[Math] Alternative approach to solve $a(x dy +2y dx)=xy dy$

Question

Find the solution of $a(x dy +2y dx)=xy dy$

My attempt:
$Mdx + Ndy = 0$ where $M=2ay, N=ax-xy,M_y=2a , N_x = a-y, \frac{N_x – M_y}{M} = \frac{-y-a}{2ay} \implies $Integrating factor $=\frac{e^{(\frac{-y}{2a})}}{\sqrt y}$

THis gives

Solution –> $2ax \sqrt y = ce^{(\frac{y}{2a})}$, c being integrating constant

I want to know is there any other simpler alternative approach to solve this???? kindly provide me alternative simpler approach

Answer 1

Just divide whole expression by $xy$ and integrate

$$a(\frac{dy}{y} + 2\frac{dx}{x})= dy$$

:)