Question

Let $y=x{e}^{-x}$ then prove that $x\frac{dy}{dx}=(1-x)y$.

Answer 1

Given,

$y=x{e}^{-x}$......(1).

Now differentiating both sides with respect to $x$ we get,

$\frac{dy}{dx}=e^{-x}-x{e}^{-x}$

or, $\frac{dy}{dx}=(1-x)e^{-x}$

or, $x\frac{dy}{dx}=(1-x)x{e}^{-x}$

or, $x\frac{dy}{dx}=(1-x)y$. [ Using (1)]