Question

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

Answer 1

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

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

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

Integrating both sides and variable separation

$\int \frac{dy}{y(1-ay)}=\int \frac{dx}{a+x}$

$\int |\frac{1}{y}+\frac{a}{1-ay}|dy=\log |x+a|+c$

$\log \left|y\right|-\log |1-ay|=\log |x+a|+\log b$ (let $c=\log b$ $b$ is constant

$\log \left|\frac{y}{1-ay}\right|=\log \left|(x+a)b\right|$

$\Rightarrow y=b(x+a)(1-ay)\Rightarrow y[1+ab(x+a\left)\right]=b(x+a)$

$y=\frac{b(x+a)}{1+ab(x+a)}$ general solution