I am studying the original paper by Ornstein and Uhlenbeck on the theory of Brownian Motion. After constructing the Fokker-Planck equation for the Langevin equation of motion, the authors arrive at the boundary problem
$\frac{\partial G}{\partial t}=\beta\frac{\partial(uG)}{\partial u}+\frac{\tau_1}{2}\frac{\partial^2G}{\partial u^2}$
with conditions $G=f(u)$ at $t=0$ and $G=0$ for $u=\pm \infty$.
I have no idea on how to solve this and I am not familiar with the theory behind it. The paper says it can be solved with the method of particular solutions. It also expresses the solution in terms of Weber's functions. Could somebody explain and give me some references please?
Thank you in advance,
Mirko