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

a family of circles whose centres are on x-axis

The correct answer is option (4) : a family of circles whose centres are on x-axis

We have,

$y\frac{dy}{dx}+x=C$

$⇒y\, dy +(x-C)dx=0$

$⇒\frac{y^2}{2}+\frac{(x-C)^2}{2}=C_1$

$⇒(x-C)^2 +y^2 =2C_1$

Clearly, it represents a family of circles having their centres on x-axis.