My 2 cents:
An eqn like y ' + p(x) y = q(x) y^n
is an example of a Bernoulli equataion which can easily be solved
by letting v = y ^ (1-n)
or y ' -2xy = x y^2
is an example of a Bernoulli equataion n = 1
Let v= y ^(-1)
dv/dx = -y^(-2)dy/dx
-y^2 dv/dx - 2xy = x y^2
dv/dx + 2x v = -x
Which can now be solved with the intergating factor e^(x^2)
d(ve^(x^2)) = -xe^(x^2)dx
v = -1/2 + ce^(-x^2)
y= 1/v = 1/ [-1/2 + ce^(-x^2)]