Wolfram Mathematica® can provide an improper solution of an Ordinary Differential Equation: An intriguing example

Abstract

Mathematica® is one of the most popular and powerful commercial softwares for scientific computation and solution of algebraic equations. With integrated symbolic computation, the user can work directly on precise models, by transforming, optimizing and solving them, only substituting approximate or specific numerical values where necessary (for instance for visualization purposes). It is developed and distributed by Wolfram Resarch (http://www.wolfram.com/company/background.html), a company founded in 1987 by Steven Wolfram, Ph.D. (http://www.stephenwolfram.com/about-sw/). Wolfram Research is now one of the world's most respected software companies, as well as a powerhouse of scientific innovation. After the first version of Mathematica® released on June 23, 1988, version 7 is now available. It now incorporates multi–core and platform–optimized numerical algorithms, making it suitable for the most computationally intensive problems. One of the key features of Mathematica® is that, in contrast to the typical fixed 16–digits limitation found in other computing systems, its numerics support platform–independent arbitrary precision across all functions. The prominent objective of this report is to show, by considering a specific application described in the next section, that sometime the use of a third–party software as a black box can lead to solutions that are in contrast with the theoretical expectation coming from the mathematical analysis. In particular, we will consider a first–order Ordinary Differential Equation (ODE) with time–variable, real coefficients. We will show that the solution calculated by Mathematica® is not formally acceptable, since it is a function to complex values. We will show that only after a proper manipulation of the original analytical formulation of the given ODE, Mathematica® is able to find the proper form of the solution.