Step 1: Understanding the Concept:
The order of a differential equation representing a family of curves is equal to the number of independent arbitrary constants (parameters) in the equation of that family.
Step 2: Key Formula or Approach:
1. Write the general equation of the family of circles.
2. Identify the number of independent arbitrary constants.
Step 3: Detailed Explanation:
The center of the circle lies on the x-axis, so let the center be \((h, 0)\).
Since the circle passes through the origin \((0, 0)\), the radius \(r\) must be the distance from \((h, 0)\) to the origin, which is \(|h|\).
The equation of the circle is:
\[ (x - h)^2 + (y - 0)^2 = h^2 \]
\[ x^2 - 2xh + h^2 + y^2 = h^2 \]
\[ x^2 + y^2 - 2xh = 0 \]
In this equation, there is only one arbitrary constant, which is \(h\). Since there is only one independent parameter, the resulting differential equation will be of the first order.
Step 4: Final Answer:
The order of the differential equation is 1.