Question 11.2.4: Determine the radius of convergence guaranteed by Theorem 1 ...

This example illustrates the fact that we must take into account complex singular points as well as real ones. Because

P(x) = \frac{x}{x² + 9}     and      Q(x) = \frac{x²}{x² + 9},

the only singular points of Eq. (6) are ±3i . The distance (in the complex plane) of each of these from 0 is 3, so a series solution of the form \sum{c_{n} x^{n}} has radius of convergence at least 3. The distance of each singular point from 4 is 5, so a series solution of the form \sum{c_{n}(x  – 4)^{n}} has radius of convergence at least 5 (see Fig. 11.2.1).