Above is a picture of Leo Frobenius, an archaeologist not to be confused with Ferdinand Frobenius, the mathematician after which one of our recent methods is named. I think the picture of Leo, however, probably represents how you might be feeling at this point. To help keep the material from Chapter 9 straight, I have put together a flowchart on the series methods we’ve been studying. The flowchart is admittedly still rather convoluted, so focus first on our two main strategies:
- At an ordinary point, look for a power series solution.
- At a regular singular point, look for a Frobenius series solution.
In the second case, we’re guaranteed to find at least one Frobenius series solution. Finding a second, linearly independent solution is the more difficult task. The bottom-right of the flowchart is dedicated to describing that solution in the various cases. For our purposes, the main takeaway from that information is that the shape of the second solution depends on the relationship between the two roots of the indicial equation. (We won’t worry about the complex Frobenius series solution.)