(1 point) A Bernoulli differential equation is one of the form
$$\frac{dy}{dx} + P(x)y = Q(x)y^n \quad (*)$$
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
$$xy' + y = 6xy^2, \quad y(1) = -4.$$
(a) This differential equation can be written in the form (*) with
$$P(x) = \qquad ,$$
$$Q(x) = \qquad , \text{ and }$$
$$n = \qquad .$$
(b) The substitution $$u = \qquad$$ will transform it into the linear equation
$$\frac{du}{dx} + \qquad u = \qquad .$$
(c) Using the substitution in part (b), we rewrite the initial condition in terms of x and u:
$$u(1) = \qquad .$$
(d) Now solve the linear equation in part (b). and find the solution that satisfies the initial condition in part (c).
$$u(x) = \qquad .$$
(e) Finally, solve for y.
$$y(x) = \qquad .$$
