If `(x-y) \ e^(x/(x-y))=a` , prove that `y(dy)/(dx)+x=2y` .

Explanation

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

Differentiate both sides w.r.t`x`

`=>(1-frac{d y}{d x}) e^{frac{x}{x-y}}+(x-y) e^{frac{x}{x-y}} times[frac{x-y-x(1-frac{d y}{d x})}{(x-y)^{2}}]=0 `

`=>e^{frac{x}{x-y}}[(1-frac{d y}{d x})+(x-y)(frac{x-y-x(1-frac{d y}{d x})}{(x-y)^{2}})]=0`

`=>(1-frac{d y}{d x})+(frac{x frac{d y}{d x}-y}{(x-y)})=0`

`=>x-y-x frac{d y}{d x}+y frac{d y}{d x}+x frac{d y}{d x}-y=0`

`=>y frac{d y}{d x}+x=2 y`