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So, I can model growth and decay if I start with assuming that the growth rate is constant:

$\frac{p'(t)}{p(t)}=\alpha$

and then I have

$p'(t)-\alpha p(t)=0$

A general linear differential equation, however, would have the form

$p'(t)-g(t) p(t)=h(t)$

So the growth rate is g(t). What is h(t)?
And what type of thing is this form of equation used to model?

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3 Answers 3

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You can read it like this: $p'(t)= \alpha p(t)+g(t)$. Then the growth is "caused" by "reproduction" plus "immigration", for example.

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So if I had something like $p'(t)=g(t)p(t)+h(t)$, g(t) might be my reproductive rate, and g(t) might be my death rate? –  Korgan Rivera Sep 9 '12 at 17:37
    
$\alpha$ could be the birth rate minus the death rate, and $g (t)$ could be immigration minus emigration. If more people are born than those who die, i.e., $\alpha > 0$, we have a demographic explosion. If more people immigrate than emigrate, then $g (t) > 0$. –  Rod Carvalho Sep 9 '12 at 17:51

I know you asked about first order, but: Second order homogeneous ODEs, e.g. $a\ddot{x} + b\dot{x} + cx = 0$, describe "unmolested" damped/driven harmonic oscillation. Introducing a function on the right hand side, e.g. $a\ddot{x} + b\dot{x} + cx = f(t)$, describes damped/driven harmonic oscillation with $f$ describing an external input. The part of the ODE which does not involve a dependent variable describes "outside influences". I am quite confident that this is the same for any order of ODE.

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If $p'(t) = g(t)p(t)+h(t)$ and $g(t) = k$ for some constant $k$, then this form is also the simplest case of feedback control of a linear time invariant system with a state term, $p(t)$, and a control term, $h(t)$.

If $g(t)$ is not constant, then you have a controlled system where the system parameters change with respect to time.

In the time invariant case, you might describe an elevator control with a single state ($z$-position) and a single control term (motor speed). In the time varying case, then you have something directly affecting the $z$-position, for instance if the elevator isn't fixed to the cable, but starts sliding down the shaft with some velocity $g'(t)$.

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