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Ordinary differential equations (ODEs) are often used to model the behavior of physical phenomena and textbooks today especially demonstrate this fact. Since only a very small collection of ODEs can be solved analytically, there is often no alternative than to use computer software to gain some insight into the behavior of solutions (and sometimes even if solution formulas are available—the formulas are often complicated!). A classic work on the numerical solution of ODEs was authored by Shampine [8]. There are some questions about the behavior of solutions of ODEs that are not quite appropriate for numerical solvers. In this paper we present examples which illustrate some of these features. However, there is no disputing the fact that the output of numerical solvers is often useful for portraying and understanding the behavior of solutions of ODEs and their utility in modeling physical phenomena, as our final example shows.

Pitfalls and Pluses in Using Numerical Software to Solve Differential Equations