Question

Solution of the differential equation
$y\left(xy+2x^2{y}^2\right)dx+x\left(xy-x^2{y}^2\right)dy=0$ is given by

• A. $2log\left|x\right|-log\left|y\right|-\frac{1}{xy}=C$
• B. $2log\left|y\right|-log\left|x\right|-\frac{1}{xy}=C$
• C. $2log\left|x\right|+log\left|y\right|+\frac{1}{xy}=C$
• D. $2log\left|y\right|+log\left|x\right|+\frac{1}{xy}=C$

Answer 1

Answer: (B)

We have, $(x{y}^2+2x^2{y}^3)dx+(x^{2}y-x^3{y}^2)dy=0$
$\Rightarrow xy\left(ydx+xdy\right)+x^2{y}^2\left(2ydx-xdy\right)=0$
$\Rightarrow \frac{d\left(xy\right)}{x^2{y}^2}+\left(\frac{2}{x}dx-\frac{1}{y}dy\right)=0$ [Dividing by $x^3{y}^3$]
On integrating, we get $-\frac{1}{xy}+2log\left|x\right|-log\left|y\right|=C$