ABSTRACT
The previous chapter probably left the reader wondering
What can be done when the basic method of Frobenius does not yield the linearly independent pair of solutions needed, and
Are there any shortcuts?
To properly answer those questions requires a good bit of analysis — some straightforward, and some not so straightforward. That analysis will be carried out in detail in the next chapter. In this chapter, are theorems summarizing the results of that analysis, chief of which is the “big theorem” describing all the basic modified power series solutions about regular singular points for second-order homogeneous linear differential equations. This naturally leads to two more topics:
The modifications to the basic method of Frobenius needed to find the “second” solutions when the basic method fails.
An analysis of the general behavior of solutions near singular points, the understanding of which may be more useful than the precise “second” solutions, and may even justify not carrying out the details of solving for that “second” solution. This is illustrated with a case study involving the Legendre equations.
