Chapter 12.5, Problem 15E

To determine

(a)

To show:

The given system,

$\begin{array}{rll}\frac{dx}{dt}& =& -y+xf\left(x,y\right)\\ \frac{dy}{dt}& =& x+yf\left(x,y\right)\end{array}$

Is asymptotically stable if $f\left(x,y\right)<0$ in domain $D$ where $f$ be continuous and have continuous first partial derivatives in $D$ containing the origin.

To determine

(b)

To show:

The given system,

$\begin{array}{rll}\frac{dx}{dt}& =& -y+xf\left(x,y\right)\\ \frac{dy}{dt}& =& x+yf\left(x,y\right)\end{array}$

Is unstable if $f\left(x,y\right)>0$ in domain $D$ where $f$ be continuous and have continuous first partial derivatives in $D$ containing the origin.