# 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 *

*and treat *

*in a similar way. We can then convert the differential equations into:*

*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.