For a linear, second order differential equation define the terms ordinary point, singular point and regular singular point.

For $a, b \in \mathbb{R}$ and $b \notin \mathbb{Z}$ consider the following differential equation

$$x \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+(b-x) \frac{\mathrm{d} y}{\mathrm{~d} x}-a y=0 .$$

Find coefficients $c_{m}(a, b)$ such that the function $y_{1}=M(x, a, b)$, where

$$M(x, a, b)=\sum_{m=0}^{\infty} c_{m}(a, b) x^{m}$$

satisfies $(*)$. By making the substitution $y=x^{1-b} u(x)$, or otherwise, find a second linearly independent solution of the form $y_{2}=x^{1-b} M(x, \alpha, \beta)$ for suitable $\alpha, \beta$.

Suppose now that $b=1$. By considering a limit of the form

$$\lim _{b \rightarrow 1} \frac{y_{2}-y_{1}}{b-1}$$

or otherwise, obtain two linearly independent solutions to $(*)$ in terms of $M$ and derivatives thereof.