$\begin{array}{lcr} f(x) &=& a_0 +& a_1x +& a_2x^2 +\cdots+& a_nx^n +\cdots \\[2ex] f'(x) &=& a_1 +& 2a_2x +& 3a_3x^2 +\cdots+& (n+1)a_{n+1}x^n +\cdots \\ xf''(x) &=& & 1\cdot2a_2 x +& 2\cdot3a_3x^2 +\cdots+& n(n+1)a_{n+1}x^n +\cdots \\ xf(x) &=& & a_0x +& a_1x^2 +\cdots+& a_{n-1}x^n +\cdots \\ \hline 0 &=& a_1 +& (4a_2+a_0)x +& (9a_3+a_1)x^2 +\cdots+& ((n+1)^2a_{n+1}+a_{n-1})x^n +\cdots \end{array}$
In order for the bottom row to be $0$, each of the coefficients must be, so we have $a_1=0$ and $(n+1)^2a_{n+1}+a_{n-1}=0$ for $n\ge 1$. Setting $k=n+1$ the latter equation becomes $a_k = -\frac{a_{k-2}}{k^2} \qquad\text{for }k\ge 2$ So all of the odd coefficients must be zero, and the even coefficients all determine each other. Every solution must be a multiple of the one with $a_0=1$, where we have $a_{2m} = \frac{(-1)^m}{2^2\cdot 4^2\cdot 6^2 \cdots (2m)^2} = \frac{(-1)^m}{m!^2 2^{2m}}$ I trust you can verify the convergence radius of this one yourself.
(This solution is the Bessel function $J_0$).