Answer to Question #292768 in Differential Equations for luna

Question

Reduce the homogeneous equation to the separated one

(x+2y)dx-xdy=0

Answer 1

$(x+2y)dx-xdy=0\\ (x+2y)dx=xdy\\ \frac{dy}{dx}=\frac{x+2y}{x}...(1)\\$

Let $y=vx$

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

From (1),

$v+x\frac{dv}{dx}=\frac{x+2vx}{x}=1+2v\\ x\frac{dv}{dx}=1+v\\ \frac{dv}{1+v}=\frac{dx}{x}\\$

Integrating both sides, we get

$ln(1+v)= lnx+ ln c\\ ln(1+v)-lnx=lnc\\ ln(\frac{1+v}{x})=lnc\\ \frac{1+v}{x}=c\\ \frac{1+\frac{y}{x}}{x}=c\\ x+y=x^2c$