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: Rearrange the equation
We start with the given equation:
\[ y \frac{dy}{dx} + x = C \]
Rearranging it gives:
\[ y \frac{dy}{dx} = C - x \]

### Step 2: Separate variables
Next, we separate the variables \(y\) and \(x\). We can rewrite the equation as:
\[ y \, dy = (C - x) \, dx \]

### Step 3: Integrate both sides
Now we will integrate both sides:
\[ \int y \, dy = \int (C - x) \, dx \]
The left side integrates to:
\[ \frac{y^2}{2} + k_1 \]
And the right side integrates to:
\[ Cx - \frac{x^2}{2} + k_2 \]
Setting \(k = k_2 - k_1\), we have:
\[ \frac{y^2}{2} = Cx - \frac{x^2}{2} + k \]

### Step 4: Rearrange the equation
Multiplying through by 2 to eliminate the fraction gives:
\[ y^2 = 2Cx - x^2 + 2k \]
Rearranging this results in:
\[ x^2 + y^2 - 2Cx = 2k \]

### Step 5: Identify the geometric representation
The equation \(x^2 + y^2 - 2Cx = 2k\) can be rewritten in the standard form of a circle. Completing the square for the \(x\) terms:
\[ (x - C)^2 + y^2 = C^2 + 2k \]
This represents a circle with center \((C, 0)\) and radius \(\sqrt{C^2 + 2k}\).

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

---