I consider an ODE of the form
$$\frac{\mathrm{d}U(u)}{\mathrm{d}u} \frac{\mathrm{d}V(v)}{\mathrm{d}v} = f(u, v). $$
The function $f$ is known, and the goal is to find $U$ and $V$, assuming that any initial or boundary conditions which turn out to be necessary have been specified. This equation arises when trying to find the Weyl rescaling upon coordinates $u$ and $v$ leading to the 2D line element $$ ds^2 = -f^{-1}(u, v) du dv.$$
Despite its apparent simplicitly I find I do not know how to solve or even classify this system: it is not a PDE since $U$ and $V$ are respectively independent of $v$ and $u$; it does not obviously separate into a pair of coupled ODEs, etc.
Anyway, I would like to solve this system, presumably numerically. Can anyone point me in the right direction?
Edit:
It has been pointed out that the form given implies $f(u,v) = g(u) h(v)$. Thus the problem separates into that of solving two coupled ODEs: $\frac{dU}{du} = -\lambda g(u)$ and $\frac{dV}{dv} = -\lambda h(v)$. Therefore, the remaining substance of the question is to find $g$ and $h$.
$$ f(u,v) = \frac{\text dU(u)}{\text du} \frac{\text dV(v)}{\text dv} = g_1(u)g_2(v) $$
Thus, if $f(u,v)$ is not expressible as $f_1 (u) f_2 (v)$ for some $f_1,f_2$, then the equation is not possible. If it is, then simply solve the two equations
$$ f_1(u) = g_1(u) \\ f_2(v) = g_2(v) $$