(a) By using a power series of the form

$$y(x)=\sum_{k=0}^{\infty} a_{k} x^{k}$$

or otherwise, find the general solution of the differential equation

$$x y^{\prime \prime}-(1-x) y^{\prime}-y=0 .$$

(b) Define the Wronskian $W(x)$ for a second order linear differential equation

$$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$

and show that $W^{\prime}+p(x) W=0$. Given a non-trivial solution $y_{1}(x)$ of $(2)$ show that $W(x)$ can be used to find a second solution $y_{2}(x)$ of $(2)$ and give an expression for $y_{2}(x)$ in the form of an integral.

(c) Consider the equation (2) with

$$p(x)=-\frac{P(x)}{x} \quad \text { and } \quad q(x)=-\frac{Q(x)}{x}$$

where $P$ and $Q$ have Taylor expansions

$$P(x)=P_{0}+P_{1} x+\ldots, \quad Q(x)=Q_{0}+Q_{1} x+\ldots$$

with $P_{0}$ a positive integer. Find the roots of the indicial equation for (2) with these assumptions. If $y_{1}(x)=1+\beta x+\ldots$ is a solution, use the method of part (b) to find the first two terms in a power series expansion of a linearly independent solution $y_{2}(x)$, expressing the coefficients in terms of $P_{0}, P_{1}$ and $\beta$.