I do not know how to determine if $\displaystyle x=0$ is a regular or irregular singular point for $\displaystyle x^{2}y'' + 2(e^{x}-1)y' + (e^{-x}cos(x))y=0$. How do I determine if $\displaystyle 2\frac{e^{x}-1}{x}$ and $\displaystyle e^{-x}cos(x)$ are analytic?