# Linear Systems equillibrium

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.

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 $$p = \gamma, r = 0$$.