# Linear Systems equillibrium

1. Apr 7, 2006

### Mynock

Hey, I'm trying to find two linear functions, f(r) and h(r) for the following system:

dp/dt = A*f(r)

dr/dt = -B*h(r)

where A and B are constants greater than zero. I'm trying to find linear functions that will tend to equillibrium, and also where

limit df(r)/dt = 0
r->0

I have been trying various linear functions and have been unable to come up with a solution. Is there a solution? and if so, what would be one? It's probalby something simple that I'm overlooking. Any help would be appreciated. Thanks.

2. Apr 8, 2006

### leach

The linear (homogeneous) functions in one variable, 'r' in this case, have the general form:

$$f(r) \ = \ \alpha\cdot r,\quad g(r) \ = \ \beta\cdot r, \qquad \alpha,\beta\in\mathbb{R}$$

So this system can be trivially solved, since the variables are separated. On one hand, you have:

$$\dot{r}\ = \ -B\beta\, r$$

so

$$r(t) \ = \ r_0e^{-B\beta t}$$

The condition for stability is obviously $$\beta > 0$$.

On the other hand:

$$\dot{p}\quad = \quad A\alpha\, r \quad = \quad A\alpha r_0 e^{-B\beta t}$$

Then,

$$p(t) \quad = \quad \gamma \ - \ \frac{A\alpha r_0}{B\beta} e^{-B\beta t}$$

where $$\gamma$$ is an integration constant. This means that, as long as $$\beta > 0$$, your system will be stable, with equilibrium point $$p = \gamma, r = 0$$.