Find the differential equation representing the curve $y = cx + c^2$, $c$ being arbitrary constant.

$y=x\left(\frac{dy}{dx}\right)+\left(\frac{dy}{dx}\right)^2$

The correct answer is Option (1) → $y=x\left(\frac{dy}{dx}\right)+\left(\frac{dy}{dx}\right)^2$

Given $y = cx + c^2$   ...(i)

Differentiating (i) w.r.t. x, we get

$\frac{dy}{dx}= C$.

Substituting this value of $c$ in (i), we get

$y = x\frac{dy}{dx}+\left(\frac{dy}{dx}\right)^2$, which is the required differential equation of the given curve.