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

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

has the general solution:
 * $\ln \dfrac x y = \dfrac {y^3} 3 + C$

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

From Differential of Logarithm of Quotient:
 * $\map \d {\ln \dfrac y x} = \dfrac {y \rd x - x \rd y} {x y}$

from which:
 * $\map \d {\ln \dfrac x y} = y^2 \rd y$

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