Question

(1 point) A Bernoulli differential equation is one of the form

dydx+P(x)y=Q(x)yn     (∗)

Observe that, if n=0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=y1−n transforms the Bernoulli equation into the linear equation

dudx+(1−n)P(x)u=(1−n)Q(x).dudx+(1−n)P(x)u=(1−n)Q(x).

Consider the initial value problem

y′=−y(1+9xy3),   y(0)=−3.

(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
dudx+ u= .

(c) Using the substitution in part (b), we rewrite the initial condition in terms of x and u:
u(0)= .

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

(e) Finally, solve for y.
1/(y(x))3=

I just need parts c - e

Answer 1

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