Question

The differential equation of all circles in a plane of radius \(r\) is \[ \text{where } y_1=\frac{dy}{dx}, \qquad y_2=\frac{d^2y}{dx^2} \]

• A. \((1+y_1^2)^2=r^2y_2^2\)
• B. \(1+y_1^2=r^2y_2\)
• C. \((1+y_1^2)^3=r^2y_2\)
• D. \((1+y_1^2)^3=r^2y_2^2\)

Hint:

Remember the radius of curvature formula: \[ \rho= \frac{(1+y'^2)^{3/2}} {|y''|} \] For a circle of radius \(r\), \[ \rho=r \] which immediately gives the required differential equation.

Answer 1

The Correct Option is D

Concept: The radius of curvature \(\rho\) of a curve \(y=f(x)\) is \[\begin{aligned} \rho = \frac{(1+y_1^2)^{3/2}} {|y_2|} \end{aligned}\] For a circle of radius \(r\), \[\begin{aligned} \rho=r \end{aligned}\] at every point.

Step 1: Use the radius of curvature formula. \[\begin{aligned} r = \frac{(1+y_1^2)^{3/2}} {|y_2|} \end{aligned}\]

Step 2: Square both sides. \[\begin{aligned} r^2 = \frac{(1+y_1^2)^3} {y_2^2} \end{aligned}\]

Step 3: Rearrange the equation. \[\begin{aligned} (1+y_1^2)^3 = r^2y_2^2 \end{aligned}\] \[\begin{aligned} \boxed{ (1+y_1^2)^3=r^2y_2^2 } \end{aligned}\] Hence, option \(\mathbf{(D)}\) is correct.