A Bernoulli differential equation is one of the form $$\frac{dy}{dx} + P(x)y = Q(x)y^n \ \ \ \ \ (*)$$
Observe that, if $n=0$ or $1$, the Bernoulli equation is linear. For other values of $n$, the substitution $u=y^{1-n}$ transforms the Bernoulli equation into the linear equation $$\frac{du}{dx} + (1-n)P(x)u = (1-n)Q(x).$$

Consider the initial value problem $$x y' + y = -3 x y^2, \ \ \ y(1)=-6.$$ (a) This differential equation can be written in the form $(*)$ with
$P(x)=$ ,
$Q(x)=$ , and
$n=$ .

(b) The substitution $u=$ will transform it into the linear equation
$\displaystyle \frac{du}{dx} +$ $u =$ .

(c) Using the substitution in part (b), we rewrite the initial condition in terms of $x$ and $u$:
$u(1)=$ .

(d) Now solve the linear equation in part (b), and find the solution that satisfies the initial condition in part (c).
$u(x)=$ .

(e) Finally, solve for $y$.
$y(x) =$ .