What is the solution of the differential equation $$a\left( {x\frac{{dy}}{{dx}} + 2y} \right) = xy\frac{{dy}}{{dx}}\,?$$

Solution :
Given differential equation is
$$\eqalign{ & a\left( {x\frac{{dy}}{{dx}} + 2y} \right) = xy\frac{{dy}}{{dx}} \cr & \Rightarrow ax\frac{{dy}}{{dx}} - xy\frac{{dy}}{{dx}} = - 2ay \cr & \Rightarrow \left( {xy - ax} \right)\frac{{dy}}{{dx}} = 2ay \cr & \Rightarrow x\left( {y - a} \right)\frac{{dy}}{{dx}} = 2ay \cr & \Rightarrow x\left( {y - a} \right)dy = 2ay\,dx \cr & \Rightarrow \frac{{\left( {y - a} \right)}}{y}dy = \frac{{2a}}{x}dx \cr & \Rightarrow \left( {1 - \frac{a}{y}} \right)dy = \frac{{2a}}{x}dx \cr & dy - \frac{a}{y}dy = \frac{{2a}}{x}dx \cr & {\text{Integrate on both side}} \cr & \int {dy - a} \int {\frac{1}{y}dy} = 2a\int {\frac{1}{x}dx} \cr & y - a\,\log \,y = 2a\,\log \,x + \log \,c \cr & \Rightarrow y = a\,\log \,{x^2}yc \Rightarrow {x^2}y = k{e^{\frac{y}{a}}} \cr} $$