Consider the first order equation $$x\frac{dy}{dx}=y(1+x).$$ The point $x=0$ is a singular point, so my understanding is that it is not possible to find a power series solution of the form $$y(x)=\sum_{n=0}^\infty a_n\,x^n.$$ However, I found $$y(x)=c\sum_{n=1}^\infty \frac{x^n}{(n-1)!},$$ which agrees with the usual solution $y(x)=x\,e^x$.
What is going on here? Am I missing something?
Your understanding is incorrect. As this example shows, a differential equation might have power series solutions around a singular point.