Question

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

Answer 1

Given,

$y=x{e}^x$

Now, differentiating both sides w.r.t. ${}^{\prime }{x}^{\prime }$ we get

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

or, $x\frac{dy}{dx}=x^2{e}^x+x{e}^x$

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

or, $x\frac{dy}{dx}=(1+x)y$