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

Answer 1

Answer: (B)

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