Consider the equation $$(x+3)y''+2y'-4(x+3)y=0.$$ I was trying to solve it by finding the solution in the form of power series. However, I stuck while trying to find any regularity in the coefficients, while Wolfram Alpha provides a pretty much nice answer. Could you please point out the direction?
There is a life hack just for you: solve the thing with Wolfram, see how it could be simplified, then solve it the way you were supposed to. What if we switch to $u(x)=(x+3)\cdot y(x)$ and look for that in the form of power series?
You could certainly use power series, but I use the expansion about the point $x=-3$ rather than $x=0$ (since you have those $(x+3)$ terms). Alternatively, let $z(x)=(x+3)y(x)$, so $z'=(x+3)y'+y$ and $z''=(x+3)y''+2y'$. You can then find a fairly simple ODE for $z$ and solve it however you like.