Differential Equations Workbook For Dummies®

9.4. Solving General Differential Equations with Regular Singular Points

This section is where you get to put together everything you practice throughout this chapter (so if you haven't reviewed the previous sections, you may want to flip back a few pages). All set? Then take a look at this general differential equation and assume that it has a regular singular point at x = 0:

Multiply by x² to make sure that the terms xp(x) and x²q(x) (at least one of which has a singular point at x = 0) can be expanded into a series:

and

The Euler equation that matches the general differential equation you're handling is

which means you can assume the Euler equation has a solution of the form

y = x^r

To handle the fact that the differential equation you're working with isn't an exact Euler equation, add a series expansion to the solution (with the assumption that the n = 0 term is the largest term and that all other terms diminish rapidly):

The result is the form of your anticipated solution, ...