First Order ODE/y dx - x dy = x y^3 dy

Theorem
The first order ODE:
 * $(1): \quad y \, \mathrm d x - x \, \mathrm d y = x y^3 \, \mathrm d y$

has the solution:
 * $\dfrac x y = \dfrac 1 y - y + C$

Proof
Rearranging, we have:
 * $\dfrac {y \, \mathrm d x - x \, \mathrm d y} {x y} = y^2 \mathrm d y$

From Differential of Logarithm of Quotient:
 * $\mathrm d \left({\ln \dfrac y x}\right) = \dfrac{y \, \mathrm d x - x \, \mathrm d y} {x y}$

from which:
 * $\mathrm d \left({\ln \dfrac x y}\right) = y^2 \mathrm d y$

Hence the result:
 * $\ln \dfrac x y = \dfrac {y^3} 3 + C$