Question

The differential equation of the family of all parabolas whose axis is the \(y\)-axis is \(\ldots\)

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

Hint:

Parabolas with the y-axis as axis are x^2 = 4a(y - b). Two constants need two differentiations.

Answer 1

The Correct Option is B

Step 1: Check with a member of the family:
Take $y = x^2$. Then $y' = 2x$ and $y'' = 2$.
Test option (B): $x\cdot2 - 2x = 0$. True.
Test option (D): $x\cdot 2 + 2x = 4x \ne 0$. False. Option (A): $2 + 2x^2 \ne 0$. False. Option (C): $2 - 2x^2 \ne 0$. False.

Step 2: Test a second member:
Take $x^2 = 4(y-1)$, so $y = \frac{x^2}{4} + 1$, $y' = \frac x2$, $y'' = \frac12$. Option (B): $x\cdot\frac12 - \frac x2 = 0$. True again.

Step 3: Reason:
Both members have vertex on the Y-axis, and only option (B) vanishes for both.

Final Answer:
Option (B). \[ \boxed{x\frac{d^2y}{dx^2}-\frac{dy}{dx}=0 \text{ (B)}} \]