I came across this question and it looked like something that is likely to have been looked into, but I couldn't find a reference.

Let $k$ be some (algebraically closed, if needed) field. There is a formal differentiation in the ring of formal power series $k[[x]]$. Let $F(x,y) \in k[[x,y]]$ be a formal series which is algebraic over $x$ and $y$. Consider a differential equation: $$ y'=F(x,y) $$ where $y$ belongs to the maximal ideal of $k[[x]]$, so $F(x,y)$ is well-defined. What is known about solutions of such ODEs?

Is it true that there exists a formal series $y \in k[[x]]$ that satisfies the equation? Is it true that it is unique?