# First Order ODE Stability

1. Sep 9, 2011

### TheBestMilk

1. The problem statement, all variables and given/known data
In the market for a certain good, the price p(t) adjusts continuously in the presence of excess supply or demand:
$\frac{dp}{dt}$ = F(D(p)-S(p)) where F(0) = 0, F'>0.
Obtain the condition for stability of the equilibrium price p* in terms of the slopes D'(p*) and S'(p*), and illustrate stable and unstable equilibria graphically for straight-line supply and demand schedules.

2. Relevant equations
General I suppose, p' = F(p)

3. The attempt at a solution
This is what I've come up with so far, but I'm not sure if it's correct or delves deeply enough:
p'=F(D(p)-S(p))
We know there would be an equilibrium point where p'=0 and therefore where F(D(p)-S(p))=0.
Since F(0) = 0, we know that at least one equilibrium exists at p=0, so p*=0. From that, we can see that p'=0=F(D(0)-S(0)) and therefore at p*, D(0) = S(0).

To test stability we would look at the sign of p'' at p*.
p'' = F'(D(p*)-S(p*))(D'(p*)-S'(p*))
Since we're given F'>0, we know that the D'(p*)-S'(p*) dictates stability.
So if D'(p*) > S'(p*) then p'' >0 and p* is unstable.
Else, if D'(p*) < S'(p*) then p'' <0 and p* is stable.

That's what I've come up with. It seems a bit simple which leads me to believe I may be missing something. Any help or insight would be greatly appreciated.

Thanks!