Explanation of step in solving homogeneous differential equation

Question

I've recently started learning homogenous differential equations after having studied equations with seperable variables. Could you explain how changing variables $u=\dfrac{y}{x} \Longrightarrow \dfrac{dy}{dx}=x\dfrac{du}{dx}+u$.

Answer 1

As @Moo said you just substitute in the original equation $$u=\frac yx \implies y=ux \implies y'=u'x+u $$ $$y'=e^{y/x}+\frac yx$$ $$u'x+u=e^u+u$$ $$u'x=e^u$$ $$x\frac {du}{dx}=e^u \implies e^{-u}du=\frac {dx}x$$ Integrate $$\int \frac {du}{e^u}=\int \frac {dx}x$$ $$-e^{-u}=\ln(x)+K$$ $$e^{-y/x}=-\ln(x)+C$$