. The solution of the differential equation  $y-x\frac{dy}{dx}=a\left(y^2+\frac{dy}{dx}\right)$ is 

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

$\Rightarrow \frac{dy}{y(1-ay)}=\frac{dx}{x+a}$  On integrating both sides

we get 

 $\begin{array}{r}\Rightarrow \log y-\log (1-ay)=\log (x+a)+\log \mathrm{c}\\ \Rightarrow \frac{y}{(1-ay)}=c(x+a)\text{ or }c(x+a)(1-ay)=y.\end{array}$