Question

If $(x-y)e^{\frac{x}{x-y}}=a$, prove that $y\frac{dy}{dx}+x=2y$.

Answer 1

$(x-y)e^{\frac{x}{x-y}}=a$

Differentiate both sides w.r.t.$x$

$\left(1-\frac{dy}{dx}\right)e^{\frac{x}{x-y}}+\left(x-y\right)e^{\frac{x}{x-y}}\times \left[\frac{x-y-x\left(1-\frac{dy}{dx}\right)}{{\left(x-y\right)}^2}\right]=0$

$e^{\frac{x}{x-y}}\left[\left(1-\frac{dy}{dx}\right)+\left(x-y\right)\left(\frac{x-y-x\left(1-\frac{dy}{dx}\right)}{{\left(x-y\right)}^2}\right)\right]=0$

$\left(1-\frac{dy}{dx}\right)+\left(\frac{x\frac{dy}{dx}-y}{\left(x-y\right)}\right)=0$

$x-y-x\frac{dy}{dx}+y\frac{dy}{dx}+x\frac{dy}{dx}-y=0$

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