Question

Find the differential equation of the family of all circles, whose center lies on the x-axis and touches the y-axis at the origin.

• A. \( 2xy \frac{dy}{dx} = y^2 - x^2 \)
• B. \( 2xy \frac{dy}{dx} = x^2 - y^2 \)
• C. \( x^2 + y^2 = 2xy \frac{dy}{dx} \)
• D. \( x^2 + y^2 = 2y \frac{dy}{dx} \)

Hint:

When differentiating implicit equations, apply the chain rule and remember to differentiate each term with respect to \( x \). For a family of circles, center and radius conditions are key.