Define the operator $F$ taking functions $f$ on $(0,\infty)^2$ into $(0,\infty)$ and returning the function $F(f)$ on $(0,\infty)^2$ defiend by $$ F(f)\triangleq \frac{1}{x^2}\left(\frac{ \frac{\partial f}{\partial t } }{ \frac{\partial^2f}{\partial x^2 } } \right) . $$
Then how $F^{-1}$ be represented as an integral operator? I've been reading a bit on Green's functions $G$ and I'm guessing that $$ F^{-1}(u)(x) = \sqrt{x} \int_0^TG(t,x)u(x)dx , $$ where $G$ is some nice Green's function.
Is my intuition on the right track? If so how can I explicitly find the Green's function and so $F^{-1}$?