Explanation

(d)
Given, $x\left( \frac{dy}{dx} \right)^{2} + 2\sqrt{xy}\frac{dy}{dx} + y = 0$
$\Rightarrow$ $\left( \sqrt{x}\ \frac{dy}{dx} + \sqrt{y} \right)^{2} = 0$
$\Rightarrow$ $\frac{1}{\sqrt{y}}dy + \frac{1}{\sqrt{x}}dx = 0$
$\Rightarrow$ $2\sqrt{y} + 2\sqrt{x} = c_{1}$
$\Rightarrow$ $\sqrt{x} + \sqrt{y} = c$