Question

The differential equation of the system of all circles of radius r in the XY plane is

• A. \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2} = r^2 \frac{d^2 y}{dx^2} \)
• B. \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right] = r^2 \frac{d^2 y}{dx^2} \)
• C. \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2} = r^2 \frac{d^2 y}{dx^2} \)
• D. \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = r^2 \frac{d^2 y}{dx^2} \)

Hint:

The differential equation for the system of circles involves the second derivative of the curve's equation and the radius.

Answer 1

The Correct Option is C

Step 1: Formula for the system of circles.
The differential equation of a system of circles is derived from the geometric properties of the circle. The correct equation is \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2} = r^2 \frac{d^2 y}{dx^2} \).

Step 2: Conclusion.
Thus, the correct answer is option (C).

Final Answer: \[ \boxed{\text{(C) } \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2} = r^2 \frac{d^2 y}{dx^2}} \]