Form the differential equation of family of standard circle

Answer

To form the differential equation of the family of standard circles, we start with the standard equation of a circle centered at the origin. The steps are as follows:

### Step-by-Step Solution:

1. **Write the Standard Equation of a Circle**:
The standard equation of a circle with radius \( r \) centered at the origin (0,0) is given by:
\[ x^2 + y^2 = r^2 \]

2. **Differentiate the Equation**:
To find the differential equation, we need to differentiate the equation with respect to \( x \). Applying implicit differentiation:
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(r^2) \]
This gives:
\[ 2x + 2y \frac{dy}{dx} = 0 \]

3. **Simplify the Equation**:
We can simplify the equation by dividing everything by 2:
\[ x + y \frac{dy}{dx} = 0 \]

4. **Rearranging the Equation**:
The final form of the differential equation representing the family of standard circles is:
\[ x + y \frac{dy}{dx} = 0 \]

### Final Answer:
The differential equation of the family of standard circles is:
\[ x + y \frac{dy}{dx} = 0 \]

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