Analyticity of solution to differential-like equation

Question

Given the following differential equation $$y'(x)=y(x)$$ It is obvious that $y$ is infinitely differentiable at any point. One would approach the problem by expanding the series of $y$ and eventually arriving at the series for $e^x$. However how could one justify that $y$ could be rewritten as its Taylor series (i.e. it is analytic). Using the theorem for uniqueness of solution one could just say that it is the solution. However in some cases one does not know whether there is a unique solution. For example $y'(x)=y(-x)$. I mean, if we want to solve this problem by findind the Taylor series for $y$, is there some way we can prove that $y$ is analytic, using only the above equation, so that replacing $y$ with its expansion is correct.