Form the differential equation of family of circles having center at origin.

Explanation

To form the differential equation of the family of circles having their center at the origin, we can follow these steps:

### Step 1: Write the equation of the family of circles
The general equation of a circle with center at the origin (0,0) and radius r is given by:
\[ x^2 + y^2 = r^2 \]

### Step 2: Differentiate the equation with respect to x
We need to differentiate both sides of the equation with respect to x. Using implicit differentiation, we get:
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(r^2) \]
This simplifies to:
\[ 2x + 2y \frac{dy}{dx} = 0 \]

### Step 3: Simplify the differentiated equation
We can divide the entire equation by 2 to simplify it:
\[ x + y \frac{dy}{dx} = 0 \]

### Step 4: Rearranging the equation
Rearranging gives us the final form of the differential equation:
\[ y \frac{dy}{dx} = -x \]

### Final Result
Thus, the differential equation of the family of circles having their center at the origin is:
\[ y \frac{dy}{dx} + x = 0 \]

---