Differential Equations Govern the Future

Mathematics is the language used to quantify anything in science.  Still, some aspect of the tool we call math is utilized in nearly every field from business, to sales, and of course, engineering.

The field of Mathematics may be divided into several branches like calculus, linear algebra, and statistics.  While all of these subjects are fascinating, let us focus on calculus, which can be summarized as "the study of change."  If no facet of the universe ever changed, we would have no use for calculus, but then again, such a universe could not support life altogether (life cannot exist without chemical reactions).

At its heart, calculus focuses on functions, which are equations describing how variables are related to one another.  The simplest kind of function consists of two variables; one variable is dependent, the other is independent.  If a taxi driver charges a customer $1.50 per minute, then the cost function for riding in the taxi would be C = 1.5t, where C is in dollars and t is in minutes.  Here, time is an independent variable, and cost is a dependent variable (as the cost depends on the duration of the cab ride).  In other words, cost is a function of time, or C = f (t).

The simple taxi function given above relates cost to time, but functions can describe the relationships between other dependent variables and time.  We could, for example, consider the temperature of a hot cup of coffee.  One can imagine that the coffee's temperature value would decrease as time goes on until it reaches the room's air temperature.  The key difference between the taxi function and the coffee function, is that the function for the ride in the taxi was specified by the taxi driver.  No one specified the temperature function of the coffee.  The coffee's temperature function is the result of a differential equation, which is also known as a governing equation - a law of nature, and in the case of the coffee, thermodynamics.