The solution of the differential equation 

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

We have,

$\begin{array}{l}y-x\frac{dy}{dx}=a\left(y^2+\frac{dy}{dx}\right)\\ \Rightarrow y-a{y}^2=(a+x)\frac{dy}{dx}\\ \Rightarrow \frac{dx}{a+x}=\frac{1}{y-a{y}^2}dy\\ \Rightarrow \int \frac{1}{a+x}dx=\int \frac{1}{y-a{y}^2}dy\\ \Rightarrow \int \frac{1}{a+x}dx=\int -\frac{1}{y(1-ay)}dy\\ \Rightarrow \int \frac{1}{a+x}dx=\int \left(\frac{1}{y}+a\frac{a}{1-ayy}\right)dy\end{array}$

$\begin{array}{l}\Rightarrow \log (a+x)=\log y-\log (1-ay)+\log C\\ \Rightarrow a+x=\frac{Cy}{1-ay}\Rightarrow (x+a)(1-ay)=Cy\end{array}$