The differential equation $y \frac{dy}{dx} + x = C$ represents

family of circles

The correct answer is Option (4) → family of circles ##

Given that, $y \frac{dy}{dx} + x = C$

$\Rightarrow y \frac{dy}{dx} = C - x$

$\Rightarrow y \cdot dy = (C - x) \, dx$

On integrating both sides, we get

$\frac{y^2}{2} = Cx - \frac{x^2}{2} + K$

$\Rightarrow \frac{x^2}{2} + \frac{y^2}{2} = Cx + K$

$\Rightarrow \frac{x^2}{2} + \frac{y^2}{2} - Cx = K$

$\Rightarrow x^2 + y^2 - 2Cx = 2K$

which represent family of circles.