Differential Equations Workbook For Dummies®

9.2. Classifying Singular Points as Regular or Irregular

NOTE

Singular points come in two different forms: regular and irregular. Regular singular points are well-behaved and defined in terms of the ratio Q(x)/P(x) and R(x)/P(x), where P(x), Q(x), and R(x) are the polynomial coefficients in the differential equation you're trying to solve.

Irregular singular points are a totally different ball game — and one that I don't get into in this chapter. As you work through the practice problems here, if the singular point in question doesn't appear to be regular, you know it's irregular.

Allow me to introduce you to this dainty differential equation:

In order for x₀ to be a regular singular point, these two relations must be true:

and

If you define

p(x) = Q(x)/P(x)

and

q(x) = R(x)/P(x)

then the two limits become

and

If both of these statements are true, then the point x₀ is a regular singular point.

In the following problems, you practice classifying singular points as regular ...