Chapter 11.5, Problem 4P

For Problem $1-8$, determine the roots of the indicial equation of the given differential equation. Also obtain the general form of two linearly independent solutions to the differential equation on an interval $\left(0,R\right)$. Finally, if $r_1-r_2$ equals a positive integer, obtain the recurrence relation and determine whether the constant $A$ in

$y_2\left(x\right)=A{y}_1\left(x\right)\ln x+x^{r_2}{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{b}_n{x}^n}$

is zero or nonzero.

$x^2{y}^{″}+x^2{y}^{\prime }-\left(2+x\right)y=0$.