Question

The solution of the differential equation $\frac{dy}{dx}+x^{5}y=x^5{y}^7$ is
(where $c$ is integration constant)

• A. $\ln |y^6-1|=x^6+c$
• B. $\ln |y^{-6}-1|=x^6+c$
• C. $\ln |y^{-6}+1|=x^{-6}+c$
• D. $\ln |y^{-6}-1|=x^{-6}+c$

Answer 1

The correct option is B $\ln |y^{-6}-1|=x^6+c$

We have,
$\frac{dy}{dx}+x^{5}y=x^5{y}^7\cdots \left(i\right)$
It is a bernoulli equation with $P\left(x\right)=x^5,Q\left(x\right)=x^5$ and $n=7$
Let, $u=y^{-6}$
$\Rightarrow y=u^{-1/6}$
$\Rightarrow \frac{dy}{dx}=-\frac{1}{6}{u}^{-7/6}\frac{du}{dx}$
From equation $\left(i\right),$
$-\frac{1}{6}{u}^{-7/6}\frac{du}{dx}+x^5{u}^{-1/6}=x^5{u}^{-7/6}$
$\Rightarrow \frac{du}{dx}-6x^{5}u=-6x^5$
$\Rightarrow \frac{du}{dx}=(u-1)6x^5$
$\Rightarrow \frac{du}{u-1}=6x^{5}dx$
Integrating both sides,
$\int \frac{du}{u-1}=\int 6x^{5}dx$
$\Rightarrow \ln |u-1|=x^6+c$
$\Rightarrow \ln |y^{-6}-1|=x^6+c$