Step 1: Understanding the Question:
The problem asks for the order of the differential equation corresponding to a specific family of curves.
The order of a differential equation derived from a general equation of a curve is equal to the number of independent arbitrary constants present in that general equation.
A circle passing through the origin \((0,0)\) with its center on the x-axis has specific geometric constraints that limit the number of constants.
Step 2: Key Formula or Approach:
Write the general equation of the circle.
The center is on the x-axis, so the center is at \((h, 0)\).
Since it passes through the origin, the distance from the center \((h, 0)\) to the origin \((0,0)\) is the radius \(r\).
Use the circle equation: \((x - h)^2 + (y - k)^2 = r^2\).
Step 3: Detailed Explanation:
Let the center of the circle be \((h, 0)\), as it is given to lie on the x-axis.
The circle passes through the origin \((0,0)\). Therefore, the radius \(r\) is the distance between \((h, 0)\) and \((0,0)\):
\[ r = \sqrt{(h - 0)^2 + (0 - 0)^2} = |h| \]
The equation of the circle is:
\[ (x - h)^2 + (y - 0)^2 = r^2 \]
\[ (x - h)^2 + y^2 = h^2 \]
Expanding the square:
\[ x^2 - 2xh + h^2 + y^2 = h^2 \]
Simplifying by cancelling \(h^2\) from both sides:
\[ x^2 + y^2 - 2xh = 0 \]
In this equation, there is only one independent arbitrary constant, which is \(h\).
According to the fundamental property of differential equations, the number of independent constants in the general solution equals the order of the differential equation.
Since there is only one constant \(h\), the order of the differential equation is 1.
Step 4: Final Answer:
The family of circles is represented by an equation containing only one arbitrary constant. Hence, the order of its differential equation is 1.