Question

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

• A. $x+y=a$
• B. $\sqrt{x}-\sqrt{y}=a$
• C. $x^2+y^2=a$
• D. $\sqrt{x}+\sqrt{y}=\sqrt{a}$

Answer 1

The correct option is D $\sqrt{x}+\sqrt{y}=\sqrt{a}$

We have
$x{\left(\frac{dy}{dx}\right)}^2+2\sqrt{xy}\frac{dy}{dx}+y=0$
$\Rightarrow {(\sqrt{x}\frac{dy}{dx}+\sqrt{y})}^2=0$
$\Rightarrow \sqrt{x}\frac{dy}{dx}+\sqrt{y}=0$
$\Rightarrow \frac{1}{\sqrt{x}}dy+\frac{1}{\sqrt{y}}dx=0$
$\Rightarrow 2\sqrt{y}+2\sqrt{x}=C$
$\Rightarrow \sqrt{x}+\sqrt{y}=\sqrt{a}=2C$