The solution of differential equation $x\, dx+y\, dy = a(x^2+y^2) dy, $ is

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

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

$x\, dy + y \, dy = a (x^2 + y^2) dy$

$⇒\frac{2xdx+2y\, dy}{x^2+y^2}=2a\, dy$

$⇒\frac{d(x^2+y^2)}{x^2+y^2}= 2a\, dy $

On integrating, we obtain

$log (x^2 + y^2) = 2ay + log \, C$

$⇒x^2 + y^2 = C\, e^{2ay}$ is the required solution.