November 24, 2002

differential equations

If you haven't taken any calculus, it will be a bit harder to keep up with this stuff. This will be a very quick run through of the basic ideas. If you have some sort of engineering degree, you'll probably want to tune in for a later post.

A *function* may be thought of as quantity that depends on another quantity. For example, you can consider you distance travelled on a trip as a function with respect to time. Given a certain time during the trip, the distance function tells you how far you have travelled.

The *derivative* of a function is the rate of change (or slope, if you prefer) of the function with respect to its underlying variable. The rate of change of the distance function, for example, refers to how quickly the distance is changing at a given time, a quantity referred to as speed (if you are only worried about how fast you are going) or velocity (if you are paying attention to what direction you are heading; a negative value would imply that you are headed back towards where you started from).

A *differential equation* expresses a mathematical relation between a function and its derivatives. Instead of distance and speed, suppose we are interested in the growth rate of bacteria in a culture. In our simple model, we will assume that every second, each bacterium splits into two, and none die. So the rate of change of the bacteria population (call that *y'(t)*) depends on the population (cal it *y(t)*) as a function of time (*t*). You could write this as the equation

y'(t) = y(t)

Those of you who know about these things will know that the exponential function ("e to the t") solves this equation.

y(t) = C * exp(t)

Here, the C is introduced as some constant. For any value of C, the differential equation is true. We can determine this constant by applying an initial condition, which is just the value of the function at one point. In this case, that means we have to know the number of bacteria for some value of *t*. If we have 100 at time 0, then C is just 100, and we can call the function 100*exp(t) the solution to the problem.

Some differential equations, given appropriate initial conditions, have a *solution* (meaning a function that satisfies the equation) and others do not. It depends on the form of the equation, and there is all sorts of theory to help determine whether or not a given problem may be solved.

Comments

...still with you... Calc 101-102....

Posted by: chris on November 27, 2002 02:22 AMPost a comment