Bang Out of Order

It does not stop there, though.

The "last post" considered only homogeneous linear differential equations (with constant coefficients) of the first order.

What about second order?

In order to explain it is necessary to recap just what a "differential" is.

The question arose because people wanted to find out how steep hills are.

How steep are they? Well, we say 1 in 10 or 1 in 4, etc, meaning that for every meter we move up vertically, we have to travel 10 or 4 meters along the road to get there. Like this:



In mathematics, we have to be more precise and say not "along the road" but " horizontally".

This is the basis of all gradients. In the diagram below, k is the gradient of each line:



BUT being mathematics, straight lines were too restrictive, as most graphs are actually curved. Enter the CALCULUS.

Calculus gave us the way to find the gradient at any point on a curve, by measuring the gradient of the "tangent" at that point:

The procedure was quite simple. Take one of the simplest curves, a parabola, whose curve is described by y = x^2



Notice how the tangents at x = 1 and x = 2 have gradients of 2 and 4 respectively. Almost like the gradients are twice the x-value of the points.

In fact, that is exactly what is happening. For every point x on y = x^2, the gradient is 2x.

By analogy, for:

y = x^3, the gradient of every point is 3x^2

y = x^4, the gradient of every point is 4x^3.

....there is obviously a pattern here.

The pattern is this:

For every curve y = x^n, the gradient for every point on these curves is (n-1)x^(n-1).

The entire process is called "differentiation", and the end result is called "the first order differential".

The math peeps didn't stop there.

What if you apply the process again?

I mean what happens if you "differentiate" the "differential"?

For example: if y = x^5, the "first order" differential is 5x^4, by the rule above.

Let's call this D1

If we differentiate 5x^4 we get 20x^3, which is called the "second order" differential.

Let's call this D2.

But what happens if we know neither D1 nor D2, nor even y?

We have a "second order differential equation", which looks like;

A*D2 + B*D1 + C*y = 0

(A, B and C are just numbers)

The whole process of doing the reverse hinges on finding a version of y that, when differentiated twice, is proportional to y itself. In principle , it is quite simple.

That may all be very weLL, but what does it all mean?

Most of math is about abstract concepts, and this seems like no exception, but in fact, this concept has applications throughout the whole of science.

More about that next time.

NOW we start!

This blog is rapidly approaching its 4th birthday, and as no-one reads it, there is no reason why I should not get a little more involved with the science.

Here is the first post, as opposed to the last post.

What is a differential equation? Those few who have read before might have had a taste of the Calculus, which describes mathematically the rate at which things increase (or decrease) using stuff called "differentials". (look it up in the index under "maths", May 28, 2007).

Here is the simplest:



There are a number of terms here that need to be identified:

f(t) = a "function"; a mathematical way of explaining something (example: 2t + 1 is always going to equal 3, if t is one, or five if t is two, etc. This makes "2t + 1" a "function")

So what is "y"?

"y" is the end result, and the be-all and end-all of this thing is to get a "y = ......"

What is "dy/dt"? It is pronounced "dee-why by dee-tea", and is the rate at which "y" changes as "t" does.

Since we do not know what "y" is yet, these equations are far more difficult to solve than "2t + 1 = 7" (what is "t" here?).

The answer to these kind of equations is not a number, but another function

The simplest answer to the above differential equation is this:



...which is a mathematical statement saying that "y" is actually an exponential function (its value increases by multiplying the previous one by the rate it is increasing).

The "A" is just any old number, and the squiggly bit before f(t) indicates a process by which you can get another "function" from the original f(t).

Most differential equations are of this type, when solved (in other words, you get a "y = ....") give "e" to the power of something (e is 2.718281828...).

The reason "e" is involved is because the definition of "e" is that the rate of change of "e" in e**t with respect to t is "e" itself.

Comments welcome! ;)

Next is "Partial differential equations"...