I am working on a post exam worksheet and I want to master before my exam tomorrow. I am a little sketchy so if you can please help i really appreciate!!

Q. Find an interval around x = 0 for which the initial value problem: (x-2)y'' +3y = x, y(0) = 0, y'(0)=0 = 1 has a unique solution.

I understand that I need to use power series. this is non cauchy-euler equation. My main concern is the power series and the shifts. Because sometimes, I think i read by ferbini's theorem, you can guess: y = sum Cnx^n or y = sum c_n(x-x0)^n

how do you know which guess to make??