Question

If $y=\frac{x}{x+2}$, prove that $x\frac{dy}{dx}=\left(1-y\right)y$

Answer 1

$\mathrm{We}\mathrm{have},y=\frac{x}{x+2}$
Differentiating with respect to x,

$$\begin{gathered} \frac{\mathit{d}y}{\mathit{d}x}=\frac{d}{dx}\left(\frac{x}{x+2}\right) \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=\frac{\left(x+2\right){\displaystyle \frac{d}{dx}}\left(x\right)-x{\displaystyle \frac{d}{dx}}\left(x+2\right)}{{\left(x+2\right)}^2} \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=\frac{x+2-x}{{\left(x+2\right)}^2} \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=\frac{x+2}{{\left(x+2\right)}^2}-\frac{x}{{\left(x+2\right)}^2} \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=\frac{1}{x+2}-\frac{x{y}^2}{x^2}\left[\because x+2=\frac{x}{y}\right] \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=\frac{y}{x}-\frac{y^2}{x} \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=\frac{1}{x}y\left(1-y\right) \\ \Rightarrow x\frac{\mathit{d}y}{\mathit{d}x}=\left(1-y\right)y \\ \mathrm{Hence}\mathrm{proved} \end{gathered}$$