Question

Solve the differential equation : $(y-x\frac{dy}{dx}-a(y^2+\frac{dy}{dx})$ ) =0

Answer 1

$\begin{array}{c}y-x\frac{dy}{dx}-a\left(y^2-\frac{dy}{dx}\right)=0\\ y-a{y}^2=\left(x+a\right)\frac{dy}{dx}\\ \Rightarrow \frac{dx}{x+a}=\frac{dy}{\left(y-a{y}^2\right)}\\ \Rightarrow {\log }_e\left(x+a\right)=2{\log }_e\left|\frac{2ay}{\sqrt{4ay-4a{y}^2}}\right|\\ \Rightarrow x+a=\frac{a^2{y}^2}{ay-a^2{y}^2}\\ \Rightarrow x+a=\frac{ay}{1-ay}Ans.\end{array}$