Differential Equations

While a derivative can be seen as a number (perhaps with units) representing the rate of change of a function when an independent variable changes, it can also be seen as a function itself.

Thus, if we have a function $f(x)$, then its derivative, $df/dx$ creates a new function, since there is a value of the derivative for every value of $x$. In math, this new function is often called $f'(x)$. In physics, we are of course familiar with this since taking the derivative of the position, $x(t)$, leads to a velocity as a function of time, $v(t) = dx/dt$, and taking the derivative of the velocity, $v(t)$, leads to an acceleration as a function of time, $a(t) = dv/dt$.  

Thinking of the derivative as a function means that we can have a new kind of equation where the derivative of a function may depend on the function itself. This leads to a differential equation. These are powerful tools for figuring out what functions look like. In science classes, we often know the mechanisms that cause change and these can lead to differential equations that tell us how a system will behave.

Differential equations are widely used in more advanced science courses, including in physics, chemistry, biology, and economics. Two places where you will see them (if you haven't already) are in chemistry (rate equations) and in biology (population dynamics).

An example of a differential equation from epidemiology

Let's see how we can build differential equations from "thinking about the physics" (or in this case, from the study of the propagation of disease).

The study of how disease spreads is called epidemiology and relies on differential equations from population dynamics. The simplest model for the spread of an infection is to assume that each organism infected has the same probability for infecting an additional organism, and that each infected organism has the same probability for getting well (or dying and being removed from the pool of organisms spreading the infection).

If $I(t)$ is the number of people infected at a given time, the rate of change in the number of cases looks like

$$\frac{dI}{dt} = (A-B)I$$

where $A$ is the rate that an infected organism infects another and $B$ is the rate at which it is removed from the infected pool by dying or getting better.

This equation tells us a lot about how $I(t)$ changes. If $A > B$, and we have a small number infected, then $dI/dt$ is positive and the number infected will grow. Since its growth rate is proportional to $I$, $I$ will increase its rate of growth with time, growing faster and faster as it gets larger and larger. This is exponential growth — faster than any power of t. (See Powers and exponentials.) If $A < B$, then $dI/dt$ is negative, and the number infected will fall.

Although this model is overly simplified, it gives us insight into what happens and how it depends on the parameters. This is why medical personnel make such an effort to isolate a patient with a highly communicative disease -- to reduce the value of $A$, hoping to turn the overall constant $A-B$ negative. This can make the difference between an infection disappearing after a few cases and a plague spreading out of control.  

An example from ethology: the relation of predator and prey

A next stage in building mathematical models using differential equations comes by coupling two variables and their derivatives. A lovely example is the Lotka-Volterra equation for the interaction of a predator and prey. This is a toy model that focuses on the relation of only two organisms: a predator and its prey. While you can easily come up with obvious corrections to this model, it provides good insights into some basic ecological and evolutionary mechanisms.

In this case, we have two variables that are functions of time: the number of predators, $x$, and the number of prey, $y$.

If there are no predators, the prey will grow in proportion to their numbers, so we would have $\frac{dy}{dt} = Ay$.

If there are no prey, the predators will die out and we might assume also at a constant rate in proportion to their number, $\frac{dx}{dt} = -Bx$. If there are both, the simplest model says the prey decrease in proportion to the product of predator and prey and the predators grow in proportion to the same product. This argument yields the model

$$\frac{dx}{dt} = Ax + Cxy$$

$$\frac{dy}{dt} = Bx - Dxy$$

where A, B, C, and D are (positive) constants that serve as parameters for the model. They are adjusted to fit the data (if possible).

These equations are much harder than our infection model since there are two variables and there is a non-linear ($xy$) term. Even this clearly oversimplified model has lots of interesting predictions and can be surprisingly good. It modeled the oscillatory populations of Canada lynx and snowshoe hares as measured by the number of furs sold to the Hudson's Bay Company in thelate  19th and early 20th centuries.

An example for physics: Newton's 2nd law

The critical equation for this course, Newton's second law, is also a differential equation. It says that the acceleration of an object is proportional to the sum of the forces the object fields, divided by its mass. Though we write is as 

$$\vec{a} = \frac{\vec{F^{net}}}{m}$$

to emphasize the physical meaning of the equation, but because the forces are typically dependent on (a function of) the object's position, we can also consider this as a differential equation, 

$$\frac{d^2x}{dt^2} = f(x)$$.

For this class, we will primarily solve equations like this numerically on a spreadsheet, using the fact that if we know a starting value of a function and how that function changes, we can find out what it is by taking small steps. (See the stepping rule section of the page What is a derivative, anyway? or Newton 2 as a stepping rule for an example of how this works.)