Monday, October 3, 2011

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.

An ordinary differential equation relates a dependent variable to an independent variable and to its own derivatives of various orders.  A derivative of a dependent variable describes its rate of change with respect to an independent variable - this is the first derivative (order one).  The second derivative of a dependent variable is the rate of change of its rate of change with respect to an independent variable (order two).

The origin of all differential equations in science come from the laws of nature.  The law of nature describing the motion of a body is Newton's Second Law, Fnet = ma.  When this relationship is first introduced to high school physics students, it is not described as a differential equation, although in reality, it is one.  The reason that this is a differential equation, is because acceleration, a, is the second derivative of position (velocity is the first derivative of position).  The solution to Newton's equation of motion is a function: the position of a body as a function of time, x = f (t).  This function is of course situation-dependent: it depends on the forces that act on the mass, and the value of the mass itself.

Let us try to find the vertical position function (altitude) for a ball dropped off of a rooftop, y = f (t).  If we ignore the aerodynamic force that acts on the ball, its differential equation of motion becomes very simple: a = - 9.8 m/s2, or simply, y'' = - 9.8, where each prime denotes a derivative with respect to time.  Integrating both sides once, we get y' = -9.8t + c1.  Integrating both sides once more, we get the general solution we are in search of: y = -4.9t2+c1t+c2, where c1 and c2 are unknown constants. 

In order to solve for those unknown constants, we need to know the initial conditions of the body; that is, we need its initial vertical position and velocity.  If the building has a height H, then f(0) = H.  As the ball is dropped in this problem, its initial velocity is zero, so f'(0) = 0.  Using these two initial condition, we can get the particular solution for the vertical position of the ball as a function of time: y = -4.9t2 + H.  We could not have arrived at our function without the initial conditions, and that makes perfect sense: you cannot know where you are going if you don't know where you are coming from.  And, it is no coincidence that we needed two initial conditions to arrive at our particular solution: the governing equation was a second-order differential equation.

The position function derived above tells us the height in meters of a ball dropped  from a building that is H (meters) tall at any time t (seconds) after it is dropped (ignoring aerodynamic effects).  This result illustrates the power of differential equations.  A law of nature defines how a dependent variable must behave.  If we solve the differential equation that defines this behaviour, we arrive at a solution.  Using known laws of nature, one can do all of this without doing any experimentation. 

If we can solve a governing differential equation for any particular situation where the initial conditions are known, we can predict its future before it occurs.  As such, fortune-telling may be done with a pen and paper rather than a crystal ball.

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