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"...