Question

The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is

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

Hint:

Always remember the degree shortcut for homogeneous power curves: if $y = C x^n$, then the differential equation is always $x \frac{dy}{dx} = n y$. For $x^2 = 4ay \implies y = \left(\frac{1}{4a}\right)x^2$, so $x \frac{dy}{dx} = 2y$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need the differential equation representing all parabolas with vertex at the origin and axis along the positive y-axis.

Step 2: Key Formula or Approach:
The standard equation is x² = 4ay; differentiate and eliminate the arbitrary parameter a.

Step 3: Detailed Explanation:
Differentiating x² = 4ay gives 2x = 4a(dy/dx). Substituting 4a = 2x/(dy/dx) back yields x² = (2x/(dy/dx))y → x(dy/dx) = 2y → x(dy/dx) – 2y = 0.

Step 4: Final Answer:
The differential equation is represented by option (D).