By Eckhard M.S. Hitzer, University of Fukui, Prof. Ryosuke Nagaoka (University of the Air, Japan)

So if you have a look at the pattern here, I will first introduce a differential equation, which describes the change of a certain amount over time, therefore the variable t for time. And it depends on a kind of growth constant. One example is the growth of bacteria. For example if you have one bacteria in the beginning and (the) it is multiplying (in one minute) after one minute you have two, after two minues you have four, after three minutes you have eight, and so on, it is multiplying. And for very big numbers it is approaching an exponential function of time. So this k is actually deciding how fast it is multiplying.

Another example, which follows the same law, but it has not growth, but the number (the amount) is getting smaller and smaller, is the example of radioactive decay. For example you have radioactive iodine at the beginning and after eight days you just have half the (initial) amount, and after further eight days you have a quarter of the (initial) amount. So it is getting less and less.

The general solution x of t, (is) follows the exponential law. So you have a constant and the constant describes what is the amount in the beginning. At the time t, then you have the exponent of k times t.

... Prof. Nagaoka ...

Yes it is also written as e to the power of k times t.

Next let us look at another very important differential equation. And now we have again on the right side the function x of t. And on the left side we have the second derivative. So it is describing not velocity, the rate of growth, but rather a kind of accelaration.

... Prof. Nagaoka ...

Yes.

And this is the example for example of a spring. If x is the elongation of the spring, then the force which is driving it back is just described by this law. And therefore also the minus sign, because if you elongate a spring, it is always a force which is trying to get back to the equilibrium. And the familiar oscillations, which you know of a spring over time, are just described by this equation. And the frequency, one over the time period of the osciallation, is just this omega here. And these oscillations are quite famously called harmonic oscillations.

Next we come to a famous equation here. And it is again describing growth over time, which is proportional to the population, for example number of people or number of bacteria. But this time the growth is not infinite. The number of people when it gets very big will slow down the growth. And therefore we have this minus beta times x squared. So it has the name in English of logistic curve. And it has many applications both in sociology and in engineering and science.

Soli Deo Gloria. Created by
Eckhard Hitzer (Fukui).

Last Modified 16 January 2004

EMS Hitzer and Prof. R. Nagaoka are not responsible for the content of external internet sites.