What does the differential equation ` y(dy)/(dx) +x =k` ( where k is a constant) represents ?

Options

A. A family of circles having centre on the y-axis
B. A family of circles having centre on the x-axis
C. A family of circles touching the x-axis
D. A family of ellipses.

Explanation

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

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

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

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

### Step 4: Forming the Equation
Putting it all together, we have:
\[ \frac{y^2}{2} = kx - \frac{x^2}{2} + C \]
Multiplying through by 2 to eliminate the fraction gives:
\[ y^2 = 2kx - x^2 + 2C \]
Rearranging this, we can write:
\[ y^2 + x^2 - 2kx - 2C = 0 \]

### Step 5: Identifying the Geometric Representation
The equation \(y^2 + x^2 - 2kx - 2C = 0\) resembles the standard form of a circle's equation:
\[ (x - g)^2 + (y - f)^2 = r^2 \]
where \(g\) and \(f\) are the coordinates of the center and \(r\) is the radius.

To find the center, we can complete the square:
\[ y^2 + (x^2 - 2kx) - 2C = 0 \]
Completing the square for \(x\):
\[ y^2 + (x - k)^2 - k^2 - 2C = 0 \]
This leads to:
\[ (y - 0)^2 + (x - k)^2 = k^2 + 2C \]
This indicates that the equation represents a family of circles centered at \((k, 0)\) with radius \(\sqrt{k^2 + 2C}\).

### Conclusion
Thus, the differential equation \( y \frac{dy}{dx} + x = k \) represents a family of circles with centers on the x-axis.

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