Question

A Bernoulli differential equation is one of the form

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

du/dx+(1−n)P(x)u=(1−n)Q(x).

Consider the initial value problem xy′+y=−8xy^2, y(1)=−1.

(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 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)=_____.

PLEASE FILL IN THE BLANKS AND BOX THE ANSWERS

Answer 1