Answer to Question #284303 in Differential Equations for Aysu

Question

Reduce the homogeneous equation to the separated one

(x+y)dx+(y-x)dy=0

Answer 1

$(x+y)+(y-x)y'=0$

Let $y(x)=xu(x).$ Then

$y'=u+xu'$

$x+xu+(ux-x)(u+xu')=0$

$x(1+u+(u-1)(u+xu')=0$

$1+u+u^2-u+xuu'-xu'=0$

$x(1-u)u'=1+u^2$

$\dfrac{1-u}{1+u^2}du=\dfrac{dx}{x}$

Integrate

$\int\dfrac{1-u}{1+u^2}du=\int \dfrac{dx}{x}$

$\tan^{-1}(u)-\dfrac{1}{2}\ln(1+u^2)=\ln x+C$

$\tan^{-1}(\dfrac{y}{x})-\dfrac{1}{2}\ln(1+\dfrac{y^2}{x^2})=\ln x+C$