The solution of differential equation $x d x+y d y=a\left(x^2+y^2\right) d y$, is

$x^2+y^2=C e^{2 a y}$

We have,

$x d x+y d y=a\left(x^2+y^2\right) d y$

$\Rightarrow \frac{2 x d x+2 y d y}{x^2+y^2}=2 a d y \Rightarrow \frac{d\left(x^2+y^2\right)}{x^2+y^2}=2 a d y$

On integrating, we obtain

$\log \left(x^2+y^2\right)=2 a y+\log C$

$\Rightarrow x^2+y^2=C e^{2 a y}$ is the required solution.