Question

Find the differential equation representing the family of circles having their centers on the Y-axis. Given that \( y_1 = \frac{dy}{dx} \) and \( y_2 = \frac{d^2y}{dx^2} \).

• A. \( y_2 = y(y_1^2 + 1) \)
• B. \( y_2 = xy(y_1^2 + 1) \)
• C. \( xy_2 = y_1(y_1^2 + 1) \)
• D. \( xy_2 = y(y_1^2 + 1) \)

Hint:

For equations of circles centered on the Y-axis, differentiate twice and simplify to obtain the required differential equation.

Answer 1

The Correct Option is C

Step 1: General equation of a circle centered on the Y-axis. A general circle with center on the Y-axis has the equation: \[ x^2 + (y - c)^2 = r^2 \] where \( c \) is the center's Y-coordinate and \( r \) is the radius. Step 2: Differentiating to obtain the first derivative. Differentiating both sides with respect to \( x \): \[ 2x + 2(y - c) \frac{dy}{dx} = 0 \] \[ x + (y - c) y_1 = 0 \] \[ y_1 = -\frac{x}{y - c} \] Step 3: Differentiating again to obtain the second derivative. Differentiating both sides again: \[ y_2 = \frac{d}{dx} \left(-\frac{x}{y - c} \right) \] Applying the quotient rule: \[ y_2 = \frac{(y - c)(-1) - (-x)y_1}{(y - c)^2} \] Simplifying, we obtain: \[ xy_2 = y_1(y_1^2 + 1) \]