The differential equation y (dy/dx) + x = c represents

The correct option is Family of circles.

To find the curves represented by the given differential equation, we need to solve it using the method of separation of variables. Let's start with the given differential equation:

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

where $c$ is a constant. Rearranging the equation to group the $x$ and $y$ terms on opposite sides, we get:

$$y\frac{dy}{dx}=c-x$$

Now, separate the variables by multiplying both sides by $dx$:

$$ydy=(c-x)dx$$

Integrate both sides of the equation to find the general solution:

$$\int ydy=\int (c-x)dx$$

Performing the integration, we get:

$$\frac{y^2}{2}=cx-\frac{x^2}{2}+c_1$$

where $c_1$ is the constant of integration. To simplify the equation, multiply the entire equation by 2:

$$y^2=2cx-x^2+2c_1$$

Rearranging the terms by bringing all variable terms to the left-hand side:

$$x^2+y^2-2cx-2c_1=0$$

Let $k=2c_1$ be another arbitrary constant. The equation becomes:

$$x^2+y^2-2cx-k=0$$

This is of the standard general form of the equation of a circle, which is given by:

$$x^2+y^2+2gx+2fy+d=0$$

Since the coefficients of $x^2$ and $y^2$ are equal (both are 1) and there is no cross-product term $xy$, the equation represents a Family of circles.