Question

The differential equation of family of circles whose centres lie on X-axis is

• A. $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + 1 = 0$
• B. $y\left(\frac{d^2y}{dx^2}\right) + \left(\frac{dy}{dx}\right)^2 + 1 = 0$
• C. $y\left(\frac{d^2y}{dx^2}\right) - \left(\frac{dy}{dx}\right)^2 - 1 = 0$
• D. $y\left(\frac{d^2y}{dx^2}\right) + \left(\frac{dy}{dx}\right)^2 - 1 = 0$

Hint:

The number of arbitrary constants in the general equation dictates the order of the resulting differential equation. Because a family of circles on the X-axis can vary in both position ($h$) and size ($r$), the equation must be differentiated exactly twice!

Answer 1

The Correct Option is B

Step 1: Write the circle family.
Centre on the X-axis means centre $(h,0)$, so $(x - h)^2 + y^2 = r^2$ with two constants $h$ and $r$.

Step 2: Differentiate once.
$$(x - h) + y\frac{dy}{dx} = 0$$

Step 3: Differentiate again to drop $h$.
$$1 + y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 0$$
\[ \boxed{y\dfrac{d^2y}{dx^2} + \left(\dfrac{dy}{dx}\right)^2 + 1 = 0} \]