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

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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:

$\ln \dfrac x y = \dfrac {y^3} 3 + C$

$\blacksquare$


Sources