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

Options

A. family of hyperbolas
B. family of parabolas
C. family of ellipses
D. family of circles

Explanation

To solve the differential equation \( y \frac{dy}{dx} + x = C \) and determine what it represents, we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ y \frac{dy}{dx} + x = C \]
We can rearrange this to isolate the term involving \( \frac{dy}{dx} \):
\[ y \frac{dy}{dx} = C - x \]

### Step 2: Separating Variables
Next, we separate the variables \( y \) and \( x \):
\[ y \, dy = (C - x) \, dx \]

### Step 3: Integrating Both Sides
Now, we integrate both sides:
\[ \int y \, dy = \int (C - x) \, dx \]
Calculating the left side:
\[ \int y \, dy = \frac{y^2}{2} \]
Calculating the right side:
\[ \int (C - x) \, dx = Cx - \frac{x^2}{2} + k \]
where \( k \) is the constant of integration.

### Step 4: Combining the Results
Now we equate the results of the integrations:
\[ \frac{y^2}{2} = Cx - \frac{x^2}{2} + k \]
To simplify, we can multiply the entire equation by 2:
\[ y^2 = 2Cx - x^2 + 2k \]

### Step 5: Rearranging into Standard Form
Rearranging gives us:
\[ y^2 + x^2 - 2Cx - 2k = 0 \]
This can be rewritten as:
\[ y^2 + (x - C)^2 = C^2 + 2k \]
This equation represents a family of circles centered at \( (C, 0) \) with radius \( \sqrt{C^2 + 2k} \).

### Conclusion
Thus, the differential equation \( y \frac{dy}{dx} + x = C \) represents a family of circles.

---