Solution of the differential equation $x\frac{dy}{dx} = y + \sqrt{x^2+y^2}$, is

$y+\sqrt{x^2+y^2}=C\, x^2$

The correct answer is option (4) : $y+\sqrt{x^2+y^2}=C\, x^2$

Substituting $y = vx $ and $\frac{dy}{dx} = v + x\frac{dv}{dx}, $ we get

$v+x\frac{dv}{dx} = v + \sqrt{1+v^2}$

$⇒\frac{1}{\sqrt{1+v^2}}dv=\frac{1}{x}dx$

On integrating, we get

$log (v + \sqrt{v^2+1}) = log x + log C$

$⇒v + \sqrt{v^2+1} = Cx $

$⇒u = \sqrt{x^2+y^2} = Cx^2$