Find a differential equation for which a series is a solution This is just a general question. Given a power series in one real variable, is it always possible to find a differential equation for which the series is a solution? If not, when is it possible?
 A: If a power series in one real variable $x \in \mathbb R$ converges for $x-x_0 < r, x_0 \in \mathbb R $ fixed, then it extends to a power series in one complex variable $z \in \mathbb C$ which converges for $|z-x_0| < r$. Hence the "natural" domain of definition of power series are complex variables. 
Considered from a certain point of view, complex analysis is the theory of convergent power series of one complex variable. Because a function $f$ is holomorphic in an open subset $U \subset \mathbb C$ iff it can be represented around each point of $U$ by a convergent power series. 
Consider a function $f:U \longrightarrow \mathbb C$ with $U \subset \mathbb C$ open. After decomposing complex numbers $z = x + i y \in U$ into real part $x$ and imaginary part $y$ and analogously decomposing $g(x,y) := f(x+iy)$ as 
$$g(x,y) = u(x,y) + i v(x,y)$$ 
with two real-valued functions $u$ and $v$, we have the Cauchy-Riemann differential equations: $f$ is holomorphic iff the pair $(u,v)$ satisfies the two partial differential equations
$$u_x = v_y, u_y = -v_x.$$
Summing up: There is a strong relation between convergent power series and differential equations. But the link is not a single ordinary differential equation. Instead it is a system of two partial differential equations. They relate the real part and the imaginary part.
