Consider the differential equation

$$x \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+(c-x) \frac{\mathrm{d} y}{\mathrm{~d} x}-y=0,$$

where $c$ is a constant with $0<c<1$. Determine two linearly independent series solutions about $x=0$, giving an explicit expression for the coefficient of the general term in each series.

Determine the solution of

$$x \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+(c-x) \frac{\mathrm{d} y}{\mathrm{~d} x}-y=x$$

for which $y(0)=0$ and $\mathrm{d} y / \mathrm{d} x$ is finite at $x=0$.