I have the following system describing a free-market economy:
$\frac{dp}{dt}=\alpha r,\qquad \frac{dr}{dt}=-\beta r$
Where $\alpha$ and $\beta$ are positive constants, $p$ is the price and $r$ the difference between demand and supply. Describe the nature of the fixed points and determine whether the price tends to these fixed points.
Is the best way to approach to consider solving for $p(t)$ and noting that $\alpha$ is $>0$ and so if the difference $r$ is positive as $t\to\infty$ then our $p\to\infty$ and vice versa for all the other possibilities. Many thanks in advance.