Question

The differential equation corresponding to the family of parabolas whose axis is along $x = 1$ is
Identify the correct option from the following:

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

Hint:

To find the differential equation of a family of curves, differentiate to eliminate parameters, ensuring the order matches the number of parameters.

Answer 1

The Correct Option is B

Step 1: Formulate the family of parabolas
Parabolas with axis along $x = 1$ (parallel to the y-axis) have the form $(y - k)^2 = 4a(x - 1)$, where $k$ and $a$ are parameters. Simplify with one parameter: $(y - k)^2 = a(x - 1)$. Step 2: Derive the differential equation
Differentiate: $2(y - k) \frac{dy}{dx} = a$. Differentiate again: $2 \left( \frac{dy}{dx} \right)^2 + 2(y - k) \frac{d^2 y}{dx^2} = 0$, so $(y - k) \frac{d^2 y}{dx^2} = -\left( \frac{dy}{dx} \right)^2$. From the first: $y - k = \frac{a}{2 \frac{dy}{dx}}}$. Substitute: $\frac{a}{2 \frac{dy}{dx}}} \frac{d^2 y}{dx^2} = -\left( \frac{dy}{dx} \right)^2$, $a \frac{d^2 y}{dx^2} = -2 \left( \frac{dy}{dx} \right)^3$. Use the first equation again: $a = 2(y - k) \frac{dy}{dx}$, but directly eliminate $a$ and $k$: $\frac{d^2 y}{dx^2} (x - 1) = \frac{dy}{dx}$. Step 3: Match with options
Rewrite: $(x - 1) \frac{d^2 y}{dx^2} - \frac{dy}{dx} = 0$, matching option (2).