Let $u(x,t)$ with $(x,t)\in\mathbb{R}\times\mathbb{R_+}$
I want to find the solution $a$ of the following PDE:
$$a"(u)=\frac{e^{b(u_x)}}{u_x}\frac{e^{c(u_t)}}{u_t}e^{a(u)}$$
where $b:\mathbb{R}\to \mathbb{R}$ and $c:\mathbb{R}\to \mathbb{R}$ are fixed functions.