I need the following result (which I believe to be true though I was too lazy to write down a complete proof). 


Let $f$ be a function of two complex variables analytic at the origin and $a\not\in\mathbb{N}$. Then the differential equation
$$z'=\frac{az}{x}+f(x,z)$$ 
has an unique solution $z(x)$ which is analytic at $x=0$.


Can anyone tell me a reference for this?



Reference request: a singular differential equation
These questions were studied for the first time by Briot and Bouquet in 19 century. For a modern reference see for example the book of E. Hille, Ordinary differential equations in the complex domain. The chapter is called Some equations of Briot and Bouguet.
Your case is actually simple: plug a formal power series for $z(x)$ and see that
all coefficients can be uniquely defined. Then prove convergence by Cauchy majorant method. Your statement is Theorem 11.1.1 on p. 403 in Hille.


