Question

Solve the differential equation
$x\frac{dy}{dx}+\frac{y^2}{x}=y$

Answer 1

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

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

$y=vx$

$\frac{dy}{dx}=v+x\frac{dv}{dx}$

$v+x\frac{dv}{dx}+v^2=v$

$x\frac{dv}{dx}=-v^2$

$\int -\frac{dv}{v^2}=\int \frac{dx}{x}$

$\frac{1}{v}=\log |x|+c$

$\frac{x}{y}=\log |x|+c$

$y=\frac{x}{\log |x|+c}$