Separation of variables

Consider population growth. Even if women have on average an unchanging number of children each year, the population will still grow (or shrink). The rate of population growth is proportional to the number of people in the population. So a simple model is:

$$\dot{p} = k p.$$

$k$ is a constant that might be positive or negative.

This can also be expressed as

$$\frac{dp}{dt} = kp.$$

We can view this equation as one that relates small changes in $dp$ and $dt$, and so, take apart those pieces, and re-arrange them. For example, like so...

$$\frac{dp}{p} = k\,dt.$$

With this kind of equation--all quantitities related to $p$ on one side and $t$ on another--we can integrate both sides:

$$\begineq \int_{p_0}^{p}\frac{1}{p'}dp' &=& \int_{t_0}^t k\,dt'\\ \[\ln p'\]_{p_0}^{p} &=& \[kt'\]_{t_0}^{t}\\ \ln p - \ln p_0 = \ln p/p_0 &=& k(t-t_0)\endeq $$

Now, raising $e$ to the power of both sides of the equation

$$p/p_0 = e^{k(t-t0)}$$

Using $p_0 = p(t=t_0)$, and $p=p(t)$, we can re-arrange this to

$$p(t) = p(t_0)e^{k(t-t0)}.$$

Exponential growth (or decay) of a population.

Problem- throwing with no air resistance

Just to re-assure yourself that this process of separation of variables works in some situation where you already know the answer... try using it to derive the height of a ball thrown straight upwards with speed $v_0$ at $t=0$ with *no air resistance* (just gravity), starting from $F=ma\to -g=dv/dt$ to get $h(t)=h_0 +v_0t-\frac{1}{2}gt^2$.

First write down Newton's law in terms of $v$ and/or $\dot{v}$, separate and integrate to get $v(t)$. Then change $v\to dh/dt$ and separate and integrate again to get $h(t)$.