Question

Solve:

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

Answer 1

Given the differential equation,

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

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

or, $\frac{dy}{y-a{y}^2}=\frac{dx}{x+a}$

or, $[\frac{1}{y}+\frac{a}{1-ay}]dy=\frac{dx}{x+a}$

Now integrating both sides we get,

$\log |y|-\log |1-ay|=\log |a+x|+\log |c|$ [ Where $c$ is integrating constant]

or, $\frac{y}{1-ay}=c(a+x)$.