Question

Y=x/x+2 prove that xdy/dx=(1-y)y

Answer 1

$\text{Given:}$

$y=\displaystyle\frac{x}{x+2}$

$\text{This can be written as}$

$y=\displaystyle\frac{(x+2)-2}{x+2}$

$y=1-\displaystyle\frac{2}{x+2}\implies\bf\,1-y=\displaystyle\frac{2}{x+2}.......(1)$

$y=1-2\displaystyle\,(x+2)^{-1}$

$\text{Differentiate with respect to x}$

$\displaystyle\frac{dy}{dx}=\frac{2}{(x+2)^2}$

$x\,\displaystyle\frac{dy}{dx}=\frac{2x}{(x+2)^2}$

$\implies\,x\displaystyle\frac{dy}{dx}=(\frac{2}{x+2})(\frac{x}{x+2})$

$\implies\bf\,x\displaystyle\frac{dy}{dx}=(1-y)y$ $\text{(Using (1))}$