The secret of all life: Differential equations
When British mathematician and educator David Acheson was in elementary school, his biology teacher, Ms. H., always gave the same test questions each week. Question 23 was What is the secret of all life? Her answer was chlorophyll. His answer is differential equations.
Most textbooks give you a definition for differential equations, but they don’t show you how to build one “in the wild” so to speak. I have my own demonstrations, but none are as good as what Professor Acheson provides in his pithy, charming book on many things math – 1089 + ALL THAT = A JOURNEY INTO MATHEMATICS.
He begins with a helpful statement of what differential equations do:
Differential equations are statements, essentially, of how a system will change after a short period of time. They tell us, in other words, how different a system will be a short time later.
The statement above tells us to do two things: 1. Choose a system to study; 2. Choose aspects, parts, variables of the system that change. For us, it is natural that the system be a brain, but to simplify things as much as possible, we shall study a brain on a spring:
The two aspects of the brain on a spring we will study are y – the height of the brain above its equilibrium point, and v – the brain’s upward velocity.
In calculus, we study how some quantity y changes with time t, and we learn to deduce the rate at which y changed with t, denoted as
But here, as with other differential equations, it is all the other way around. In the equations of motion for the [brain] we have some rather obscure information about the rates of change and our task is to deduce how y and v depend on time t.
To see the general idea, note first that at time t the [brain] is at height y and has upward velocity v. And a little later, at time t + δt, its height and upward velocity will have changed slightly to y + δy and v + δv. (Remember that δ, which denotes “a little bit” of some quantity, can be as small as we need it to be. This is where we are headed, to a demonstration, via a computer simulation, of what happens when we make δ smaller and smaller.)
Suppose, then, that we approximate
These are formulae for the small changes in y and v that occur after a small increase in time of amount δt.
And, most importantly, they allow us to write the ‘new’ values of y and v ( i.e. y and δy and v and δv in terms of the ‘old’ ones:
To visually see what happens as the values of δ change, go to Dr. Acheson’s website: http://home.jesus.ox.ac.uk/~dacheson/index.htmland click on 1089 and All That. You can download a program in BASIC which will allow you to experiment with different values of δ.
For those readers whose gag reflex is particularly active when near maths, I say unto you, relax and give it a try. There is a path just for you into an enchanting world. For me it has been the realization that mathematics is a language. An equation is a poem, a mathematical theme is a drama with no beginning and no end. The notation is often very beautiful. The mind is at play. Find your way.