\text{ = ax}}^{\text{c}}{\text{+ by}}^{\text{c}}$. To show that all the trajectories starting in the positive quadrant are attracted to the invariant line $\text{x + y = 1}$. To draw the phase portraits in the $\left(\text{x, y}\right)$ plane for $\text{c = 1, c > 1,}$ and $\text{c < 1}$.

Concept Introduction:

It can be shown that $\text{x}\left(\text{t}\right)\text{ + y}\left(\text{t}\right)\text{ = 1}$ for all $\text{t}$, by adding the given equations.

The Jacobian matrix at a general point $\left(\text{x, y}\right)$ is given by

$\text{J = }\left(\begin{array}{cc}\frac{\partial \dot{\text{x}}}{\partial \text{x}}& \frac{\partial \dot{\text{x}}}{\partial \text{y}}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}& \frac{\partial \dot{\text{y}}}{\partial \text{y}}\end{array}\right)$