## ABSTRACT

The partial sums of a power series solution about an ordinary point of a differential equation provide fairly accurate approximations to the equation’s solutions at any point x near that ordinary point. However, these power series typically converge slower and slower as x approaches a singular point. As a result, the power series solutions derived in the previous two chapters usually say little about the solutions near singular points. This is unfortunate because, in some applications, the behavior of the solutions near certain singular points can be an important issue.

Fortunately, in many of these applications, the singular point in question is not that “bad” a singular point. In these applications, the differential equation is “similar” to an easily solved Euler equation, at least in some interval about that singular point. The allows the algebraic method discussed in the previous chapters to be modified so as to obtain “modified” power series solutions about these points. The basic process for generating these modified power series solutions is called the “method of Frobenius”, and is what is developed and demonstrated in this and the next two chapters. This chapter motivates and provides an introductory discussion of this important method.