I need help to find an analytical solution to:
$$p''(x)-k_1xp'(x)-k_2p(x)=0 \text{ where } k_1,k_2\in\mathbb R^+$$
with boundary conditions $p'(0)=0, p(r)=p(-r)=k_3$ where $r,k_3\in\mathbb R^+$. Mathematica gives as solution the ratio of two Hypergeometric $_1F_1$ (Kummer) functions but I'd like to know if Mathematica is right and a possible solution procedure.
Thanks a lot.