Question

y = xlogy Show that xdy/dx = y^2/(y-x)

Answer 1

Step-by-step explanation:

We have,

$\rm \:y = x. log(y)$

Differentiating both sides w.r.t x, we get,

$\rm \: \frac{dy}{dx} = log(y) + x. \frac{1}{y} \frac{dy}{dx}$

$\rm \: \implies\frac{dy}{dx} = \frac{y}{x} + \frac{x}{y} \frac{dy}{dx}$

$\rm \: \implies\frac{dy}{dx} - \frac{x}{y} \frac{dy}{dx} = \frac{y}{x}$

$\rm \: \implies \bigg(1- \frac{x}{y} \bigg)\frac{dy}{dx} = \frac{y}{x}$

$\rm \: \implies \bigg( \frac{y - x}{y} \bigg)\frac{dy}{dx} = \frac{y}{x}$

$\rm \: \implies \frac{dy}{dx} = \frac{y}{x} . \frac{y}{y - x}$

$\rm \: \implies x\frac{dy}{dx} = \frac{ {y}^{2} }{y - x}$