Y as Dependent Variable

Problem 04
$y' = y - xy^3e^{-2x}$
 

Solution 04
$y' = y - xy^3e^{-2x}$

$\dfrac{dy}{dx} - y = -xe^{-2x}y^3$

$dy - y~dx = -xe^{-2x}y^3~dx$       → Bernoulli's equation

$dy + Py~dx = Qy^n~dx$
 

Integrating factor,
$u = e^{(1 - n)\int P~dx} = e^{-2\int (-1)~dx}$

$u = e^{2\int dx} = e^{2x}$
 

Thus,
$\displaystyle zu = (1 - n)\int Qu~dx + C$