Solution of the differential equation $x\frac{dy}{dx} = y + \sqrt{x^2+y^2}$, is
Answer & explanation
Correct answer: option 4
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 $
$⇒u = \sqrt{x^2+y^2} = Cx^2$