Friday, March 28, 2008

If You Can Solve Only One Differential Equation...

If you can solve only one differential equation, let it be dy/dt = k y. This equation states that the rate of increase of a quantity y is proportional to the present amount of y. The solution is the exponential function, y=exp(kt). Exponential growth is extremely important in medicine and biology, and in the 4th Edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I devote the entire Chapter 2 to this topic:

"The exponential function is one of the most important and widely occurring functions in physics and biology. In biology, it may describe the growth of bacteria or animal populations, the decrease of the number of bacteria in response to a sterilization procedure, the growth of a tumor, or the absorption or excretion of a drug....In physics, the exponential function describes the decay of radioactive nuclei, the emission of light by atoms, the absorption of light as it passes through matter, the change in voltage or current in some electrical circuits, the variation of temperature with time as a warm object cools, and the rate of some chemical reactions."

Albert Bartlett has written a fascinating collection of essays about the exponential function: The Essential Exponential! For the Future of Our Planet. He claims that "the greatest shortcoming of the human race is our inability to understand the exponential function." You can see Bartlett talking about the exponential and its implications for population growth on YouTube.

The exponential function is often written using the number e=2.718... (If you want better precision, go to Google and search for "e"). This may be the most famous number, besides pi, that is not an integer. If you would like to read about the history of e, try Eli Maor's delightful book e: The Story of a Number .