# Bernoulli's (differential) equation.

SrEstroncio
I have been doing some self study on differential equations using Tom Apostol's Calculus Vol. 1. and I got stuck on a problem (problem 12, section 8.5, vol. 1).

## Homework Statement

Let K be a non zero constant. Suppose P and Q are continuous in an open interval I.
Let $$a\in I$$ and b a real number. Let $$v=g(x)$$ be the only solution to the initial value problem $$v' +kP(x)v=kQ(x)$$ in I, with $$g(a)=b$$. If $$n\not= 1$$ and $$k=1-n$$, prove the function $$y=f(x)$$ not identically zero in I, is a solution to $$y'+P(x)y=Q(x)y^n$$ and $${f(a)}^k=b$$ in I, if and only if the k-th power of $$f$$ is equal to $$g$$ in I.

## Homework Equations

The general solution of a linear first order differential equation is probably used.

## The Attempt at a Solution

I have tried several manipulations and substitutions, but I am kinda lost. I would appreciate any help.

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