Chapter 11.5, Problem 5P

To determine

The roots of the indicial equation and obtain the general form of two linearly independent solutions to the differential equation on an interval $\left(0,R\right)$ and if $r_1-r_2$ equals a positive integer, obtain the recurrence relation and check whether the constant $A$ in $y_2\left(x\right)=A{y}_1\left(x\right)\ln x+x^{r_2}{\displaystyle \sum _{n=0}^{\infty }{b}_n{x}^n}$ is zero or nonzero.