## ABSTRACT

The “big theorem on the Frobenius method”, which described all the possible general solutions to a second-order homogeneous linear differential equation about its regular singular points, was the main topic of the previous chapter. There the theorem was stated and used. What was not there was all the work needed to verify that theorem. That work is presented in this chapter.

This work begins in a rather obvious manner — by applying the basic Frobenius method to a generic second-order homogeneous linear differential equation (in “reduced form”) with a regular singular point, and then closely examining the results, chief of which is a general recursion formula for the coefficients of the modified power series. This, along with a theorem on convergence that is also developed, shows precisely when the basic method succeeds and why it fails for certain cases. After that, the alternative solution formulas for when the basic method fails are derived and verified. Dealing with these cases is the challenging part (and is not done in most other texts on differential equations).