First Order ODE/(1 - x y) y' = y^2

Theorem
The first order ODE:
 * $(1): \quad \left({1 - x y}\right) y' = y^2$

has the solution:
 * $x y = \ln y + C$

Proof
Let $(1)$ be rearranged as:

It can be seen that $(2)$ is a linear first order ODE in the form:
 * $\dfrac {\mathrm d x}{\mathrm d y} + P \left({y}\right) x = Q \left({y}\right)$

where:
 * $P \left({y}\right) = \dfrac 1 y$
 * $Q \left({y}\right) = \dfrac 1 {y^2}$

Thus:

Thus from Solution by Integrating Factor, $(2)$ can be rewritten as: