I was wondering whether for any convergent real power series (or a Frobenius series) we can find (or prove that there exists) a corresponding differential equation that characterizes it. I am aware of Hölder's theorem. So, in effect I am looking for results in these lines but, of course , for real analytic functions. (Generally, my question reads: Can real analytic functions be characterized by differential equations?)

P.S.: I feel these statements are rather vague. But I am eager to hear your comments/answers.