Chapter 11.2, Problem 17P

Consider the differential equation

$x{y^{\prime }}^{\prime }-\left(x-1\right)y^{\prime }-xy=0\left(\mathrm{11.2.18}\right)$

(a) Is $x=0$ an ordinary point?

(b) Determine the first three nonzero terms in each of two linearly independent series solutions to Equation $\left(\mathrm{11.2.18}\right)$ centered at $x=1$.

[Hint: Make the change of variables $z=x-1$ and obtain a series solution in powers of $z$.]

Give a lower bound on the radius of convergence of each of your solutions.