Question

The solution of the differential equation
$\left\{1+x\sqrt{\left(x^{2}+y^{2}\right)}\right\}dx+\left\{\sqrt{\left(x^{2}+y^{2}\right)-1}\right\}y\,dy=0$ is

• A. $x^{2}+\frac{y^{2}}{2}+\frac{1}{3}\left(x^{2}+y^{2}\right)^{3/2}=C$
• B. $x-\frac{y^{2}}{3}+\frac{1}{2}\left(x^{2}+y^{2}\right)^{1/2}=C$
• C. $x-\frac{y^{2}}{2}+\frac{1}{3}\left(x^{2}+y^{2}\right)^{3/2}=C$
• D. None of these

Answer 1

Answer: (C)

Rearranging the equation, we ha
$dx-y\,dy+\sqrt{\left(x^{2}+y^{2}\right)}\left(xdx+y\,dy\right)=0$
$\Rightarrow dx-y\,dy+\frac{1}{2} \sqrt{\left(x^{2}+y^{2}\right)}d\left(x^{2}+y^{2}\right)=0$
On integrating, we get
$x-\frac{y^{2}}{2}+\frac{1}{2} \int\,\sqrt{t}\,dt=C, \left\{t=\sqrt{\left(x^{2}+y^{2}\right)}\right\}$
or $x-\frac{y^{2}}{2}+\frac{1}{2}\left(x^{2}+y^{2}\right)^{3/ 2}=C$