I am trying to solve
$u_t +x^2 u_x = 0\hspace{0.5 cm} x>0, t\in \mathbb{R}$
$u(0,t)=g(t)\hspace{0.5 cm} t\in \mathbb{R}$
and by the method of the characteristics I have found that it has as a general solution: $u(x,t)=f(\frac{1}{x}+t)$
where $f$ is an arbitrary function. But I have no idea how to find the particular solution given the boundary condition. Well, when I substitute $ x = 0 $ in the general solution the term $\frac{1}{x}$ is indeterminate.
tThen i think i'm doing something wrong. They could tell me what I'm doing wrong or have a suggestion to solve the problem. Thanks